From d5abe972591693eff4b58a5e38927ba260bfc379 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Wed, 20 Mar 2024 22:31:23 +0000 Subject: [PATCH] build based on 22ff5b4 --- previews/PR1300/api/index.html | 46 ++++--- previews/PR1300/dev_docs/index.html | 2 +- previews/PR1300/generated/autodiff/index.html | 2 +- previews/PR1300/generated/box/index.html | 8 +- .../PR1300/generated/custom_rule/index.html | 6 +- previews/PR1300/index.html | 124 ++++++++++++++++-- previews/PR1300/internal_api/index.html | 2 +- previews/PR1300/pullbacks/index.html | 4 +- previews/PR1300/search/index.html | 2 +- previews/PR1300/search_index.js | 2 +- 10 files changed, 155 insertions(+), 43 deletions(-) diff --git a/previews/PR1300/api/index.html b/previews/PR1300/api/index.html index f4ea8fd565..4d4ff6904a 100644 --- a/previews/PR1300/api/index.html +++ b/previews/PR1300/api/index.html @@ -1,5 +1,5 @@ -API · Enzyme.jl
Edit on GitHub

API

Types and constants

Functions and macros

Documentation

Enzyme.gradient! — Method
gradient!(::ReverseMode, dx, f, x)

Compute the gradient of an array-input function f using reverse mode, storing the derivative result in an existing array dx.

Example:

f(x) = x[1]*x[2]
+API · Enzyme.jl
Edit on GitHub
API
Types and constants
Functions and macros
Documentation
Enzyme.gradient! — Method
gradient!(::ReverseMode, dx, f, x)
Compute the gradient of an array-input function f using reverse mode, storing the derivative result in an existing array dx.
Example:
f(x) = x[1]*x[2]
 
 dx = [0.0, 0.0]
 gradient!(Reverse, dx, f, [2.0, 3.0])
@@ -8,13 +8,13 @@
 
 2-element Vector{Float64}:
  3.0
- 2.0
source
Enzyme.gradient — Method
gradient(::ForwardMode, f, x; shadow=onehot(x))
Compute the gradient of an array-input function f using forward mode. The optional keyword argument shadow is a vector of one-hot vectors of type x which are used to forward-propagate into the return. For performance reasons, this should be computed once, outside the call to gradient, rather than within this call.
Example:
f(x) = x[1]*x[2]
+ 2.0
source
Enzyme.gradient — Method
gradient(::ForwardMode, f, x; shadow=onehot(x))
Compute the gradient of an array-input function f using forward mode. The optional keyword argument shadow is a vector of one-hot vectors of type x which are used to forward-propagate into the return. For performance reasons, this should be computed once, outside the call to gradient, rather than within this call.
Example:
f(x) = x[1]*x[2]
 
 grad = gradient(Forward, f, [2.0, 3.0])
 
 # output
 
-(3.0, 2.0)
source
Enzyme.gradient — Method
gradient(::ReverseMode, f, x)
Compute the gradient of an array-input function f using reverse mode. This will allocate and return new array with the gradient result.
Example:
f(x) = x[1]*x[2]
+(3.0, 2.0)
source
Enzyme.gradient — Method
gradient(::ReverseMode, f, x)
Compute the gradient of an array-input function f using reverse mode. This will allocate and return new array with the gradient result.
Example:
f(x) = x[1]*x[2]
 
 grad = gradient(Reverse, f, [2.0, 3.0])
 
@@ -22,13 +22,13 @@
 
 2-element Vector{Float64}:
  3.0
- 2.0
source
Enzyme.gradient — Method
gradient(::ForwardMode, f, x, ::Val{chunk}; shadow=onehot(x))
Compute the gradient of an array-input function f using vector forward mode. Like gradient, except it uses a chunk size of chunk to compute chunk derivatives in a single call.
Example:
f(x) = x[1]*x[2]
+ 2.0
source
Enzyme.gradient — Method
gradient(::ForwardMode, f, x, ::Val{chunk}; shadow=onehot(x))
Compute the gradient of an array-input function f using vector forward mode. Like gradient, except it uses a chunk size of chunk to compute chunk derivatives in a single call.
Example:
f(x) = x[1]*x[2]
 
 grad = gradient(Forward, f, [2.0, 3.0], Val(2))
 
 # output
 
-(3.0, 2.0)
source
Enzyme.jacobian — Method
jacobian(::ForwardMode, f, x; shadow=onehot(x))
 jacobian(::ForwardMode, f, x, ::Val{chunk}; shadow=onehot(x))
Compute the jacobian of an array-input function f using (potentially vector) forward mode. This is a simple rename of the gradient function, and all relevant arguments apply here.
Example:
f(x) = [x[1]*x[2], x[2]]
 
 grad = jacobian(Forward, f, [2.0, 3.0])
@@ -37,7 +37,7 @@
 
 2×2 Matrix{Float64}:
  3.0  2.0
- 0.0  1.0
source
Enzyme.jacobian — Method
jacobian(::ReverseMode, f, x, ::Val{num_outs}, ::Val{chunk})
Compute the jacobian of an array-input function f using (potentially vector) reverse mode. The chunk argument denotes the chunk size to use and num_outs denotes the number of outputs f will return in an array.
Example:
f(x) = [x[1]*x[2], x[2]]
+ 0.0  1.0
source
Enzyme.jacobian — Method
jacobian(::ReverseMode, f, x, ::Val{num_outs}, ::Val{chunk})
Compute the jacobian of an array-input function f using (potentially vector) reverse mode. The chunk argument denotes the chunk size to use and num_outs denotes the number of outputs f will return in an array.
Example:
f(x) = [x[1]*x[2], x[2]]
 
 grad = jacobian(Reverse, f, [2.0, 3.0], Val(2))
 
@@ -45,7 +45,7 @@
 
 2×2 Matrix{Float64}:
  3.0  2.0
- 0.0  1.0
source
EnzymeCore.autodiff — Method
autodiff(::ForwardMode, f, Activity, args...)
Auto-differentiate function f at arguments args using forward mode.
args may be numbers, arrays, structs of numbers, structs of arrays and so on. Enzyme will only differentiate in respect to arguments that are wrapped in a Duplicated or similar argument. Non-annotated arguments will automatically be treated as Const. Unlike reverse mode in autodiff, Active arguments are not allowed here, since all derivative results of immutable objects will be returned and should instead use Duplicated or variants like DuplicatedNoNeed.
Activity is the Activity of the return value, it may be:
  • Const if the return is not to be differentiated with respect to
  • Duplicated, if the return is being differentiated with respect to and both the original value and the derivative return are desired
  • DuplicatedNoNeed, if the return is being differentiated with respect to and only the derivative return is desired.
  • BatchDuplicated, like Duplicated, but computing multiple derivatives at once. All batch sizes must be the same for all arguments.
  • BatchDuplicatedNoNeed, like DuplicatedNoNeed, but computing multiple derivatives at one. All batch sizes must be the same for all arguments.
Example returning both original return and derivative:
f(x) = x*x
+ 0.0  1.0
source
EnzymeCore.autodiff — Method
autodiff(::ForwardMode, f, Activity, args::Vararg{Annotation, Nargs})
Auto-differentiate function f at arguments args using forward mode.
args may be numbers, arrays, structs of numbers, structs of arrays and so on. Enzyme will only differentiate in respect to arguments that are wrapped in a Duplicated or similar argument. Unlike reverse mode in autodiff, Active arguments are not allowed here, since all derivative results of immutable objects will be returned and should instead use Duplicated or variants like DuplicatedNoNeed.
Activity is the Activity of the return value, it may be:
  • Const if the return is not to be differentiated with respect to
  • Duplicated, if the return is being differentiated with respect to and both the original value and the derivative return are desired
  • DuplicatedNoNeed, if the return is being differentiated with respect to and only the derivative return is desired.
  • BatchDuplicated, like Duplicated, but computing multiple derivatives at once. All batch sizes must be the same for all arguments.
  • BatchDuplicatedNoNeed, like DuplicatedNoNeed, but computing multiple derivatives at one. All batch sizes must be the same for all arguments.
Example returning both original return and derivative:
f(x) = x*x
 res, ∂f_∂x = autodiff(Forward, f, Duplicated, Duplicated(3.14, 1.0))
 
 # output
@@ -55,12 +55,12 @@
 
 # output
 
-(6.28,)
source
EnzymeCore.autodiff — Method
autodiff(::ReverseMode, f, Activity, args...)
Auto-differentiate function f at arguments args using reverse mode.
Limitations:
  • f may only return a Real (of a built-in/primitive type) or nothing, not an array, struct, BigFloat, etc. To handle vector-valued return types, use a mutating f! that returns nothing and stores it's return value in one of the arguments, which must be wrapped in a Duplicated.
args may be numbers, arrays, structs of numbers, structs of arrays and so on. Enzyme will only differentiate in respect to arguments that are wrapped in an Active (for arguments whose derivative result must be returned rather than mutated in place, such as primitive types and structs thereof) or Duplicated (for mutable arguments like arrays, Refs and structs thereof). Non-annotated arguments will automatically be treated as Const.
Activity is the Activity of the return value, it may be Const or Active.
Example:
a = 4.2
+(6.28,)
source
EnzymeCore.autodiff — Method
autodiff(::ReverseMode, f, Activity, args::Vararg{Annotation, Nargs})
Auto-differentiate function f at arguments args using reverse mode.
Limitations:
  • f may only return a Real (of a built-in/primitive type) or nothing, not an array, struct, BigFloat, etc. To handle vector-valued return types, use a mutating f! that returns nothing and stores it's return value in one of the arguments, which must be wrapped in a Duplicated.
args may be numbers, arrays, structs of numbers, structs of arrays and so on. Enzyme will only differentiate in respect to arguments that are wrapped in an Active (for arguments whose derivative result must be returned rather than mutated in place, such as primitive types and structs thereof) or Duplicated (for mutable arguments like arrays, Refs and structs thereof).
Activity is the Activity of the return value, it may be Const or Active.
Example:
a = 4.2
 b = [2.2, 3.3]; ∂f_∂b = zero(b)
 c = 55; d = 9
 
 f(a, b, c, d) = a * √(b[1]^2 + b[2]^2) + c^2 * d^2
-∂f_∂a, _, _, ∂f_∂d = autodiff(Reverse, f, Active, Active(a), Duplicated(b, ∂f_∂b), c, Active(d))[1]
+∂f_∂a, _, _, ∂f_∂d = autodiff(Reverse, f, Active, Active(a), Duplicated(b, ∂f_∂b), Const(c), Active(d))[1]
 
 # output
 
@@ -68,7 +68,7 @@
 
 # output
 
-((6.0,), 9.0)
Note
Enzyme gradients with respect to integer values are zero. Active will automatically convert plain integers to floating point values, but cannot do so for integer values in tuples and structs.
source
EnzymeCore.autodiff_deferred — Method
autodiff_deferred(::ForwardMode, f, Activity, args...)
Same as autodiff(::ForwardMode, ...) but uses deferred compilation to support usage in GPU code, as well as high-order differentiation.
source
EnzymeCore.autodiff_deferred — Method
autodiff_deferred(::ReverseMode, f, Activity, args...)
Same as autodiff but uses deferred compilation to support usage in GPU code, as well as high-order differentiation.
source
EnzymeCore.autodiff_deferred_thunk — Method
autodiff_deferred_thunk(::ReverseModeSplit, ftype, Activity, argtypes...)
Provide the split forward and reverse pass functions for annotated function type ftype when called with args of type argtypes when using reverse mode.
Activity is the Activity of the return value, it may be Const, Active, or Duplicated (or its variants DuplicatedNoNeed, BatchDuplicated, and BatchDuplicatedNoNeed).
The forward function will return a tape, the primal (or nothing if not requested), and the shadow (or nothing if not a Duplicated variant), and tapes the corresponding type arguements provided.
The reverse function will return the derivative of Active arguments, updating the Duplicated arguments in place. The same arguments to the forward pass should be provided, followed by the adjoint of the return (if the return is active), and finally the tape from the forward pass.
Example:

+((6.0,), 9.0)
Note
Enzyme gradients with respect to integer values are zero. Active will automatically convert plain integers to floating point values, but cannot do so for integer values in tuples and structs.
source
EnzymeCore.autodiff — Method
autodiff(mode::Mode, f, ::Type{A}, args::Vararg{Annotation, Nargs})
Like autodiff but will try to extend f to an annotation, if needed.
source
EnzymeCore.autodiff_deferred — Method
autodiff_deferred(::ForwardMode, f, Activity, args::Vararg{Annotation, Nargs})
Same as autodiff(::ForwardMode, f, Activity, args) but uses deferred compilation to support usage in GPU code, as well as high-order differentiation.
source
EnzymeCore.autodiff_deferred — Method
autodiff_deferred(::ReverseMode, f, Activity, args::Vararg{Annotation, Nargs})
Same as autodiff but uses deferred compilation to support usage in GPU code, as well as high-order differentiation.
source
EnzymeCore.autodiff_deferred_thunk — Method
autodiff_deferred_thunk(::ReverseModeSplit, ftype, Activity, argtypes::Vararg{Type{<:Annotation}, Nargs})
Provide the split forward and reverse pass functions for annotated function type ftype when called with args of type argtypes when using reverse mode.
Activity is the Activity of the return value, it may be Const, Active, or Duplicated (or its variants DuplicatedNoNeed, BatchDuplicated, and BatchDuplicatedNoNeed).
The forward function will return a tape, the primal (or nothing if not requested), and the shadow (or nothing if not a Duplicated variant), and tapes the corresponding type arguements provided.
The reverse function will return the derivative of Active arguments, updating the Duplicated arguments in place. The same arguments to the forward pass should be provided, followed by the adjoint of the return (if the return is active), and finally the tape from the forward pass.
Example:

 A = [2.2]; ∂A = zero(A)
 v = 3.3
 
@@ -78,7 +78,8 @@
     res
 end
 
-forward, reverse = autodiff_deferred_thunk(ReverseSplitWithPrimal, Const{typeof(f)}, Active, Duplicated{typeof(A)}, Active{typeof(v)})
+TapeType = tape_type(ReverseSplitWithPrimal, Const{typeof(f)}, Active, Duplicated{typeof(A)}, Active{typeof(v)})
+forward, reverse = autodiff_deferred_thunk(ReverseSplitWithPrimal, TapeType, Const{typeof(f)}, Active, Active{Float64}, Duplicated{typeof(A)}, Active{typeof(v)})
 
 tape, result, shadow_result  = forward(Const(f), Duplicated(A, ∂A), Active(v))
 _, ∂v = reverse(Const(f), Duplicated(A, ∂A), Active(v), 1.0, tape)[1]
@@ -87,7 +88,7 @@
 
 # output
 
-(7.26, 2.2, [3.3])
source
EnzymeCore.autodiff_thunk — Method
autodiff_thunk(::ForwardMode, ftype, Activity, argtypes...)
Provide the thunk forward mode function for annotated function type ftype when called with args of type argtypes.
Activity is the Activity of the return value, it may be Const or Duplicated (or its variants DuplicatedNoNeed, BatchDuplicated, andBatchDuplicatedNoNeed).
The forward function will return the primal (if requested) and the shadow (or nothing if not a Duplicated variant).
Example returning both original return and derivative:
a = 4.2
+(7.26, 2.2, [3.3])
source
EnzymeCore.autodiff_thunk — Method
autodiff_thunk(::ForwardMode, ftype, Activity, argtypes::Vararg{Type{<:Annotation}, Nargs})
Provide the thunk forward mode function for annotated function type ftype when called with args of type argtypes.
Activity is the Activity of the return value, it may be Const or Duplicated (or its variants DuplicatedNoNeed, BatchDuplicated, andBatchDuplicatedNoNeed).
The forward function will return the primal (if requested) and the shadow (or nothing if not a Duplicated variant).
Example returning both original return and derivative:
a = 4.2
 b = [2.2, 3.3]; ∂f_∂b = zero(b)
 c = 55; d = 9
 
@@ -107,7 +108,7 @@
 
 # output
 
-(6.28,)
source
EnzymeCore.autodiff_thunk — Method
autodiff_thunk(::ReverseModeSplit, ftype, Activity, argtypes...)
Provide the split forward and reverse pass functions for annotated function type ftype when called with args of type argtypes when using reverse mode.
Activity is the Activity of the return value, it may be Const, Active, or Duplicated (or its variants DuplicatedNoNeed, BatchDuplicated, and BatchDuplicatedNoNeed).
The forward function will return a tape, the primal (or nothing if not requested), and the shadow (or nothing if not a Duplicated variant), and tapes the corresponding type arguements provided.
The reverse function will return the derivative of Active arguments, updating the Duplicated arguments in place. The same arguments to the forward pass should be provided, followed by the adjoint of the return (if the return is active), and finally the tape from the forward pass.
Example:

+(6.28,)
source
EnzymeCore.autodiff_thunk — Method
autodiff_thunk(::ReverseModeSplit, ftype, Activity, argtypes::Vararg{Type{<:Annotation}, Nargs})
Provide the split forward and reverse pass functions for annotated function type ftype when called with args of type argtypes when using reverse mode.
Activity is the Activity of the return value, it may be Const, Active, or Duplicated (or its variants DuplicatedNoNeed, BatchDuplicated, and BatchDuplicatedNoNeed).
The forward function will return a tape, the primal (or nothing if not requested), and the shadow (or nothing if not a Duplicated variant), and tapes the corresponding type arguements provided.
The reverse function will return the derivative of Active arguments, updating the Duplicated arguments in place. The same arguments to the forward pass should be provided, followed by the adjoint of the return (if the return is active), and finally the tape from the forward pass.
Example:

 A = [2.2]; ∂A = zero(A)
 v = 3.3
 
@@ -126,14 +127,17 @@
 
 # output
 
-(7.26, 2.2, [3.3])
source
EnzymeCore.Active — Type
Active(x)
Mark a function argument x of autodiff as active, Enzyme will auto-differentiate in respect Active arguments.
Note
Enzyme gradients with respect to integer values are zero. Active will automatically convert plain integers to floating point values, but cannot do so for integer values in tuples and structs.
source
EnzymeCore.BatchDuplicated — Type
BatchDuplicated(x, ∂f_∂xs)
Like Duplicated, except contains several shadows to compute derivatives for all at once. Argument ∂f_∂xs should be a tuple of the several values of type x.
source
EnzymeCore.Const — Type
Const(x)
Mark a function argument x of autodiff as constant, Enzyme will not auto-differentiate in respect Const arguments.
source
EnzymeCore.Duplicated — Type
Duplicated(x, ∂f_∂x)
Mark a function argument x of autodiff as duplicated, Enzyme will auto-differentiate in respect to such arguments, with dx acting as an accumulator for gradients (so $\partial f / \partial x$ will be added to) ∂f_∂x.
source
EnzymeCore.FFIABI — Type
struct FFIABI <: ABI
Foreign function call ABI. JIT the differentiated function, then inttoptr call the address.
source
EnzymeCore.Mode — Type
abstract type Mode
Abstract type for what differentiation mode will be used.
source
EnzymeCore.ReverseMode — Type
struct ReverseMode{ReturnPrimal,ABI} <: Mode{ABI}
Reverse mode differentiation.
  • ReturnPrimal: Should Enzyme return the primal return value from the augmented-forward.
source
EnzymeCore.ReverseModeSplit — Type
struct ReverseModeSplit{ReturnPrimal,ReturnShadow,Width,ModifiedBetween,ABI} <: Mode{ABI}
Reverse mode differentiation.
  • ReturnPrimal: Should Enzyme return the primal return value from the augmented-forward.
  • ReturnShadow: Should Enzyme return the shadow return value from the augmented-forward.
  • Width: Batch Size (0 if to be automatically derived)
  • ModifiedBetween: Tuple of each argument's modified between state (true if to be automatically derived).
source
EnzymeCore.make_zero — Function
make_zero(::Type{T}, seen::IdDict, prev::T, ::Val{copy_if_inactive}=Val(false))::T
+(7.26, 2.2, [3.3])
source
EnzymeCore.Active — Type
Active(x)
Mark a function argument x of autodiff as active, Enzyme will auto-differentiate in respect Active arguments.
Note
Enzyme gradients with respect to integer values are zero. Active will automatically convert plain integers to floating point values, but cannot do so for integer values in tuples and structs.
source
EnzymeCore.BatchDuplicated — Type
BatchDuplicated(x, ∂f_∂xs)
Like Duplicated, except contains several shadows to compute derivatives for all at once. Argument ∂f_∂xs should be a tuple of the several values of type x.
source
EnzymeCore.Const — Type
Const(x)
Mark a function argument x of autodiff as constant, Enzyme will not auto-differentiate in respect Const arguments.
source
EnzymeCore.Duplicated — Type
Duplicated(x, ∂f_∂x)
Mark a function argument x of autodiff as duplicated, Enzyme will auto-differentiate in respect to such arguments, with dx acting as an accumulator for gradients (so $\partial f / \partial x$ will be added to) ∂f_∂x.
source
EnzymeCore.FFIABI — Type
struct FFIABI <: ABI
Foreign function call ABI. JIT the differentiated function, then inttoptr call the address.
source
EnzymeCore.Mode — Type
abstract type Mode
Abstract type for what differentiation mode will be used.
source
EnzymeCore.ReverseMode — Type
struct ReverseMode{ReturnPrimal,ABI,Holomorphic} <: Mode{ABI}
Reverse mode differentiation.
  • ReturnPrimal: Should Enzyme return the primal return value from the augmented-forward.
  • ABI: What runtime ABI to use
  • Holomorphic: Whether the complex result function is holomorphic and we should compute d/dz
source
EnzymeCore.ReverseModeSplit — Type
struct ReverseModeSplit{ReturnPrimal,ReturnShadow,Width,ModifiedBetween,ABI} <: Mode{ABI}
Reverse mode differentiation.
  • ReturnPrimal: Should Enzyme return the primal return value from the augmented-forward.
  • ReturnShadow: Should Enzyme return the shadow return value from the augmented-forward.
  • Width: Batch Size (0 if to be automatically derived)
  • ModifiedBetween: Tuple of each argument's modified between state (true if to be automatically derived).
source
EnzymeCore.compiler_job_from_backend — Function
compiler_job_from_backend(::KernelAbstractions.Backend, F::Type, TT:Type)::GPUCompiler.CompilerJob
Returns a GPUCompiler CompilerJob from a backend as specified by the first argument to the function.
For example, in CUDA one would do:
function EnzymeCore.compiler_job_from_backend(::CUDABackend, @nospecialize(F::Type), @nospecialize(TT::Type))
+    mi = GPUCompiler.methodinstance(F, TT)
+    return GPUCompiler.CompilerJob(mi, CUDA.compiler_config(CUDA.device()))
+end
source
EnzymeCore.make_zero — Function
make_zero(::Type{T}, seen::IdDict, prev::T, ::Val{copy_if_inactive}=Val(false))::T
 
 Recursively make a zero'd copy of the value `prev` of type `T`. The argument `copy_if_inactive` specifies
-what to do if the type `T` is guaranteed to be inactive, use the primal (the default) or still copy the value.
source
EnzymeCore.make_zero — Method
make_zero(prev::T)
+what to do if the type `T` is guaranteed to be inactive, use the primal (the default) or still copy the value.
source
EnzymeCore.EnzymeRules.AugmentedReturn — Type
AugmentedReturn(primal, shadow, tape)
Augment the primal return value of a function with its shadow, as well as any additional information needed to correctly  compute the reverse pass, stored in tape.
Unless specified by the config that a variable is not overwritten, rules must assume any arrays/data structures/etc are  overwritten between the forward and the reverse pass. Any floats or variables passed by value are always preserved as is  (as are the arrays themselves, just not necessarily the values in the array).
See also augmented_primal.
source
EnzymeCore.EnzymeRules.Config — Type
Config{NeedsPrimal, NeedsShadow, Width, Overwritten}
-ConfigWidth{Width} = Config{<:Any,<:Any, Width}
Configuration type to dispatch on in custom reverse rules (see augmented_primal and reverse).
  • NeedsPrimal and NeedsShadow: boolean values specifying whether the primal and shadow (resp.) should be returned. 
  • Width: an integer that specifies the number of adjoints/shadows simultaneously being propagated.
  • Overwritten: a tuple of booleans of whether each argument (including the function itself) is modified between the   forward and reverse pass (true if potentially modified between).
Getters for the four type parameters are provided by needs_primal, needs_shadow, width, and overwritten.
source
EnzymeCore.EnzymeRules.augmented_primal — Function
augmented_primal(::Config, func::Annotation{typeof(f)}, RT::Type{<:Annotation}, args::Annotation...)
Must return an AugmentedReturn type.
  • The primal must be the same type of the original return if needs_primal(config), otherwise nothing.
  • The shadow must be nothing if needs_shadow(config) is false. If width is 1, the shadow should be the same type of the original return. If the width is greater than 1, the shadow should be NTuple{original return, width}.
  • The tape can be any type (including Nothing) and is preserved for the reverse call.
source
EnzymeCore.EnzymeRules.forward — Function
forward(func::Annotation{typeof(f)}, RT::Type{<:Annotation}, args::Annotation...)
Calculate the forward derivative. The first argument func is the callable for which the rule applies to. Either wrapped in a Const), or a Duplicated if it is a closure. The second argument is the return type annotation, and all other arguments are the annotated function arguments.
source
EnzymeCore.EnzymeRules.inactive — Function
inactive(func::typeof(f), args...)
Mark a particular function as always being inactive in both its return result and the function call itself.
source
EnzymeCore.EnzymeRules.inactive_noinl — Function
inactive_noinl(func::typeof(f), args...)
Mark a particular function as always being inactive in both its return result and the function call itself, but do not prevent inlining of the function.
source
EnzymeCore.EnzymeRules.reverse — Function
reverse(::Config, func::Annotation{typeof(f)}, dret::Active, tape, args::Annotation...)
-reverse(::Config, func::Annotation{typeof(f)}, ::Type{<:Annotation), tape, args::Annotation...)
Takes gradient of derivative, activity annotation, and tape. If there is an active return dret is passed as Active{T} with the derivative of the active return val. Otherwise dret is passed as Type{Duplicated{T}}, etc.
source
EnzymeTestUtils.@test_msg — Macro
@test_msg msg condion kws...
This is per Test.@test condion kws... except that if it fails it also prints the msg. If msg=="" then this is just like @test, nothing is printed
Examles
julia> @test_msg "It is required that the total is under 10" sum(1:1000) < 10;
+Helper function to recursively make zero.
source
EnzymeCore.EnzymeRules.AugmentedReturn — Type
AugmentedReturn(primal, shadow, tape)
Augment the primal return value of a function with its shadow, as well as any additional information needed to correctly  compute the reverse pass, stored in tape.
Unless specified by the config that a variable is not overwritten, rules must assume any arrays/data structures/etc are  overwritten between the forward and the reverse pass. Any floats or variables passed by value are always preserved as is  (as are the arrays themselves, just not necessarily the values in the array).
See also augmented_primal.
source
EnzymeCore.EnzymeRules.Config — Type
Config{NeedsPrimal, NeedsShadow, Width, Overwritten}
+ConfigWidth{Width} = Config{<:Any,<:Any, Width}
Configuration type to dispatch on in custom reverse rules (see augmented_primal and reverse).
  • NeedsPrimal and NeedsShadow: boolean values specifying whether the primal and shadow (resp.) should be returned. 
  • Width: an integer that specifies the number of adjoints/shadows simultaneously being propagated.
  • Overwritten: a tuple of booleans of whether each argument (including the function itself) is modified between the   forward and reverse pass (true if potentially modified between).
Getters for the four type parameters are provided by needs_primal, needs_shadow, width, and overwritten.
source
EnzymeCore.EnzymeRules.augmented_primal — Function
augmented_primal(::Config, func::Annotation{typeof(f)}, RT::Type{<:Annotation}, args::Annotation...)
Must return an AugmentedReturn type.
  • The primal must be the same type of the original return if needs_primal(config), otherwise nothing.
  • The shadow must be nothing if needs_shadow(config) is false. If width is 1, the shadow should be the same type of the original return. If the width is greater than 1, the shadow should be NTuple{original return, width}.
  • The tape can be any type (including Nothing) and is preserved for the reverse call.
source
EnzymeCore.EnzymeRules.forward — Function
forward(func::Annotation{typeof(f)}, RT::Type{<:Annotation}, args::Annotation...)
Calculate the forward derivative. The first argument func is the callable for which the rule applies to. Either wrapped in a Const), or a Duplicated if it is a closure. The second argument is the return type annotation, and all other arguments are the annotated function arguments.
source
EnzymeCore.EnzymeRules.inactive — Function
inactive(func::typeof(f), args...)
Mark a particular function as always being inactive in both its return result and the function call itself.
source
EnzymeCore.EnzymeRules.inactive_noinl — Function
inactive_noinl(func::typeof(f), args...)
Mark a particular function as always being inactive in both its return result and the function call itself, but do not prevent inlining of the function.
source
EnzymeCore.EnzymeRules.reverse — Function
reverse(::Config, func::Annotation{typeof(f)}, dret::Active, tape, args::Annotation...)
+reverse(::Config, func::Annotation{typeof(f)}, ::Type{<:Annotation), tape, args::Annotation...)
Takes gradient of derivative, activity annotation, and tape. If there is an active return dret is passed as Active{T} with the derivative of the active return val. Otherwise dret is passed as Type{Duplicated{T}}, etc.
source
EnzymeTestUtils.@test_msg — Macro
@test_msg msg condion kws...
This is per Test.@test condion kws... except that if it fails it also prints the msg. If msg=="" then this is just like @test, nothing is printed
Examles
julia> @test_msg "It is required that the total is under 10" sum(1:1000) < 10;
 Test Failed at REPL[1]:1
   Expression: sum(1:1000) < 10
   Problem: It is required that the total is under 10
@@ -155,7 +159,7 @@
   Test Failed at REPL[153]:1
     Expression: sum(1:1000) < 10
      Evaluated: 500500 < 10
-  ERROR: There was an error during testing
source
EnzymeTestUtils.test_forward — Method
test_forward(f, Activity, args...; kwargs...)
Test Enzyme.autodiff of f in Forward-mode against finite differences.
f has all constraints of the same argument passed to Enzyme.autodiff, with several additional constraints:
  • If it mutates one of its arguments, it must return that argument.
Arguments
  • Activity: the activity of the return value of f
  • args: Each entry is either an argument to f, an activity type accepted by autodiff,   or a tuple of the form (arg, Activity), where Activity is the activity type of   arg. If the activity type specified requires a tangent, a random tangent will be   automatically generated.
Keywords
  • fdm=FiniteDifferences.central_fdm(5, 1): The finite differences method to use.
  • fkwargs: Keyword arguments to pass to f.
  • rtol: Relative tolerance for isapprox.
  • atol: Absolute tolerance for isapprox.
  • testset_name: Name to use for a testset in which all tests are evaluated.
Examples
Here we test a rule for a function of scalars. Because we don't provide an activity annotation for y, it is assumed to be Const.
using Enzyme, EnzymeTestUtils
+  ERROR: There was an error during testing
source
EnzymeTestUtils.test_forward — Method
test_forward(f, Activity, args...; kwargs...)
Test Enzyme.autodiff of f in Forward-mode against finite differences.
f has all constraints of the same argument passed to Enzyme.autodiff, with several additional constraints:
  • If it mutates one of its arguments, it must return that argument.
Arguments
  • Activity: the activity of the return value of f
  • args: Each entry is either an argument to f, an activity type accepted by autodiff,   or a tuple of the form (arg, Activity), where Activity is the activity type of   arg. If the activity type specified requires a tangent, a random tangent will be   automatically generated.
Keywords
  • fdm=FiniteDifferences.central_fdm(5, 1): The finite differences method to use.
  • fkwargs: Keyword arguments to pass to f.
  • rtol: Relative tolerance for isapprox.
  • atol: Absolute tolerance for isapprox.
  • testset_name: Name to use for a testset in which all tests are evaluated.
Examples
Here we test a rule for a function of scalars. Because we don't provide an activity annotation for y, it is assumed to be Const.
using Enzyme, EnzymeTestUtils
 
 x, y = randn(2)
 for Tret in (Const, Duplicated, DuplicatedNoNeed), Tx in (Const, Duplicated)
@@ -167,7 +171,7 @@
     Ty in (Const, BatchDuplicated)
 
     test_forward(*, Tret, (x, Tx), (y, Ty))
-end
source
EnzymeTestUtils.test_reverse — Method
test_reverse(f, Activity, args...; kwargs...)
Test Enzyme.autodiff_thunk of f in ReverseSplitWithPrimal-mode against finite differences.
f has all constraints of the same argument passed to Enzyme.autodiff_thunk, with several additional constraints:
  • If it mutates one of its arguments, it must not also return that argument.
  • If the return value is a struct, then all floating point numbers contained in the struct   or its fields must be in arrays.
Arguments
  • Activity: the activity of the return value of f.
  • args: Each entry is either an argument to f, an activity type accepted by autodiff,   or a tuple of the form (arg, Activity), where Activity is the activity type of   arg. If the activity type specified requires a shadow, one will be automatically   generated.
Keywords
  • fdm=FiniteDifferences.central_fdm(5, 1): The finite differences method to use.
  • fkwargs: Keyword arguments to pass to f.
  • rtol: Relative tolerance for isapprox.
  • atol: Absolute tolerance for isapprox.
  • testset_name: Name to use for a testset in which all tests are evaluated.
Examples
Here we test a rule for a function of scalars. Because we don't provide an activity annotation for y, it is assumed to be Const.
using Enzyme, EnzymeTestUtils
+end
source
EnzymeTestUtils.test_reverse — Method
test_reverse(f, Activity, args...; kwargs...)
Test Enzyme.autodiff_thunk of f in ReverseSplitWithPrimal-mode against finite differences.
f has all constraints of the same argument passed to Enzyme.autodiff_thunk, with several additional constraints:
  • If it mutates one of its arguments, it must not also return that argument.
  • If the return value is a struct, then all floating point numbers contained in the struct   or its fields must be in arrays.
Arguments
  • Activity: the activity of the return value of f.
  • args: Each entry is either an argument to f, an activity type accepted by autodiff,   or a tuple of the form (arg, Activity), where Activity is the activity type of   arg. If the activity type specified requires a shadow, one will be automatically   generated.
Keywords
  • fdm=FiniteDifferences.central_fdm(5, 1): The finite differences method to use.
  • fkwargs: Keyword arguments to pass to f.
  • rtol: Relative tolerance for isapprox.
  • atol: Absolute tolerance for isapprox.
  • testset_name: Name to use for a testset in which all tests are evaluated.
Examples
Here we test a rule for a function of scalars. Because we don't provide an activity annotation for y, it is assumed to be Const.
using Enzyme, EnzymeTestUtils
 
 x = randn()
 y = randn()
@@ -176,4 +180,4 @@
 end
Here we test a rule for a function of an array in batch reverse-mode:
x = randn(3)
 for Tret in (Const, Active), Tx in (Const, BatchDuplicated)
     test_reverse(prod, Tret, (x, Tx))
-end
source
Enzyme.API.runtimeActivity! — Method
runtimeActivity!(val::Bool)
Enzyme runs an activity analysis which deduces which values, instructions, etc are necessary to be differentiated and therefore involved in the differentiation procedure. This runs at compile time. However, there may be implementation flaws in this analysis that means that Enzyme cannot deduce that an inactive (const) value is actually const. Alternatively, there may be some data which is conditionally active, depending on which runtime branch is taken. In these cases Enzyme conservatively presumes the value is active.
However, in certain cases, an insufficiently aggressive activity analysis may result in derivative errors – for example by mistakenly using the primal (const) argument and mistaking it for the duplicated shadow. As a result this may result in incorrect results, or accidental updates to the primal.
This flag enables runntime activity which tells all load/stores to check at runtime whether the value they are updating is indeed active (in addition to the compile-time activity analysis). This will remedy these such errors, but at a performance penalty of performing such checks.
It is on the Enzyme roadmap to add a PotentiallyDuplicated style activity, in addition to the current Const and Duplicated styles that will disable the need for this, which does  not require the check when a value is guaranteed active, but still supports runtime-based activity information.
This function takes an argument to set the runtime activity value, true means it is on, and false means off. By default it is off.
source

+end
source
Enzyme.API.runtimeActivity! — Method
runtimeActivity!(val::Bool)

Enzyme runs an activity analysis which deduces which values, instructions, etc are necessary to be differentiated and therefore involved in the differentiation procedure. This runs at compile time. However, there may be implementation flaws in this analysis that means that Enzyme cannot deduce that an inactive (const) value is actually const. Alternatively, there may be some data which is conditionally active, depending on which runtime branch is taken. In these cases Enzyme conservatively presumes the value is active.

However, in certain cases, an insufficiently aggressive activity analysis may result in derivative errors – for example by mistakenly using the primal (const) argument and mistaking it for the duplicated shadow. As a result this may result in incorrect results, or accidental updates to the primal.

This flag enables runntime activity which tells all load/stores to check at runtime whether the value they are updating is indeed active (in addition to the compile-time activity analysis). This will remedy these such errors, but at a performance penalty of performing such checks.

It is on the Enzyme roadmap to add a PotentiallyDuplicated style activity, in addition to the current Const and Duplicated styles that will disable the need for this, which does not require the check when a value is guaranteed active, but still supports runtime-based activity information.

This function takes an argument to set the runtime activity value, true means it is on, and false means off. By default it is off.

source
diff --git a/previews/PR1300/dev_docs/index.html b/previews/PR1300/dev_docs/index.html index b74993be54..087edfab4b 100644 --- a/previews/PR1300/dev_docs/index.html +++ b/previews/PR1300/dev_docs/index.html @@ -11,4 +11,4 @@ julia -e "using Pkg; pkg\"add LLVM_full_jll@${LLVM_MAJOR_VER}\"" LLVM_DIR=`julia -e "using LLVM_full_jll; print(LLVM_full_jll.artifact_dir)"` echo "LLVM_DIR=$LLVM_DIR" -cmake ../enzyme/ -G Ninja -DENZYME_EXTERNAL_SHARED_LIB=ON -DLLVM_DIR=${LLVM_DIR} -DLLVM_EXTERNAL_LIT=${LLVM_DIR}/tools/lit/lit.py

Manual build of Julia

cmake ../enzyme/ -G Ninja -DENZYME_EXTERNAL_SHARED_LIB=ON -DLLVM_DIR=${PATH_TO_BUILDDIR_OF_JULIA}/usr/lib/cmake/llvm/
+cmake ../enzyme/ -G Ninja -DENZYME_EXTERNAL_SHARED_LIB=ON -DLLVM_DIR=${LLVM_DIR} -DLLVM_EXTERNAL_LIT=${LLVM_DIR}/tools/lit/lit.py

Manual build of Julia

cmake ../enzyme/ -G Ninja -DENZYME_EXTERNAL_SHARED_LIB=ON -DLLVM_DIR=${PATH_TO_BUILDDIR_OF_JULIA}/usr/lib/cmake/llvm/
diff --git a/previews/PR1300/generated/autodiff/index.html b/previews/PR1300/generated/autodiff/index.html index 41efaf7444..252d974ba5 100644 --- a/previews/PR1300/generated/autodiff/index.html +++ b/previews/PR1300/generated/autodiff/index.html @@ -66,4 +66,4 @@ hess[1][2] == 1.0
true

as well as the second row/column

hess[2][1] == 1.0
 
-hess[2][2] == 0.0
true

This page was generated using Literate.jl.

+hess[2][2] == 0.0
true

This page was generated using Literate.jl.

diff --git a/previews/PR1300/generated/box/index.html b/previews/PR1300/generated/box/index.html index 30b0680a05..d042113feb 100644 --- a/previews/PR1300/generated/box/index.html +++ b/previews/PR1300/generated/box/index.html @@ -233,7 +233,7 @@ Duplicated([Tbar; Sbar], dstate_old), Duplicated(out_now, dout_now), Duplicated(out_old, dout_old), - parameters, + Const(parameters), Const(10*parameters.day) )
((nothing, nothing, nothing, nothing, nothing, nothing),)

In order to run Enzyme on one_step_forward, we've needed to provide quite a few placeholders, and wrap everything in Duplicated as all components of our function are vectors, not scalars. Let's go through and see what Enzyme did with all of those placeholders.

First we can look at what happened to the zero vectors out_now and out_old:

@show out_now, out_old
([20.101970893653334, 0.9646957730133332, 1.0, 35.50026715349918, 34.49973284650082, 34.5], [20.025492723413333, 0.9911739432533333, 1.0, 35.500066788374795, 34.499933211625205, 34.5])

Comparing to the results of forward func:

@show states_before_smoother[2], states_after_smoother[2]
([20.101970893653334, 0.9646957730133332, 1.0, 35.50026715349918, 34.49973284650082, 34.5], [20.025492723413333, 0.9911739432533333, 1.0, 35.500066788374795, 34.499933211625205, 34.5])

we see that Enzyme has computed and stored exactly the output of the forward step. Next, let's look at dstate_now:

@show dstate_now
6-element Vector{Float64}:
  0.41666666666666663
@@ -251,7 +251,7 @@
     Duplicated([Tbar; Sbar], dstate_old_new),
     Duplicated(out_now, dout_now),
     Duplicated(out_old, dout_old),
-    parameters,
+    Const(parameters),
     Const(10*parameters.day)
 )
((nothing, nothing, nothing, nothing, nothing, nothing),)

Now checking dstate_now and dstate_old we see

@show dstate_now_new
6-element Vector{Float64}:
  0.8333333333333333
@@ -281,7 +281,7 @@
             Duplicated(states_after_smoother[j], dstate_old),
             Duplicated(zeros(6), dout_now),
             Duplicated(zeros(6), dout_old),
-            parameters,
+            Const(parameters),
             Const(10*parameters.day)
         )
 
@@ -362,4 +362,4 @@
  9.732210923954578e-5
  0.0026401658789532625
  0.015152571729925521
- 0.03212933056407103

and we get down to a percent difference on the order of $1e^{-5}$, showing Enzyme calculated the correct derivative. Success!

This page was generated using Literate.jl.

+ 0.03212933056407103

and we get down to a percent difference on the order of $1e^{-5}$, showing Enzyme calculated the correct derivative. Success!

This page was generated using Literate.jl.

diff --git a/previews/PR1300/generated/custom_rule/index.html b/previews/PR1300/generated/custom_rule/index.html index 0648e12fe6..51a8ac9070 100644 --- a/previews/PR1300/generated/custom_rule/index.html +++ b/previews/PR1300/generated/custom_rule/index.html @@ -91,7 +91,7 @@ end # Return an AugmentedReturn object with shadow = nothing return AugmentedReturn(primal, nothing, tape) -end
augmented_primal (generic function with 8 methods)

Let's unpack our signature for augmented_primal :

Now, let's unpack the body of our augmented_primal rule:

Now, we write a method for EnzymeRules.reverse:

function reverse(config::ConfigWidth{1}, func::Const{typeof(f)}, dret::Active, tape,
+end
augmented_primal (generic function with 9 methods)

Let's unpack our signature for augmented_primal :

Now, let's unpack the body of our augmented_primal rule:

Now, we write a method for EnzymeRules.reverse:

function reverse(config::ConfigWidth{1}, func::Const{typeof(f)}, dret::Active, tape,
                  y::Duplicated, x::Duplicated)
     println("In custom reverse rule.")
     # retrieve x value, either from original x or from tape if x may have been overwritten.
@@ -102,7 +102,7 @@
     x.dval .+= 2 .* xval .* y.dval
     y.dval .= 0
     return (nothing, nothing)
-end
reverse (generic function with 8 methods)

Let's make a few observations about our reverse rule:

Finally, let's see our reverse rule in action!

x  = [3.0, 1.0]
+end
reverse (generic function with 9 methods)

Let's make a few observations about our reverse rule:

Finally, let's see our reverse rule in action!

x  = [3.0, 1.0]
 dx = [0.0, 0.0]
 y  = [0.0, 0.0]
 dy = [0.0, 0.0]
@@ -165,4 +165,4 @@
             test_reverse(fun, RT, (x, Tx), (y, Ty))
         end
     end
-end
Test.DefaultTestSet("f rules", Any[Test.DefaultTestSet("forward", Any[Test.DefaultTestSet("RT = Const, Tx = Const, Ty = Const", Any[Test.DefaultTestSet("test_forward: g with return activity Const on (::Vector{Float64}, Const), (::Vector{Float64}, Const)", Any[], 7, false, false, true, 1.708527515984778e9, 1.708527517962745e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.708527513430784e9, 1.708527517962752e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Const, Tx = Const, Ty = Duplicated", Any[Test.DefaultTestSet("test_forward: g with return activity Const on (::Vector{Float64}, Const), (::Vector{Float64}, Duplicated)", Any[], 7, false, false, true, 1.708527517963903e9, 1.708527519271361e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.708527517962804e9, 1.708527519271367e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Const, Tx = Duplicated, Ty = Const", Any[Test.DefaultTestSet("test_forward: g with return activity Const on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Const)", Any[], 7, false, false, true, 1.708527519272438e9, 1.708527520426653e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.708527519271403e9, 1.708527520426658e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Const, Tx = Duplicated, Ty = Duplicated", Any[Test.DefaultTestSet("test_forward: g with return activity Const on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Duplicated)", Any[], 7, false, false, true, 1.708527520427763e9, 1.708527521654913e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.708527520426704e9, 1.708527521654918e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = DuplicatedNoNeed, Tx = Const, Ty = Const", Any[Test.DefaultTestSet("test_forward: g with return activity DuplicatedNoNeed on (::Vector{Float64}, Const), (::Vector{Float64}, Const)", Any[], 6, false, false, true, 1.708527521656028e9, 1.708527523046988e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.708527521654963e9, 1.708527523046993e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = DuplicatedNoNeed, Tx = Const, Ty = Duplicated", Any[Test.DefaultTestSet("test_forward: g with return activity DuplicatedNoNeed on (::Vector{Float64}, Const), (::Vector{Float64}, Duplicated)", Any[], 6, false, false, true, 1.70852752304809e9, 1.708527525092253e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.708527523047038e9, 1.708527525092258e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = DuplicatedNoNeed, Tx = Duplicated, Ty = Const", Any[Test.DefaultTestSet("test_forward: g with return activity DuplicatedNoNeed on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Const)", Any[], 6, false, false, true, 1.708527525093373e9, 1.708527526504367e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.708527525092289e9, 1.708527526504371e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = DuplicatedNoNeed, Tx = Duplicated, Ty = Duplicated", Any[Test.DefaultTestSet("test_forward: g with return activity DuplicatedNoNeed on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Duplicated)", Any[], 6, false, false, true, 1.708527526505497e9, 1.70852752825799e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.708527526504402e9, 1.708527528257994e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Duplicated, Tx = Const, Ty = Const", Any[Test.DefaultTestSet("test_forward: g with return activity Duplicated on (::Vector{Float64}, Const), (::Vector{Float64}, Const)", Any[], 7, false, false, true, 1.708527528259073e9, 1.708527529288653e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.708527528258024e9, 1.708527529288658e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Duplicated, Tx = Const, Ty = Duplicated", Any[Test.DefaultTestSet("test_forward: g with return activity Duplicated on (::Vector{Float64}, Const), (::Vector{Float64}, Duplicated)", Any[], 7, false, false, true, 1.708527529289745e9, 1.708527530382497e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.708527529288687e9, 1.708527530382501e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Duplicated, Tx = Duplicated, Ty = Const", Any[Test.DefaultTestSet("test_forward: g with return activity Duplicated on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Const)", Any[], 7, false, false, true, 1.708527530383594e9, 1.708527531507317e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.708527530382529e9, 1.708527531507322e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Duplicated, Tx = Duplicated, Ty = Duplicated", Any[Test.DefaultTestSet("test_forward: g with return activity Duplicated on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Duplicated)", Any[], 7, false, false, true, 1.708527531508461e9, 1.708527532715627e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.70852753150735e9, 1.708527532715631e9, false, "custom_rule.md")], 0, false, false, true, 1.70852751343071e9, 1.708527532715632e9, false, "custom_rule.md"), Test.DefaultTestSet("reverse", Any[Test.DefaultTestSet("RT = Active, Tx = Duplicated, Ty = Duplicated, fun = g", Any[Test.DefaultTestSet("test_reverse: g with return activity Active on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Duplicated)", Any[], 11, false, false, true, 1.70852753284811e9, 1.708527536341405e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_reverse.jl")], 0, false, false, true, 1.708527532715689e9, 1.70852753634141e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Active, Tx = Duplicated, Ty = Duplicated, fun = h", Any[Test.DefaultTestSet("test_reverse: h with return activity Active on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Duplicated)", Any[], 11, false, false, true, 1.70852753634214e9, 1.708527540147945e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_reverse.jl")], 0, false, false, true, 1.708527536341445e9, 1.708527540147949e9, false, "custom_rule.md")], 0, false, false, true, 1.708527532715651e9, 1.708527540147951e9, false, "custom_rule.md")], 0, false, false, true, 1.708527513410021e9, 1.708527540147951e9, false, "custom_rule.md")

In any package that implements Enzyme rules using EnzymeRules, it is recommended to add EnzymeTestUtils as a test dependency to test the rules.

This page was generated using Literate.jl.

+end
Test.DefaultTestSet("f rules", Any[Test.DefaultTestSet("forward", Any[Test.DefaultTestSet("RT = Const, Tx = Const, Ty = Const", Any[Test.DefaultTestSet("test_forward: g with return activity Const on (::Vector{Float64}, Const), (::Vector{Float64}, Const)", Any[], 7, false, false, true, 1.710973836649167e9, 1.71097383890532e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.710973834041321e9, 1.710973838905328e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Const, Tx = Const, Ty = Duplicated", Any[Test.DefaultTestSet("test_forward: g with return activity Const on (::Vector{Float64}, Const), (::Vector{Float64}, Duplicated)", Any[], 7, false, false, true, 1.710973838906592e9, 1.710973840240632e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.710973838905382e9, 1.710973840240638e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Const, Tx = Duplicated, Ty = Const", Any[Test.DefaultTestSet("test_forward: g with return activity Const on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Const)", Any[], 7, false, false, true, 1.710973840241874e9, 1.710973841401706e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.710973840240691e9, 1.710973841401711e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Const, Tx = Duplicated, Ty = Duplicated", Any[Test.DefaultTestSet("test_forward: g with return activity Const on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Duplicated)", Any[], 7, false, false, true, 1.710973841402782e9, 1.71097384266981e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.710973841401755e9, 1.710973842669815e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = DuplicatedNoNeed, Tx = Const, Ty = Const", Any[Test.DefaultTestSet("test_forward: g with return activity DuplicatedNoNeed on (::Vector{Float64}, Const), (::Vector{Float64}, Const)", Any[], 6, false, false, true, 1.710973842671204e9, 1.71097384410062e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.710973842669871e9, 1.710973844100626e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = DuplicatedNoNeed, Tx = Const, Ty = Duplicated", Any[Test.DefaultTestSet("test_forward: g with return activity DuplicatedNoNeed on (::Vector{Float64}, Const), (::Vector{Float64}, Duplicated)", Any[], 6, false, false, true, 1.710973844101898e9, 1.710973846178704e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.710973844100676e9, 1.710973846178709e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = DuplicatedNoNeed, Tx = Duplicated, Ty = Const", Any[Test.DefaultTestSet("test_forward: g with return activity DuplicatedNoNeed on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Const)", Any[], 6, false, false, true, 1.710973846179807e9, 1.710973847570577e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.710973846178741e9, 1.710973847570583e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = DuplicatedNoNeed, Tx = Duplicated, Ty = Duplicated", Any[Test.DefaultTestSet("test_forward: g with return activity DuplicatedNoNeed on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Duplicated)", Any[], 6, false, false, true, 1.710973847571973e9, 1.710973849339177e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.710973847570614e9, 1.710973849339181e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Duplicated, Tx = Const, Ty = Const", Any[Test.DefaultTestSet("test_forward: g with return activity Duplicated on (::Vector{Float64}, Const), (::Vector{Float64}, Const)", Any[], 7, false, false, true, 1.710973849340235e9, 1.710973850359327e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.71097384933921e9, 1.710973850359331e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Duplicated, Tx = Const, Ty = Duplicated", Any[Test.DefaultTestSet("test_forward: g with return activity Duplicated on (::Vector{Float64}, Const), (::Vector{Float64}, Duplicated)", Any[], 7, false, false, true, 1.7109738503604e9, 1.710973851457519e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.71097385035936e9, 1.710973851457523e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Duplicated, Tx = Duplicated, Ty = Const", Any[Test.DefaultTestSet("test_forward: g with return activity Duplicated on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Const)", Any[], 7, false, false, true, 1.710973851458587e9, 1.710973852580822e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.710973851457551e9, 1.710973852580828e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Duplicated, Tx = Duplicated, Ty = Duplicated", Any[Test.DefaultTestSet("test_forward: g with return activity Duplicated on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Duplicated)", Any[], 7, false, false, true, 1.71097385258233e9, 1.710973853786377e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_forward.jl")], 0, false, false, true, 1.710973852580861e9, 1.71097385378638e9, false, "custom_rule.md")], 0, false, false, true, 1.710973834041267e9, 1.710973853786381e9, false, "custom_rule.md"), Test.DefaultTestSet("reverse", Any[Test.DefaultTestSet("RT = Active, Tx = Duplicated, Ty = Duplicated, fun = g", Any[Test.DefaultTestSet("test_reverse: g with return activity Active on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Duplicated)", Any[], 11, false, false, true, 1.710973853918987e9, 1.710973857495458e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_reverse.jl")], 0, false, false, true, 1.710973853786446e9, 1.710973857495464e9, false, "custom_rule.md"), Test.DefaultTestSet("RT = Active, Tx = Duplicated, Ty = Duplicated, fun = h", Any[Test.DefaultTestSet("test_reverse: h with return activity Active on (::Vector{Float64}, Duplicated), (::Vector{Float64}, Duplicated)", Any[], 11, false, false, true, 1.710973857496413e9, 1.710973861359235e9, false, "/home/runner/work/Enzyme.jl/Enzyme.jl/lib/EnzymeTestUtils/src/test_reverse.jl")], 0, false, false, true, 1.710973857495503e9, 1.710973861359238e9, false, "custom_rule.md")], 0, false, false, true, 1.710973853786405e9, 1.71097386135924e9, false, "custom_rule.md")], 0, false, false, true, 1.710973834037112e9, 1.71097386135924e9, false, "custom_rule.md")

In any package that implements Enzyme rules using EnzymeRules, it is recommended to add EnzymeTestUtils as a test dependency to test the rules.

This page was generated using Literate.jl.

diff --git a/previews/PR1300/index.html b/previews/PR1300/index.html index 90d75929cc..763ceb8d66 100644 --- a/previews/PR1300/index.html +++ b/previews/PR1300/index.html @@ -1,9 +1,9 @@ -Home · Enzyme.jl
Edit on GitHub

Enzyme

Documentation for Enzyme.jl, the Julia bindings for Enzyme.

Enzyme performs automatic differentiation (AD) of statically analyzable LLVM. It is highly-efficient and its ability to perform AD on optimized code allows Enzyme to meet or exceed the performance of state-of-the-art AD tools.

Enzyme.jl can be installed in the usual way Julia packages are installed:

] add Enzyme

The Enzyme binary dependencies will be installed automatically via Julia's binary artifact system.

The Enzyme.jl API revolves around the function autodiff. For some common operations, Enzyme additionally wraps autodiff in several convenience functions; e.g., gradient and jacobian.

The tutorial below covers the basic usage of these functions. For a complete overview of Enzyme's functionality, see the API documentation. Also see Implementing pullbacks on how to implement back-propagation for functions with non-scalar results.

Getting started

julia> rosenbrock(x, y) = (1.0 - x)^2 + 100.0 * (y - x^2)^2
+Home · Enzyme.jl
Edit on GitHub
Enzyme
Documentation for Enzyme.jl, the Julia bindings for Enzyme.
Enzyme performs automatic differentiation (AD) of statically analyzable LLVM. It is highly-efficient and its ability to perform AD on optimized code allows Enzyme to meet or exceed the performance of state-of-the-art AD tools.
Enzyme.jl can be installed in the usual way Julia packages are installed:
] add Enzyme
The Enzyme binary dependencies will be installed automatically via Julia's binary artifact system.
The Enzyme.jl API revolves around the function autodiff. For some common operations, Enzyme additionally wraps autodiff in several convenience functions; e.g., gradient and jacobian.
The tutorial below covers the basic usage of these functions. For a complete overview of Enzyme's functionality, see the API documentation. Also see Implementing pullbacks on how to implement back-propagation for functions with non-scalar results.
Getting started
julia> rosenbrock(x, y) = (1.0 - x)^2 + 100.0 * (y - x^2)^2
 rosenbrock (generic function with 1 method)
 
 julia> rosenbrock_inp(x) = (1.0 - x[1])^2 + 100.0 * (x[2] - x[1]^2)^2
-rosenbrock_inp (generic function with 1 method)
Reverse mode
The return value of reverse mode autodiff is a tuple that contains as a first value the derivative value of the active inputs and optionally the primal return value.
julia> autodiff(Reverse, rosenbrock, Active, Active(1.0), Active(2.0))
+rosenbrock_inp (generic function with 1 method)
Reverse mode
The return value of reverse mode autodiff is a tuple that contains as a first value the derivative value of the active inputs and optionally the primal return value.
julia> autodiff(Reverse, rosenbrock, Active, Active(1.0), Active(2.0))
 ((-400.0, 200.0),)
 
 julia> autodiff(ReverseWithPrimal, rosenbrock, Active, Active(1.0), Active(2.0))
@@ -50,7 +50,7 @@
 julia> dx_1 = [1.0, 0.0]; dx_2 = [0.0, 1.0];
 
 julia> autodiff(Forward, rosenbrock_inp, BatchDuplicated, BatchDuplicated(x, (dx_1, dx_2)))
-(400.0, (var"1" = -800.0, var"2" = 400.0))
Convenience functions
Note
While the convenience functions discussed below use autodiff internally, they are generally more limited in their functionality. Beyond that, these convenience functions may also come with performance penalties; especially if one makes a closure of a multi-argument function instead of calling the appropriate multi-argument autodiff function directly.
Key convenience functions for common derivative computations are gradient (and its inplace variant gradient!) and jacobian. Like autodiff, the mode (forward or reverse) is determined by the first argument.
The functions gradient and gradient! compute the gradient of function with vector input and scalar return.
julia> gradient(Reverse, rosenbrock_inp, [1.0, 2.0])
+(400.0, (var"1" = -800.0, var"2" = 400.0))
Convenience functions
Note
While the convenience functions discussed below use autodiff internally, they are generally more limited in their functionality. Beyond that, these convenience functions may also come with performance penalties; especially if one makes a closure of a multi-argument function instead of calling the appropriate multi-argument autodiff function directly.
Key convenience functions for common derivative computations are gradient (and its inplace variant gradient!) and jacobian. Like autodiff, the mode (forward or reverse) is determined by the first argument.
The functions gradient and gradient! compute the gradient of function with vector input and scalar return.
julia> gradient(Reverse, rosenbrock_inp, [1.0, 2.0])
 2-element Vector{Float64}:
  -400.0
   200.0
@@ -98,7 +98,7 @@
        jacobian(Forward, foo, [1.0, 2.0], chunk_size)
 2×2 Matrix{Float64}:
  -400.0  200.0
-    2.0    1.0
Caveats / Known-issues
Activity of temporary storage
If you pass in any temporary storage which may be involved in an active computation to a function you want to differentiate, you must also pass in a duplicated temporary storage for use in computing the derivatives. 
function f(x, tmp, k, n)
+    2.0    1.0
Caveats / Known-issues
Activity of temporary storage / Activity Unstable Code
If you pass in any temporary storage which may be involved in an active computation to a function you want to differentiate, you must also pass in a duplicated temporary storage for use in computing the derivatives. For example, consider the following function which uses a temporary buffer to compute the result.
function f(x, tmp, k, n)
     tmp[1] = 1.0
     for i in 1:n
         tmp[k] *= x
@@ -108,15 +108,48 @@
 
 # output
 
-f (generic function with 1 method)
Enzyme.autodiff(Reverse, f, Active(1.2), Const(Vector{Float64}(undef, 1)), Const(1), Const(5))  # Incorrect
+f (generic function with 1 method)
Marking the argument for tmp as Const (aka non-differentiable) means that Enzyme believes that all variables loaded from or stored into tmp must also be non-differentiable, since all values inside a non-differentiable variable must also by definition be non-differentiable.
Enzyme.autodiff(Reverse, f, Active(1.2), Const(Vector{Float64}(undef, 1)), Const(1), Const(5))  # Incorrect
 
 # output
 
-((0.0, nothing, nothing, nothing),)
Enzyme.autodiff(Reverse, f, Active(1.2), Duplicated(Vector{Float64}(undef, 1), Vector{Float64}(undef, 1)), Const(1), Const(5))  # Correct (returns 10.367999999999999 == 1.2^4 * 5)
+((0.0, nothing, nothing, nothing),)
Passing in a dupliacted (e.g. differentiable) variable for tmp now leads to the correct answer.
Enzyme.autodiff(Reverse, f, Active(1.2), Duplicated(Vector{Float64}(undef, 1), Vector{Float64}(undef, 1)), Const(1), Const(5))  # Correct (returns 10.367999999999999 == 1.2^4 * 5)
 
 # output
 
-((10.367999999999999, nothing, nothing, nothing),)
CUDA.jl support
CUDA.jl is only supported on Julia v1.7.0 and onwards. On v1.6, attempting to differentiate CUDA kernel functions will not use device overloads correctly and thus returns fundamentally wrong results.
Sparse Arrays
At the moment there is limited support for sparse linear algebra operations. Sparse arrays may be used, but care must be taken because backing arrays drop zeros in Julia (unless told not to).
using SparseArrays
+((10.367999999999999, nothing, nothing, nothing),)
However, even if we ignore the semantic guarantee provided by marking tmp as constant, another issue arises. When computing the original function, intermediate computations (like in f above) can use tmp for temporary storage. When computing the derivative, Enzyme also needs additional temporary storage space for the corresponding derivative variables as well. If tmp is marked as Const, Enzyme does not have any temporary storage space for the derivatives!
Recent versions of Enzyme will attempt to error when they detect these latter types of situations, which we will refer to as activity unstable. This term is chosen to mirror the Julia notion of type-unstable code (e.g. where a type is not known at compile time). If an expression is activity unstable, it could either be constant, or active, depending on data not known at compile time. For example, consider the following:
function g(cond, active_var, constant_var)
+  if cond
+    return active_var
+  else
+    return constant_var
+end
+
+Enzyme.autodiff(Forward, g, Const(condition), Duplicated(x, dx), Const(y))
The returned value here could either by constant or duplicated, depending on the runtime-defined value of cond. If cond is true, Enzyme simply returns the shadow of active_var as the derivative. However, if cond is false, there is no derivative shadow for constant_var and Enzyme will throw a "Mismatched activity" error. For some simple types, e.g. a float Enzyme can circumvent this issue, for example by returning the float 0. Similarly, for some types like the Symbol type, which are never differentiable, such a shadow value will never be used, and Enzyme can return the original "primal" value as its derivative.  However, for arbitrary data structures, Enzyme presently has no generic mechanism to resolve this.
For example consider a third function:
function h(cond, active_var, constant_var)
+  return [g(cond, active_var, constant_var), g(cond, active_var, constant_var)]
+end
+
+Enzyme.autodiff(Forward, h, Const(condition), Duplicated(x, dx), Const(y))
Enzyme provides a nice utility Enzyme.make_zero which takes a data structure and constructs a deepcopy of the data structure with all of the floats set to zero and non-differentiable types like Symbols set to their primal value. If Enzyme gets into such a "Mismatched activity" situation where it needs to return a differentiable data structure from a constant variable, it could try to resolve this situation by constructing a new shadow data structure, such as with Enzyme.make_zero. However, this still can lead to incorrect results. In the case of h above, suppose that active_var and consant_var are both arrays, which are mutable (aka in-place) data types. This means that the return of h is going to either be result = [active_var, active_var] or result = [constant_var, constant_var].  Thus an update to result[1][1] would also change result[2][1] since result[1] and result[2] are the same array. 
If one created a new zero'd copy of each return from g, this would mean that the derivative dresult would have one copy made for the first element, and a second copy made for the second element. This could lead to incorrect results, and is unfortunately not a general resolution. However, for non-mutable variables (e.g. like floats) or non-differrentiable types (e.g. like Symbols) this problem can never arise.
Instead, Enzyme has a special mode known as "Runtime Activity" which can handle these types of situations. It can come with a minor performance reduction, and is therefore off by default. It can be enabled with Enzyme.API.runtimeActivity!(true) right after importing Enzyme for the first time. 
The way Enzyme's runtime activity resolves this issue is to return the original primal variable as the derivative whenever it needs to denote the fact that a variable is a constant. As this issue can only arise with mutable variables, they must be represented in memory via a pointer. All addtional loads and stores will now be modified to first check if the primal pointer is the same as the shadow pointer, and if so, treat it as a constant. Note that this check is not saying that the same arrays contain the same values, but rather the same backing memory represents both the primal and the shadow (e.g. a === b or equivalently pointer(a) == pointer(b)). 
Enabling runtime activity does therefore, come with a sharp edge, which is that if the computed derivative of a function is mutable, one must also check to see if the primal and shadow represent the same pointer, and if so the true derivative of the function is actually zero.
Generally, the preferred solution to these type of activity unstable codes should be to make your variables all activity-stable (e.g. always containing differentiable memory or always containing non-differentiable memory). However, with care, Enzyme does support "Runtime Activity" as a way to differentiate these programs without having to modify your code.
Mixed Activity
Sometimes in Reverse mode (but not forward mode), you may see an error Type T has mixed internal activity types for some type. This error arises when a variable in a computation cannot be fully represented as either a Duplicated or Active variable.
Active variables are used for immutable variables (like Float64), whereas Duplicated variables are used for mutable variables (like Vector{Float64}). Speciically, since Active variables are immutable, functions with Active inputs will return the adjoint of that variable. In contrast Duplicated variables will have their derivatives +='d in place.
This error indicates that you have a type, like Tuple{Float, Vector{Float64}} that has immutable components and mutable components. Therefore neither Active nor Duplicated can be used for this type.
Internally, by virtue of working at the LLVM level, most Julia types are represented as pointers, and this issue does not tend to arise within code fully differentiated by Enzyme internally. However, when a program needs to interact with Julia API's (e.g. as arguments to a custom rule, a type unstable call, or the outermost function being differentiated), Enzyme must adhere to Julia's notion of immutability and will throw this error rather than risk an incorrect result.
For example, consider the following code, which has a type unstable call to myfirst, passing in a mixed type Tuple{Float64, Vector{Float64}}.
@noinline function myfirst(tup::T) where T
+    return tup[1]
+end
+
+function f(x::Float64)
+    vec = [x]
+    tup = (x, vec)
+    Base.inferencebarrier(myfirst)(tup)::Float64
+end
+
+Enzyme.autodiff(Reverse, f, Active, Active(3.1))
When this situation arises, it is often easiest to resolve it by adding a level of indirection to ensure the entire variable is mutable. For example, one could enclose this variable in a reference, such as Ref{Tuple{Float, Vector{Float64}}}, like as follows.
@noinline function myfirst_ref(tup_ref::T) where T
+    tup = tup_ref[]
+    return tup[1]
+end
+
+function f2(x::Float64)
+    vec = [x]
+    tup = (x, vec)
+    tup_ref = Ref(tup)
+    Base.inferencebarrier(myfirst_ref)(tup_ref)::Float64
+end
+
+Enzyme.autodiff(Reverse, f2, Active, Active(3.1))
CUDA.jl support
CUDA.jl is only supported on Julia v1.7.0 and onwards. On v1.6, attempting to differentiate CUDA kernel functions will not use device overloads correctly and thus returns fundamentally wrong results.
Sparse Arrays
At the moment there is limited support for sparse linear algebra operations. Sparse arrays may be used, but care must be taken because backing arrays drop zeros in Julia (unless told not to).
using SparseArrays
 a = sparse([2.0])
 da1 = sparse([0.0]) # Incorrect: SparseMatrixCSC drops explicit zeros
 Enzyme.autodiff(Reverse, sum, Active, Duplicated(a, da1))
@@ -131,4 +164,79 @@
 # output
 
 1-element SparseVector{Float64, Int64} with 1 stored entry:
-  [1]  =  1.0
Sometimes, determining how to perform this zeroing can be complicated. That is why Enzyme provides a helper function Enzyme.make_zero that does this automatically.

+  [1]  =  1.0

Sometimes, determining how to perform this zeroing can be complicated. That is why Enzyme provides a helper function Enzyme.make_zero that does this automatically.

Complex Numbers

Differentiation of a function which returns a complex number is ambiguous, because there are several different gradients which may be desired. Rather than assume a specific of these conventions and potentially result in user error when the resulting derivative is not the desired one, Enzyme forces users to specify the desired convention by returning a real number instead.

Consider the function f(z) = z*z. If we were to differentiate this and have real inputs and outputs, the derivative f'(z) would be unambiguously 2*z. However, consider breaking down a complex number down into real and imaginary parts. Suppose now we were to call f with the explicit real and imaginary components, z = x + i y. This means that f is a function that takes an input of two values and returns two values f(x, y) = u(x, y) + i v(x, y). In the case of z*z this means that u(x,y) = x*x-y*y and v(x,y) = 2*x*y.

If we were to look at all first-order derivatives in total, we would end up with a 2x2 matrix (i.e. Jacobian), the derivative of each output wrt each input. Let's try to compute this, first by hand, then with Enzyme.

grad u(x, y) = [d/dx u, d/dy u] = [d/dx x*x-y*y, d/dy x*x-y*y] = [2*x, -2*y];
+grad v(x, y) = [d/dx v, d/dy v] = [d/dx 2*x*y, d/dy 2*x*y] = [2*y, 2*x];

Reverse mode differentiation computes the derivative of all inputs with respect to a single output by propagating the derivative of the return to its inputs. Here, we can explicitly differentiate with respect to the real and imaginary results, respectively, to find this matrix.

f(z) = z * z
+
+# a fixed input to use for testing
+z = 3.1 + 2.7im
+
+grad_u = Enzyme.autodiff(Reverse, z->real(f(z)), Active, Active(z))[1][1]
+grad_v = Enzyme.autodiff(Reverse, z->imag(f(z)), Active, Active(z))[1][1]
+
+(grad_u, grad_v)
+# output
+(6.2 - 5.4im, 5.4 + 6.2im)

This is somewhat inefficient, since we need to call the forward pass twice, once for the real part, once for the imaginary. We can solve this using batched derivatives in Enzyme, which computes several derivatives for the same function all in one go. To make it work, we're going to need to use split mode, which allows us to provide a custom derivative return value.

fwd, rev = Enzyme.autodiff_thunk(ReverseSplitNoPrimal, Const{typeof(f)}, Active, Active{ComplexF64})
+
+# Compute the reverse pass seeded with a differntial return of 1.0 + 0.0im
+grad_u = rev(Const(f), Active(z), 1.0 + 0.0im, fwd(Const(f), Active(z))[1])[1][1]
+# Compute the reverse pass seeded with a differntial return of 0.0 + 1.0im
+grad_v = rev(Const(f), Active(z), 0.0 + 1.0im, fwd(Const(f), Active(z))[1])[1][1]
+
+(grad_u, grad_v)
+
+# output
+(6.2 - 5.4im, 5.4 + 6.2im)

Now let's make this batched

fwd, rev = Enzyme.autodiff_thunk(ReverseSplitWidth(ReverseSplitNoPrimal, Val(2)), Const{typeof(f)}, Active, Active{ComplexF64})
+
+# Compute the reverse pass seeded with a differential return of 1.0 + 0.0im and 0.0 + 1.0im in one go!
+rev(Const(f), Active(z), (1.0 + 0.0im, 0.0 + 1.0im), fwd(Const(f), Active(z))[1])[1][1]
+
+# output
+(6.2 - 5.4im, 5.4 + 6.2im)

In contrast, Forward mode differentiation computes the derivative of all outputs with respect to a single input by providing a differential input. Thus we need to seed the shadow input with either 1.0 or 1.0im, respectively. This will compute the transpose of the matrix we found earlier.

d/dx f(x, y) = d/dx [u(x,y), v(x,y)] = d/dx [x*x-y*y, 2*x*y] = [ 2*x, 2*y];
+d/dy f(x, y) = d/dy [u(x,y), v(x,y)] = d/dy [x*x-y*y, 2*x*y] = [-2*y, 2*x];
d_dx = Enzyme.autodiff(Forward, f, Duplicated(z, 1.0+0.0im))[1]
+d_dy = Enzyme.autodiff(Forward, f, Duplicated(z, 0.0+1.0im))[1]
+
+(d_dx, d_dy)
+
+# output
+(6.2 + 5.4im, -5.4 + 6.2im)

Again, we can go ahead and batch this.

Enzyme.autodiff(Forward, f, BatchDuplicated(z, (1.0+0.0im, 0.0+1.0im)))[1]
+
+# output
+(var"1" = 6.2 + 5.4im, var"2" = -5.4 + 6.2im)

Taking Jacobians with respect to the real and imaginary results is fine, but for a complex scalar function it would be really nice to have a single complex derivative. More concretely, in this case when differentiating z*z, it would be nice to simply return 2*z. However, there are four independent variables in the 2x2 jacobian, but only two in a complex number.

Complex differentiation is often viewed in the lens of directional derivatives. For example, what is the derivative of the function as the real input increases, or as the imaginary input increases. Consider the derivative along the real axis, $\texttt{lim}_{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}$. This simplifies to $\texttt{lim}_{\Delta x \rightarrow 0} \frac{u(x+\Delta x, y)-u(x, y) + i \left[ v(x+\Delta x, y)-v(x, y)\right]}{\Delta x} = \frac{\partial}{\partial x} u(x,y) + i\frac{\partial}{\partial x} v(x,y)$. This is exactly what we computed by seeding forward mode with a shadow of 1.0 + 0.0im.

For completeness, we can also consider the derivative along the imaginary axis $\texttt{lim}_{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{i\Delta y}$. Here this simplifies to $\texttt{lim}_{u(x, y+\Delta y)-u(x, y) + i \left[ v(x, y+\Delta y)-v(x, y)\right]}{i\Delta y} = -i\frac{\partial}{\partial y} u(x,y) + \frac{\partial}{\partial y} v(x,y)$. Except for the $i$ in the denominator of the limit, this is the same as the result of Forward mode, when seeding x with a shadow of 0.0 + 1.0im. We can thus compute the derivative along the real axis by multiplying our second Forward mode call by -im.

d_real = Enzyme.autodiff(Forward, f, Duplicated(z, 1.0+0.0im))[1]
+d_im   = -im * Enzyme.autodiff(Forward, f, Duplicated(z, 0.0+1.0im))[1]
+
+(d_real, d_im)
+
+# output
+(6.2 + 5.4im, 6.2 + 5.4im)

Interestingly, the derivative of z*z is the same when computed in either axis. That is because this function is part of a special class of functions that are invariant to the input direction, called holomorphic.

Thus, for holomorphic functions, we can simply seed Forward-mode AD with a shadow of one for whatever input we are differenitating. This is nice since seeding the shadow with an input of one is exactly what we'd do for real-valued funtions as well.

Reverse-mode AD, however, is more tricky. This is because holomorphic functions are invariant to the direction of differentiation (aka the derivative inputs), not the direction of the differential return.

However, if a function is holomorphic, the two derivative functions we computed above must be the same. As a result, $\frac{\partial}{\partial x} u = \frac{\partial}{\partial y} v$ and $\frac{\partial}{\partial y} u = -\frac{\partial}{\partial x} v$.

We saw earlier, that performing reverse-mode AD with a return seed of 1.0 + 0.0im yielded [d/dx u, d/dy u]. Thus, for a holomorphic function, a real-seeded Reverse-mode AD computes [d/dx u, -d/dx v], which is the complex conjugate of the derivative.

conj(grad_u)
+
+# output
+
+6.2 + 5.4im

In the case of a scalar-input scalar-output function, that's sufficient. However, most of the time one uses reverse mode, it involves either several inputs or outputs, perhaps via memory. This case requires additional handling to properly sum all the partial derivatives from the use of each input and apply the conjugate operator at only the ones relevant to the differential return.

For simplicity, Enzyme provides a helper utlity ReverseHolomorphic which performs Reverse mode properly here, assuming that the function is indeed holomorphic and thus has a well-defined single derivative.

Enzyme.autodiff(ReverseHolomorphic, f, Active, Active(z))[1][1]
+
+# output
+
+6.2 + 5.4im

For even non-holomorphic functions, complex analysis allows us to define $\frac{\partial}{\partial z} = \frac{1}{2}\left(\frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right)$. For non-holomorphic functions, this allows us to compute d/dz. Let's consider myabs2(z) = z * conj(z). We can compute the derivative wrt z of this in Forward mode as follows, which as one would expect results in a result of conj(z):

myabs2(z) = z * conj(z)
+
+dabs2_dx, dabs2_dy = Enzyme.autodiff(Forward, myabs2, BatchDuplicated(z, (1.0 + 0.0im, 0.0 + 1.0im)))[1]
+(dabs2_dx - im * dabs2_dy) / 2
+
+# output
+
+3.1 - 2.7im

Similarly, we can compute d/d conj(z) = d/dx + i d/dy.

(dabs2_dx + im * dabs2_dy) / 2
+
+# output
+
+3.1 + 2.7im

Computing this in Reverse mode is more tricky. Let's expand f in terms of u and v. $\frac{\partial}{\partial z} f = \frac12 \left( [u_x + i v_x] - i [u_y + i v_y] \right) = \frac12 \left( [u_x + v_y] + i [v_x - u_y] \right)$. Thus d/dz = (conj(grad_u) + im * conj(grad_v))/2.

abs2_fwd, abs2_rev = Enzyme.autodiff_thunk(ReverseSplitWidth(ReverseSplitNoPrimal, Val(2)), Const{typeof(myabs2)}, Active, Active{ComplexF64})
+
+# Compute the reverse pass seeded with a differential return of 1.0 + 0.0im and 0.0 + 1.0im in one go!
+gradabs2_u, gradabs2_v = abs2_rev(Const(myabs2), Active(z), (1.0 + 0.0im, 0.0 + 1.0im), abs2_fwd(Const(myabs2), Active(z))[1])[1][1]
+
+(conj(gradabs2_u) + im * conj(gradabs2_v)) / 2
+
+# output
+
+3.1 - 2.7im

For d/d conj(z), $\frac12 \left( [u_x + i v_x] + i [u_y + i v_y] \right) = \frac12 \left( [u_x - v_y] + i [v_x + u_y] \right)$. Thus d/d conj(z) = (grad_u + im * grad_v)/2.

(gradabs2_u + im * gradabs2_v) / 2
+
+# output
+
+3.1 + 2.7im

Note: when writing rules for complex scalar functions, in reverse mode one needs to conjugate the differential return, and similarly the true result will be the conjugate of that value (in essence you can think of reverse-mode AD as working in the conjugate space).

diff --git a/previews/PR1300/internal_api/index.html b/previews/PR1300/internal_api/index.html index 786e344820..3bdfb337b9 100644 --- a/previews/PR1300/internal_api/index.html +++ b/previews/PR1300/internal_api/index.html @@ -1,2 +1,2 @@ -Internal API · Enzyme.jl
Edit on GitHub

Internal API

Note

This is the documentation of Enzymes's internal API. The internal API is not subject to semantic versioning and may change at any time and without deprecation.

+Internal API · Enzyme.jl
Edit on GitHub

Internal API

Note

This is the documentation of Enzymes's internal API. The internal API is not subject to semantic versioning and may change at any time and without deprecation.

diff --git a/previews/PR1300/pullbacks/index.html b/previews/PR1300/pullbacks/index.html index 6cc4c62dc1..97483f424d 100644 --- a/previews/PR1300/pullbacks/index.html +++ b/previews/PR1300/pullbacks/index.html @@ -1,5 +1,5 @@ -Implementing pullbacks · Enzyme.jl
Edit on GitHub

Implementing pullbacks

Enzyme's autodiff function can only handle functions with scalar output. To implement pullbacks (back-propagation of gradients/tangents) for array-valued functions, use a mutating function that returns nothing and stores it's result in one of the arguments, which must be passed wrapped in a Duplicated.

Example

Given a function mymul! that performs the equivalent of R = A * B for matrices A and B, and given a gradient (tangent) ∂z_∂R, we can compute ∂z_∂A and ∂z_∂B like this:

using Enzyme, Random
+Implementing pullbacks · Enzyme.jl
Edit on GitHub
Implementing pullbacks
Enzyme's autodiff function can only handle functions with scalar output. To implement pullbacks (back-propagation of gradients/tangents) for array-valued functions, use a mutating function that returns nothing and stores it's result in one of the arguments, which must be passed wrapped in a Duplicated.
Example
Given a function mymul! that performs the equivalent of R = A * B for matrices A and B, and given a gradient (tangent) ∂z_∂R, we can compute ∂z_∂A and ∂z_∂B like this:
using Enzyme, Random
 
 function mymul!(R, A, B)
     @assert axes(A,2) == axes(B,1)
@@ -27,4 +27,4 @@
 
 Enzyme.autodiff(Reverse, mymul!, Const, Duplicated(R, ∂z_∂R), Duplicated(A, ∂z_∂A), Duplicated(B, ∂z_∂B))
((nothing, nothing, nothing),)
Now we have:
R ≈ A * B            &&
 ∂z_∂A ≈ ∂z_∂R0 * B'   &&  # equivalent to Zygote.pullback(*, A, B)[2](∂z_∂R)[1]
-∂z_∂B ≈ A' * ∂z_∂R0       # equivalent to Zygote.pullback(*, A, B)[2](∂z_∂R)[2]
true
Note that the result of the backpropagation is added to ∂z_∂A and ∂z_∂B, they act as accumulators for gradient information.

+∂z_∂B ≈ A' * ∂z_∂R0       # equivalent to Zygote.pullback(*, A, B)[2](∂z_∂R)[2]
true

Note that the result of the backpropagation is added to ∂z_∂A and ∂z_∂B, they act as accumulators for gradient information.

diff --git a/previews/PR1300/search/index.html b/previews/PR1300/search/index.html index b35d181b5c..47bced4a4a 100644 --- a/previews/PR1300/search/index.html +++ b/previews/PR1300/search/index.html @@ -1,2 +1,2 @@ -Search · Enzyme.jl

Loading search...

    +Search · Enzyme.jl

    Loading search...

      diff --git a/previews/PR1300/search_index.js b/previews/PR1300/search_index.js index 27e25236ff..3e5618c07a 100644 --- a/previews/PR1300/search_index.js +++ b/previews/PR1300/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"EditURL = \"../../../examples/autodiff.jl\"","category":"page"},{"location":"generated/autodiff/#AutoDiff-API","page":"AutoDiff API","title":"AutoDiff API","text":"","category":"section"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"The goal of this tutorial is to give users already familiar with automatic differentiation (AD) an overview of the Enzyme differentiation API for the following differentiation modes","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"Reverse mode\nForward mode\nForward over reverse mode\nVector Forward over reverse mode","category":"page"},{"location":"generated/autodiff/#Defining-a-function","page":"AutoDiff API","title":"Defining a function","text":"","category":"section"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"Enzyme differentiates arbitrary multivariate vector functions as the most general case in automatic differentiation","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"f mathbbR^n rightarrow mathbbR^m y = f(x)","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"For simplicity we define a vector function with m=1. However, this tutorial can easily be applied to arbitrary m in mathbbN.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"using Enzyme\n\nfunction f(x::Array{Float64}, y::Array{Float64})\n y[1] = x[1] * x[1] + x[2] * x[1]\n return nothing\nend;\nnothing #hide","category":"page"},{"location":"generated/autodiff/#Reverse-mode","page":"AutoDiff API","title":"Reverse mode","text":"","category":"section"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"The reverse model in AD is defined as","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"beginaligned\ny = f(x) \nbarx = bary cdot nabla f(x)\nendaligned","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"bar denotes an adjoint variable. Note that executing an AD in reverse mode computes both y and the adjoint barx.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"x = [2.0, 2.0]\nbx = [0.0, 0.0]\ny = [0.0]\nby = [1.0];\nnothing #hide","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"Enzyme stores the value and adjoint of a variable in an object of type Duplicated where the first element represent the value and the second the adjoint. Evaluating the reverse model using Enzyme is done via the following call.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"Enzyme.autodiff(Reverse, f, Duplicated(x, bx), Duplicated(y, by));\nnothing #hide","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"This yields the gradient of f in bx at point x = [2.0, 2.0]. by is called the seed and has to be set to 10 in order to compute the gradient. Let's save the gradient for later.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"g = copy(bx)","category":"page"},{"location":"generated/autodiff/#Forward-mode","page":"AutoDiff API","title":"Forward mode","text":"","category":"section"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"The forward model in AD is defined as","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"beginaligned\ny = f(x) \ndoty = nabla f(x) cdot dotx\nendaligned","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"To obtain the first element of the gradient using the forward model we have to seed dotx with dotx = 1000","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"x = [2.0, 2.0]\ndx = [1.0, 0.0]\ny = [0.0]\ndy = [0.0];\nnothing #hide","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"In the forward mode the second element of Duplicated stores the tangent.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"Enzyme.autodiff(Forward, f, Duplicated(x, dx), Duplicated(y, dy));\nnothing #hide","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"We can now verify that indeed the reverse mode and forward mode yield the same result for the first component of the gradient. Note that to acquire the full gradient one needs to execute the forward model a second time with the seed dx set to [0.0,1.0].","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"Let's verify whether the reverse and forward model agree.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"g[1] == dy[1]","category":"page"},{"location":"generated/autodiff/#Forward-over-reverse","page":"AutoDiff API","title":"Forward over reverse","text":"","category":"section"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"The forward over reverse (FoR) model is obtained by applying the forward model to the reverse model using the chain rule for the product in the adjoint statement.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"beginaligned\ny = f(x) \ndoty = nabla f(x) cdot dotx \nbarx = bary cdot nabla f(x) \ndotbarx = bary cdot nabla^2 f(x) cdot dotx + dotbary cdot nabla f(x)\nendaligned","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"To obtain the first column/row of the Hessian nabla^2 f(x) we have to seed dotbary with 00, bary with 10 and dotx with 10 00.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"y = [0.0]\nx = [2.0, 2.0]\n\ndy = [0.0]\ndx = [1.0, 0.0]\n\nbx = [0.0, 0.0]\nby = [1.0]\ndbx = [0.0, 0.0]\ndby = [0.0]\n\nEnzyme.autodiff(\n Forward,\n (x,y) -> Enzyme.autodiff_deferred(Reverse, f, x, y),\n Duplicated(Duplicated(x, bx), Duplicated(dx, dbx)),\n Duplicated(Duplicated(y, by), Duplicated(dy, dby)),\n)","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"The FoR model also computes the forward model from before, giving us again the first component of the gradient.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"g[1] == dy[1]","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"In addition we now have the first row/column of the Hessian.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"dbx[1] == 2.0\ndbx[2] == 1.0","category":"page"},{"location":"generated/autodiff/#Vector-forward-over-reverse","page":"AutoDiff API","title":"Vector forward over reverse","text":"","category":"section"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"The vector FoR allows us to propagate several tangents at once through the second-order model by computing the derivative of the gradient at multiple points at once. We begin by defining a helper function for the gradient. Since we will not need the original results (stored in y), we can mark it DuplicatedNoNeed. Specifically, this will perform the following:","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"beginaligned\nbarx = barx + bary cdot nabla f(x) \nbary = 0\nendaligned","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"function grad(x, dx, y, dy)\n Enzyme.autodiff_deferred(Reverse, f, Duplicated(x, dx), DuplicatedNoNeed(y, dy))\n nothing\nend","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"To compute the conventional gradient, we would call this function with our given inputs, dy = [1.0], and dx = [0.0, 0.0]. Since y is not needed, we can just set it to an undef vector.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"x = [2.0, 2.0]\ny = Vector{Float64}(undef, 1)\ndx = [0.0, 0.0]\ndy = [1.0]\n\ngrad(x, dx, y, dy)","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"dx now contains the gradient","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"@show dx","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"To compute the hessian, we need to take the dervative of this gradient function at every input. Following the same seeding strategy as before, we now seed both in the vx[1]=[1.0, 0.0] and vx[2]=[0.0, 1.0] direction. These tuples have to be put into a BatchDuplicated type. We then compute the forward mode derivative at all these points.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"vx = ([1.0, 0.0], [0.0, 1.0])\nhess = ([0.0, 0.0], [0.0, 0.0])\ndx = [0.0, 0.0]\ndy = [1.0]\n\nEnzyme.autodiff(Enzyme.Forward, grad,\n Enzyme.BatchDuplicated(x, vx),\n Enzyme.BatchDuplicated(dx, hess),\n Const(y),\n Const(dy))","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"Again we obtain the first-order gradient. 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The optional keyword argument shadow is a vector of one-hot vectors of type x which are used to forward-propagate into the return. For performance reasons, this should be computed once, outside the call to gradient, rather than within this call.\n\nExample:\n\nf(x) = x[1]*x[2]\n\ngrad = gradient(Forward, f, [2.0, 3.0])\n\n# output\n\n(3.0, 2.0)\n\n\n\n\n\n","category":"method"},{"location":"api/#Enzyme.gradient-Tuple{ReverseMode, Any, Any}","page":"API","title":"Enzyme.gradient","text":"gradient(::ReverseMode, f, x)\n\nCompute the gradient of an array-input function f using reverse mode. This will allocate and return new array with the gradient result.\n\nExample:\n\nf(x) = x[1]*x[2]\n\ngrad = gradient(Reverse, f, [2.0, 3.0])\n\n# output\n\n2-element Vector{Float64}:\n 3.0\n 2.0\n\n\n\n\n\n","category":"method"},{"location":"api/#Enzyme.gradient-Union{Tuple{chunk}, Tuple{X}, Tuple{F}, Tuple{ForwardMode, F, X, Val{chunk}}} where {F, X, chunk}","page":"API","title":"Enzyme.gradient","text":"gradient(::ForwardMode, f, x, ::Val{chunk}; shadow=onehot(x))\n\nCompute the gradient of an array-input function f using vector forward mode. Like gradient, except it uses a chunk size of chunk to compute chunk derivatives in a single call.\n\nExample:\n\nf(x) = x[1]*x[2]\n\ngrad = gradient(Forward, f, [2.0, 3.0], Val(2))\n\n# output\n\n(3.0, 2.0)\n\n\n\n\n\n","category":"method"},{"location":"api/#Enzyme.jacobian-Tuple{ForwardMode, Any, Any}","page":"API","title":"Enzyme.jacobian","text":"jacobian(::ForwardMode, f, x; shadow=onehot(x))\njacobian(::ForwardMode, f, x, ::Val{chunk}; shadow=onehot(x))\n\nCompute the jacobian of an array-input function f using (potentially vector) forward mode. This is a simple rename of the gradient function, and all relevant arguments apply here.\n\nExample:\n\nf(x) = [x[1]*x[2], x[2]]\n\ngrad = jacobian(Forward, f, [2.0, 3.0])\n\n# output\n\n2×2 Matrix{Float64}:\n 3.0 2.0\n 0.0 1.0\n\n\n\n\n\n","category":"method"},{"location":"api/#Enzyme.jacobian-Union{Tuple{RABI}, Tuple{ReturnPrimal}, Tuple{n_out_val}, Tuple{chunk}, Tuple{X}, Tuple{F}, Tuple{ReverseMode{ReturnPrimal, RABI}, F, X, Val{n_out_val}, Val{chunk}}} where {F, X, chunk, n_out_val, ReturnPrimal, RABI<:EnzymeCore.ABI}","page":"API","title":"Enzyme.jacobian","text":"jacobian(::ReverseMode, f, x, ::Val{num_outs}, ::Val{chunk})\n\nCompute the jacobian of an array-input function f using (potentially vector) reverse mode. The chunk argument denotes the chunk size to use and num_outs denotes the number of outputs f will return in an array.\n\nExample:\n\nf(x) = [x[1]*x[2], x[2]]\n\ngrad = jacobian(Reverse, f, [2.0, 3.0], Val(2))\n\n# output\n\n2×2 Matrix{Float64}:\n 3.0 2.0\n 0.0 1.0\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff-Union{Tuple{A}, Tuple{FA}, Tuple{RABI}, Tuple{ForwardMode{RABI}, FA, Type{A}, Vararg{Any}}} where {RABI<:EnzymeCore.ABI, FA<:Annotation, A<:Annotation}","page":"API","title":"EnzymeCore.autodiff","text":"autodiff(::ForwardMode, f, Activity, args...)\n\nAuto-differentiate function f at arguments args using forward mode.\n\nargs may be numbers, arrays, structs of numbers, structs of arrays and so on. Enzyme will only differentiate in respect to arguments that are wrapped in a Duplicated or similar argument. Non-annotated arguments will automatically be treated as Const. Unlike reverse mode in autodiff, Active arguments are not allowed here, since all derivative results of immutable objects will be returned and should instead use Duplicated or variants like DuplicatedNoNeed.\n\nActivity is the Activity of the return value, it may be:\n\nConst if the return is not to be differentiated with respect to\nDuplicated, if the return is being differentiated with respect to and both the original value and the derivative return are desired\nDuplicatedNoNeed, if the return is being differentiated with respect to and only the derivative return is desired.\nBatchDuplicated, like Duplicated, but computing multiple derivatives at once. All batch sizes must be the same for all arguments.\nBatchDuplicatedNoNeed, like DuplicatedNoNeed, but computing multiple derivatives at one. All batch sizes must be the same for all arguments.\n\nExample returning both original return and derivative:\n\nf(x) = x*x\nres, ∂f_∂x = autodiff(Forward, f, Duplicated, Duplicated(3.14, 1.0))\n\n# output\n\n(9.8596, 6.28)\n\nExample returning just the derivative:\n\nf(x) = x*x\n∂f_∂x = autodiff(Forward, f, DuplicatedNoNeed, Duplicated(3.14, 1.0))\n\n# output\n\n(6.28,)\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff-Union{Tuple{CMode}, Tuple{FA}, Tuple{CMode, FA, Vararg{Any}}} where {FA<:Annotation, CMode<:EnzymeCore.Mode}","page":"API","title":"EnzymeCore.autodiff","text":"autodiff(mode::Mode, f, args...)\n\nLike autodiff but will try to guess the activity of the return value.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff-Union{Tuple{CMode}, Tuple{F}, Tuple{CMode, F, Vararg{Any}}} where {F, CMode<:EnzymeCore.Mode}","page":"API","title":"EnzymeCore.autodiff","text":"autodiff(mode::Mode, f, ::Type{A}, args...)\n\nLike autodiff but will try to extend f to an annotation, if needed.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff-Union{Tuple{RABI}, Tuple{ReturnPrimal}, Tuple{A}, Tuple{FA}, Tuple{ReverseMode{ReturnPrimal, RABI}, FA, Type{A}, Vararg{Any}}} where {FA<:Annotation, A<:Annotation, ReturnPrimal, RABI<:EnzymeCore.ABI}","page":"API","title":"EnzymeCore.autodiff","text":"autodiff(::ReverseMode, f, Activity, args...)\n\nAuto-differentiate function f at arguments args using reverse mode.\n\nLimitations:\n\nf may only return a Real (of a built-in/primitive type) or nothing, not an array, struct, BigFloat, etc. To handle vector-valued return types, use a mutating f! that returns nothing and stores it's return value in one of the arguments, which must be wrapped in a Duplicated.\n\nargs may be numbers, arrays, structs of numbers, structs of arrays and so on. Enzyme will only differentiate in respect to arguments that are wrapped in an Active (for arguments whose derivative result must be returned rather than mutated in place, such as primitive types and structs thereof) or Duplicated (for mutable arguments like arrays, Refs and structs thereof). Non-annotated arguments will automatically be treated as Const.\n\nActivity is the Activity of the return value, it may be Const or Active.\n\nExample:\n\na = 4.2\nb = [2.2, 3.3]; ∂f_∂b = zero(b)\nc = 55; d = 9\n\nf(a, b, c, d) = a * √(b[1]^2 + b[2]^2) + c^2 * d^2\n∂f_∂a, _, _, ∂f_∂d = autodiff(Reverse, f, Active, Active(a), Duplicated(b, ∂f_∂b), c, Active(d))[1]\n\n# output\n\n(3.966106403010388, nothing, nothing, 54450.0)\n\nhere, autodiff returns a tuple (partial fpartial a partial fpartial d), while partial fpartial b will be added to ∂f_∂b (but not returned). c will be treated as Const(c).\n\nOne can also request the original returned value of the computation.\n\nExample:\n\nEnzyme.autodiff(ReverseWithPrimal, x->x*x, Active(3.0))\n\n# output\n\n((6.0,), 9.0)\n\nnote: Note\nEnzyme gradients with respect to integer values are zero. Active will automatically convert plain integers to floating point values, but cannot do so for integer values in tuples and structs.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff_deferred-Union{Tuple{A}, Tuple{FA}, Tuple{ForwardMode, FA, Type{A}, Vararg{Any}}} where {FA<:Annotation, A<:Annotation}","page":"API","title":"EnzymeCore.autodiff_deferred","text":"autodiff_deferred(::ForwardMode, f, Activity, args...)\n\nSame as autodiff(::ForwardMode, ...) but uses deferred compilation to support usage in GPU code, as well as high-order differentiation.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff_deferred-Union{Tuple{CMode}, Tuple{F}, Tuple{CMode, F, Vararg{Any}}} where {F, CMode<:EnzymeCore.Mode}","page":"API","title":"EnzymeCore.autodiff_deferred","text":"autodiff_deferred(mode::Mode, f, ::Type{A}, args...)\n\nLike autodiff_deferred but will try to extend f to an annotation, if needed.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff_deferred-Union{Tuple{ReturnPrimal}, Tuple{A}, Tuple{FA}, Tuple{ReverseMode{ReturnPrimal}, FA, Type{A}, Vararg{Any}}} where {FA<:Annotation, A<:Annotation, ReturnPrimal}","page":"API","title":"EnzymeCore.autodiff_deferred","text":"autodiff_deferred(::ReverseMode, f, Activity, args...)\n\nSame as autodiff but uses deferred compilation to support usage in GPU code, as well as high-order differentiation.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff_deferred_thunk-Union{Tuple{RABI}, Tuple{ModifiedBetweenT}, Tuple{Width}, Tuple{ReturnShadow}, Tuple{ReturnPrimal}, Tuple{A}, Tuple{FA}, Tuple{EnzymeCore.ReverseModeSplit{ReturnPrimal, ReturnShadow, Width, ModifiedBetweenT, RABI}, Type{FA}, Type{A}, Vararg{Any}}} where {FA<:Annotation, A<:Annotation, ReturnPrimal, ReturnShadow, Width, ModifiedBetweenT, RABI<:EnzymeCore.ABI}","page":"API","title":"EnzymeCore.autodiff_deferred_thunk","text":"autodiff_deferred_thunk(::ReverseModeSplit, ftype, Activity, argtypes...)\n\nProvide the split forward and reverse pass functions for annotated function type ftype when called with args of type argtypes when using reverse mode.\n\nActivity is the Activity of the return value, it may be Const, Active, or Duplicated (or its variants DuplicatedNoNeed, BatchDuplicated, and BatchDuplicatedNoNeed).\n\nThe forward function will return a tape, the primal (or nothing if not requested), and the shadow (or nothing if not a Duplicated variant), and tapes the corresponding type arguements provided.\n\nThe reverse function will return the derivative of Active arguments, updating the Duplicated arguments in place. The same arguments to the forward pass should be provided, followed by the adjoint of the return (if the return is active), and finally the tape from the forward pass.\n\nExample:\n\n\nA = [2.2]; ∂A = zero(A)\nv = 3.3\n\nfunction f(A, v)\n res = A[1] * v\n A[1] = 0\n res\nend\n\nforward, reverse = autodiff_deferred_thunk(ReverseSplitWithPrimal, Const{typeof(f)}, Active, Duplicated{typeof(A)}, Active{typeof(v)})\n\ntape, result, shadow_result = forward(Const(f), Duplicated(A, ∂A), Active(v))\n_, ∂v = reverse(Const(f), Duplicated(A, ∂A), Active(v), 1.0, tape)[1]\n\nresult, ∂v, ∂A \n\n# output\n\n(7.26, 2.2, [3.3])\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff_thunk-Union{Tuple{RABI}, Tuple{A}, Tuple{FA}, Tuple{ForwardMode{RABI}, Type{FA}, Type{A}, Vararg{Any}}} where {FA<:Annotation, A<:Annotation, RABI<:EnzymeCore.ABI}","page":"API","title":"EnzymeCore.autodiff_thunk","text":"autodiff_thunk(::ForwardMode, ftype, Activity, argtypes...)\n\nProvide the thunk forward mode function for annotated function type ftype when called with args of type argtypes.\n\nActivity is the Activity of the return value, it may be Const or Duplicated (or its variants DuplicatedNoNeed, BatchDuplicated, andBatchDuplicatedNoNeed).\n\nThe forward function will return the primal (if requested) and the shadow (or nothing if not a Duplicated variant).\n\nExample returning both original return and derivative:\n\na = 4.2\nb = [2.2, 3.3]; ∂f_∂b = zero(b)\nc = 55; d = 9\n\nf(x) = x*x\nforward = autodiff_thunk(Forward, Const{typeof(f)}, Duplicated, Duplicated{Float64})\nres, ∂f_∂x = forward(Const(f), Duplicated(3.14, 1.0))\n\n# output\n\n(9.8596, 6.28)\n\nExample returning just the derivative:\n\na = 4.2\nb = [2.2, 3.3]; ∂f_∂b = zero(b)\nc = 55; d = 9\n\nf(x) = x*x\nforward = autodiff_thunk(Forward, Const{typeof(f)}, DuplicatedNoNeed, Duplicated{Float64})\n∂f_∂x = forward(Const(f), Duplicated(3.14, 1.0))\n\n# output\n\n(6.28,)\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff_thunk-Union{Tuple{RABI}, Tuple{ModifiedBetweenT}, Tuple{Width}, Tuple{ReturnShadow}, Tuple{ReturnPrimal}, Tuple{A}, Tuple{FA}, Tuple{EnzymeCore.ReverseModeSplit{ReturnPrimal, ReturnShadow, Width, ModifiedBetweenT, RABI}, Type{FA}, Type{A}, Vararg{Any}}} where {FA<:Annotation, A<:Annotation, ReturnPrimal, ReturnShadow, Width, ModifiedBetweenT, RABI<:EnzymeCore.ABI}","page":"API","title":"EnzymeCore.autodiff_thunk","text":"autodiff_thunk(::ReverseModeSplit, ftype, Activity, argtypes...)\n\nProvide the split forward and reverse pass functions for annotated function type ftype when called with args of type argtypes when using reverse mode.\n\nActivity is the Activity of the return value, it may be Const, Active, or Duplicated (or its variants DuplicatedNoNeed, BatchDuplicated, and BatchDuplicatedNoNeed).\n\nThe forward function will return a tape, the primal (or nothing if not requested), and the shadow (or nothing if not a Duplicated variant), and tapes the corresponding type arguements provided.\n\nThe reverse function will return the derivative of Active arguments, updating the Duplicated arguments in place. The same arguments to the forward pass should be provided, followed by the adjoint of the return (if the return is active), and finally the tape from the forward pass.\n\nExample:\n\n\nA = [2.2]; ∂A = zero(A)\nv = 3.3\n\nfunction f(A, v)\n res = A[1] * v\n A[1] = 0\n res\nend\n\nforward, reverse = autodiff_thunk(ReverseSplitWithPrimal, Const{typeof(f)}, Active, Duplicated{typeof(A)}, Active{typeof(v)})\n\ntape, result, shadow_result = forward(Const(f), Duplicated(A, ∂A), Active(v))\n_, ∂v = reverse(Const(f), Duplicated(A, ∂A), Active(v), 1.0, tape)[1]\n\nresult, ∂v, ∂A \n\n# output\n\n(7.26, 2.2, [3.3])\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.ABI","page":"API","title":"EnzymeCore.ABI","text":"abstract type ABI\n\nAbstract type for what ABI will be used.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.Active","page":"API","title":"EnzymeCore.Active","text":"Active(x)\n\nMark a function argument x of autodiff as active, Enzyme will auto-differentiate in respect Active arguments.\n\nnote: Note\nEnzyme gradients with respect to integer values are zero. Active will automatically convert plain integers to floating point values, but cannot do so for integer values in tuples and structs.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.Annotation","page":"API","title":"EnzymeCore.Annotation","text":"abstract type Annotation{T}\n\nAbstract type for autodiff function argument wrappers like Const, Active and Duplicated.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.BatchDuplicated","page":"API","title":"EnzymeCore.BatchDuplicated","text":"BatchDuplicated(x, ∂f_∂xs)\n\nLike Duplicated, except contains several shadows to compute derivatives for all at once. Argument ∂f_∂xs should be a tuple of the several values of type x.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.BatchDuplicatedNoNeed","page":"API","title":"EnzymeCore.BatchDuplicatedNoNeed","text":"BatchDuplicatedNoNeed(x, ∂f_∂xs)\n\nLike DuplicatedNoNeed, except contains several shadows to compute derivatives for all at once. Argument ∂f_∂xs should be a tuple of the several values of type x.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.Const","page":"API","title":"EnzymeCore.Const","text":"Const(x)\n\nMark a function argument x of autodiff as constant, Enzyme will not auto-differentiate in respect Const arguments.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.Duplicated","page":"API","title":"EnzymeCore.Duplicated","text":"Duplicated(x, ∂f_∂x)\n\nMark a function argument x of autodiff as duplicated, Enzyme will auto-differentiate in respect to such arguments, with dx acting as an accumulator for gradients (so partial f partial x will be added to) ∂f_∂x.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.DuplicatedNoNeed","page":"API","title":"EnzymeCore.DuplicatedNoNeed","text":"DuplicatedNoNeed(x, ∂f_∂x)\n\nLike Duplicated, except also specifies that Enzyme may avoid computing the original result and only compute the derivative values.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.FFIABI","page":"API","title":"EnzymeCore.FFIABI","text":"struct FFIABI <: ABI\n\nForeign function call ABI. JIT the differentiated function, then inttoptr call the address.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.ForwardMode","page":"API","title":"EnzymeCore.ForwardMode","text":"struct Forward <: Mode\n\nForward mode differentiation\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.InlineABI","page":"API","title":"EnzymeCore.InlineABI","text":"struct InlineABI <: ABI\n\nInlining function call ABI. \n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.Mode","page":"API","title":"EnzymeCore.Mode","text":"abstract type Mode\n\nAbstract type for what differentiation mode will be used.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.ReverseMode","page":"API","title":"EnzymeCore.ReverseMode","text":"struct ReverseMode{ReturnPrimal,ABI} <: Mode{ABI}\n\nReverse mode differentiation.\n\nReturnPrimal: Should Enzyme return the primal return value from the augmented-forward.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.ReverseModeSplit","page":"API","title":"EnzymeCore.ReverseModeSplit","text":"struct ReverseModeSplit{ReturnPrimal,ReturnShadow,Width,ModifiedBetween,ABI} <: Mode{ABI}\n\nReverse mode differentiation.\n\nReturnPrimal: Should Enzyme return the primal return value from the augmented-forward.\nReturnShadow: Should Enzyme return the shadow return value from the augmented-forward.\nWidth: Batch Size (0 if to be automatically derived)\nModifiedBetween: Tuple of each argument's modified between state (true if to be automatically derived).\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.make_zero","page":"API","title":"EnzymeCore.make_zero","text":"make_zero(::Type{T}, seen::IdDict, prev::T, ::Val{copy_if_inactive}=Val(false))::T\n\nRecursively make a zero'd copy of the value `prev` of type `T`. The argument `copy_if_inactive` specifies\nwhat to do if the type `T` is guaranteed to be inactive, use the primal (the default) or still copy the value.\n\n\n\n\n\n","category":"function"},{"location":"api/#EnzymeCore.make_zero-Union{Tuple{copy_if_inactive}, Tuple{T}, Tuple{T, Val{copy_if_inactive}}} where {T, copy_if_inactive}","page":"API","title":"EnzymeCore.make_zero","text":"make_zero(prev::T)\n\nHelper function to recursively make zero.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.EnzymeRules.AugmentedReturn","page":"API","title":"EnzymeCore.EnzymeRules.AugmentedReturn","text":"AugmentedReturn(primal, shadow, tape)\n\nAugment the primal return value of a function with its shadow, as well as any additional information needed to correctly compute the reverse pass, stored in tape.\n\nUnless specified by the config that a variable is not overwritten, rules must assume any arrays/data structures/etc are overwritten between the forward and the reverse pass. Any floats or variables passed by value are always preserved as is (as are the arrays themselves, just not necessarily the values in the array).\n\nSee also augmented_primal.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.EnzymeRules.Config","page":"API","title":"EnzymeCore.EnzymeRules.Config","text":"Config{NeedsPrimal, NeedsShadow, Width, Overwritten}\nConfigWidth{Width} = Config{<:Any,<:Any, Width}\n\nConfiguration type to dispatch on in custom reverse rules (see augmented_primal and reverse).\n\nNeedsPrimal and NeedsShadow: boolean values specifying whether the primal and shadow (resp.) should be returned. \nWidth: an integer that specifies the number of adjoints/shadows simultaneously being propagated.\nOverwritten: a tuple of booleans of whether each argument (including the function itself) is modified between the forward and reverse pass (true if potentially modified between).\n\nGetters for the four type parameters are provided by needs_primal, needs_shadow, width, and overwritten.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.EnzymeRules.augmented_primal","page":"API","title":"EnzymeCore.EnzymeRules.augmented_primal","text":"augmented_primal(::Config, func::Annotation{typeof(f)}, RT::Type{<:Annotation}, args::Annotation...)\n\nMust return an AugmentedReturn type.\n\nThe primal must be the same type of the original return if needs_primal(config), otherwise nothing.\nThe shadow must be nothing if needs_shadow(config) is false. If width is 1, the shadow should be the same type of the original return. If the width is greater than 1, the shadow should be NTuple{original return, width}.\nThe tape can be any type (including Nothing) and is preserved for the reverse call.\n\n\n\n\n\n","category":"function"},{"location":"api/#EnzymeCore.EnzymeRules.forward","page":"API","title":"EnzymeCore.EnzymeRules.forward","text":"forward(func::Annotation{typeof(f)}, RT::Type{<:Annotation}, args::Annotation...)\n\nCalculate the forward derivative. The first argument func is the callable for which the rule applies to. Either wrapped in a Const), or a Duplicated if it is a closure. The second argument is the return type annotation, and all other arguments are the annotated function arguments.\n\n\n\n\n\n","category":"function"},{"location":"api/#EnzymeCore.EnzymeRules.inactive","page":"API","title":"EnzymeCore.EnzymeRules.inactive","text":"inactive(func::typeof(f), args...)\n\nMark a particular function as always being inactive in both its return result and the function call itself.\n\n\n\n\n\n","category":"function"},{"location":"api/#EnzymeCore.EnzymeRules.inactive_noinl","page":"API","title":"EnzymeCore.EnzymeRules.inactive_noinl","text":"inactive_noinl(func::typeof(f), args...)\n\nMark a particular function as always being inactive in both its return result and the function call itself, but do not prevent inlining of the function.\n\n\n\n\n\n","category":"function"},{"location":"api/#EnzymeCore.EnzymeRules.inactive_type-Tuple{Type}","page":"API","title":"EnzymeCore.EnzymeRules.inactive_type","text":"inactive_type(::Type{Ty})\n\nMark a particular type Ty as always being inactive.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.EnzymeRules.reverse","page":"API","title":"EnzymeCore.EnzymeRules.reverse","text":"reverse(::Config, func::Annotation{typeof(f)}, dret::Active, tape, args::Annotation...)\nreverse(::Config, func::Annotation{typeof(f)}, ::Type{<:Annotation), tape, args::Annotation...)\n\nTakes gradient of derivative, activity annotation, and tape. If there is an active return dret is passed as Active{T} with the derivative of the active return val. Otherwise dret is passed as Type{Duplicated{T}}, etc.\n\n\n\n\n\n","category":"function"},{"location":"api/#EnzymeTestUtils.ExprAndMsg","page":"API","title":"EnzymeTestUtils.ExprAndMsg","text":"A cunning hack to carry extra message along with the original expression in a test\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeTestUtils.@test_msg-Tuple{Any, Any, Vararg{Any}}","page":"API","title":"EnzymeTestUtils.@test_msg","text":"@test_msg msg condion kws...\n\nThis is per Test.@test condion kws... except that if it fails it also prints the msg. If msg==\"\" then this is just like @test, nothing is printed\n\nExamles\n\njulia> @test_msg \"It is required that the total is under 10\" sum(1:1000) < 10;\nTest Failed at REPL[1]:1\n Expression: sum(1:1000) < 10\n Problem: It is required that the total is under 10\n Evaluated: 500500 < 10\nERROR: There was an error during testing\n\n\njulia> @test_msg \"It is required that the total is under 10\" error(\"not working at all\");\nError During Test at REPL[2]:1\n Test threw exception\n Expression: error(\"not working at all\")\n Problem: It is required that the total is under 10\n \"not working at all\"\n Stacktrace:\n\njulia> a = \"\";\n\njulia> @test_msg a sum(1:1000) < 10;\n Test Failed at REPL[153]:1\n Expression: sum(1:1000) < 10\n Evaluated: 500500 < 10\n ERROR: There was an error during testing\n\n\n\n\n\n","category":"macro"},{"location":"api/#EnzymeTestUtils.are_activities_compatible-Tuple{Any, Vararg{Any}}","page":"API","title":"EnzymeTestUtils.are_activities_compatible","text":"are_activities_compatible(Tret, activities...) -> Bool\n\nReturn true if return activity type Tret and activity types activities are compatible.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeTestUtils.test_forward-Tuple{Any, Any, Vararg{Any}}","page":"API","title":"EnzymeTestUtils.test_forward","text":"test_forward(f, Activity, args...; kwargs...)\n\nTest Enzyme.autodiff of f in Forward-mode against finite differences.\n\nf has all constraints of the same argument passed to Enzyme.autodiff, with several additional constraints:\n\nIf it mutates one of its arguments, it must return that argument.\n\nArguments\n\nActivity: the activity of the return value of f\nargs: Each entry is either an argument to f, an activity type accepted by autodiff, or a tuple of the form (arg, Activity), where Activity is the activity type of arg. If the activity type specified requires a tangent, a random tangent will be automatically generated.\n\nKeywords\n\nfdm=FiniteDifferences.central_fdm(5, 1): The finite differences method to use.\nfkwargs: Keyword arguments to pass to f.\nrtol: Relative tolerance for isapprox.\natol: Absolute tolerance for isapprox.\ntestset_name: Name to use for a testset in which all tests are evaluated.\n\nExamples\n\nHere we test a rule for a function of scalars. Because we don't provide an activity annotation for y, it is assumed to be Const.\n\nusing Enzyme, EnzymeTestUtils\n\nx, y = randn(2)\nfor Tret in (Const, Duplicated, DuplicatedNoNeed), Tx in (Const, Duplicated)\n test_forward(*, Tret, (x, Tx), y)\nend\n\nHere we test a rule for a function of an array in batch forward-mode:\n\nx = randn(3)\ny = randn()\nfor Tret in (Const, BatchDuplicated, BatchDuplicatedNoNeed),\n Tx in (Const, BatchDuplicated),\n Ty in (Const, BatchDuplicated)\n\n test_forward(*, Tret, (x, Tx), (y, Ty))\nend\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeTestUtils.test_reverse-Tuple{Any, Any, Vararg{Any}}","page":"API","title":"EnzymeTestUtils.test_reverse","text":"test_reverse(f, Activity, args...; kwargs...)\n\nTest Enzyme.autodiff_thunk of f in ReverseSplitWithPrimal-mode against finite differences.\n\nf has all constraints of the same argument passed to Enzyme.autodiff_thunk, with several additional constraints:\n\nIf it mutates one of its arguments, it must not also return that argument.\nIf the return value is a struct, then all floating point numbers contained in the struct or its fields must be in arrays.\n\nArguments\n\nActivity: the activity of the return value of f.\nargs: Each entry is either an argument to f, an activity type accepted by autodiff, or a tuple of the form (arg, Activity), where Activity is the activity type of arg. If the activity type specified requires a shadow, one will be automatically generated.\n\nKeywords\n\nfdm=FiniteDifferences.central_fdm(5, 1): The finite differences method to use.\nfkwargs: Keyword arguments to pass to f.\nrtol: Relative tolerance for isapprox.\natol: Absolute tolerance for isapprox.\ntestset_name: Name to use for a testset in which all tests are evaluated.\n\nExamples\n\nHere we test a rule for a function of scalars. Because we don't provide an activity annotation for y, it is assumed to be Const.\n\nusing Enzyme, EnzymeTestUtils\n\nx = randn()\ny = randn()\nfor Tret in (Const, Active), Tx in (Const, Active)\n test_reverse(*, Tret, (x, Tx), y)\nend\n\nHere we test a rule for a function of an array in batch reverse-mode:\n\nx = randn(3)\nfor Tret in (Const, Active), Tx in (Const, BatchDuplicated)\n test_reverse(prod, Tret, (x, Tx))\nend\n\n\n\n\n\n","category":"method"},{"location":"api/#Enzyme.API.runtimeActivity!-Tuple{Bool}","page":"API","title":"Enzyme.API.runtimeActivity!","text":"runtimeActivity!(val::Bool)\n\nEnzyme runs an activity analysis which deduces which values, instructions, etc are necessary to be differentiated and therefore involved in the differentiation procedure. This runs at compile time. However, there may be implementation flaws in this analysis that means that Enzyme cannot deduce that an inactive (const) value is actually const. Alternatively, there may be some data which is conditionally active, depending on which runtime branch is taken. In these cases Enzyme conservatively presumes the value is active.\n\nHowever, in certain cases, an insufficiently aggressive activity analysis may result in derivative errors – for example by mistakenly using the primal (const) argument and mistaking it for the duplicated shadow. As a result this may result in incorrect results, or accidental updates to the primal.\n\nThis flag enables runntime activity which tells all load/stores to check at runtime whether the value they are updating is indeed active (in addition to the compile-time activity analysis). This will remedy these such errors, but at a performance penalty of performing such checks.\n\nIt is on the Enzyme roadmap to add a PotentiallyDuplicated style activity, in addition to the current Const and Duplicated styles that will disable the need for this, which does not require the check when a value is guaranteed active, but still supports runtime-based activity information.\n\nThis function takes an argument to set the runtime activity value, true means it is on, and false means off. By default it is off.\n\n\n\n\n\n","category":"method"},{"location":"api/#Enzyme.API.runtimeActivity-Tuple{}","page":"API","title":"Enzyme.API.runtimeActivity","text":"runtimeActivity()\n\nGets the current value of the runtime activity. See runtimeActivity! for more information.\n\n\n\n\n\n","category":"method"},{"location":"pullbacks/#Implementing-pullbacks","page":"Implementing pullbacks","title":"Implementing pullbacks","text":"","category":"section"},{"location":"pullbacks/","page":"Implementing pullbacks","title":"Implementing pullbacks","text":"Enzyme's autodiff function can only handle functions with scalar output. To implement pullbacks (back-propagation of gradients/tangents) for array-valued functions, use a mutating function that returns nothing and stores it's result in one of the arguments, which must be passed wrapped in a Duplicated.","category":"page"},{"location":"pullbacks/#Example","page":"Implementing pullbacks","title":"Example","text":"","category":"section"},{"location":"pullbacks/","page":"Implementing pullbacks","title":"Implementing pullbacks","text":"Given a function mymul! that performs the equivalent of R = A * B for matrices A and B, and given a gradient (tangent) ∂z_∂R, we can compute ∂z_∂A and ∂z_∂B like this:","category":"page"},{"location":"pullbacks/","page":"Implementing pullbacks","title":"Implementing pullbacks","text":"using Enzyme, Random\n\nfunction mymul!(R, A, B)\n @assert axes(A,2) == axes(B,1)\n @inbounds @simd for i in eachindex(R)\n R[i] = 0\n end\n @inbounds for j in axes(B, 2), i in axes(A, 1)\n @inbounds @simd for k in axes(A,2)\n R[i,j] += A[i,k] * B[k,j]\n end\n end\n nothing\nend\n\nRandom.seed!(1234)\nA = rand(5, 3)\nB = rand(3, 7)\n\nR = zeros(size(A,1), size(B,2))\n∂z_∂R = rand(size(R)...) # Some gradient/tangent passed to us\n∂z_∂R0 = copyto!(similar(∂z_∂R), ∂z_∂R) # exact copy for comparison\n\n∂z_∂A = zero(A)\n∂z_∂B = zero(B)\n\nEnzyme.autodiff(Reverse, mymul!, Const, Duplicated(R, ∂z_∂R), Duplicated(A, ∂z_∂A), Duplicated(B, ∂z_∂B))","category":"page"},{"location":"pullbacks/","page":"Implementing pullbacks","title":"Implementing pullbacks","text":"Now we have:","category":"page"},{"location":"pullbacks/","page":"Implementing pullbacks","title":"Implementing pullbacks","text":"R ≈ A * B &&\n∂z_∂A ≈ ∂z_∂R0 * B' && # equivalent to Zygote.pullback(*, A, B)[2](∂z_∂R)[1]\n∂z_∂B ≈ A' * ∂z_∂R0 # equivalent to Zygote.pullback(*, A, B)[2](∂z_∂R)[2]","category":"page"},{"location":"pullbacks/","page":"Implementing pullbacks","title":"Implementing pullbacks","text":"Note that the result of the backpropagation is added to ∂z_∂A and ∂z_∂B, they act as accumulators for gradient information.","category":"page"},{"location":"dev_docs/#Enzyme-developer-documentation","page":"For developers","title":"Enzyme developer documentation","text":"","category":"section"},{"location":"dev_docs/#Development-of-Enzyme-and-Enzyme.jl-together","page":"For developers","title":"Development of Enzyme and Enzyme.jl together","text":"","category":"section"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"Normally Enzyme.jl downloads and install Enzyme for the user automatically since Enzyme needs to be built against Julia bundeled LLVM. In case that you are making updates to Enzyme and want to test them against Enzyme.jl the instructions below should help you get started.","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"Start Julia in your development copy of Enzyme.jl","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"~/s/Enzyme (master)> julia --project=.","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"Then create a development copy of Enzyme_jll and activate it within.","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"julia> using Enzyme_jll\njulia> Enzyme_jll.dev_jll()\n[ Info: Enzyme_jll dev'ed out to ${JULIA_PKG_DEVDIR}/Enzyme_jll with pre-populated override directory\n(Enzyme) pkg> dev Enzyme_jll\nPath `${JULIA_PKG_DEVDIR}/Enzyme_jll` exists and looks like the correct package. Using existing path.","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"After restarting Julia:","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"julia> Enzyme_jll.dev_jll()\njulia> Enzyme_jll.libEnzyme_path\n\"${JULIA_PKG_DEVDIR}/Enzyme_jll/override/lib/LLVMEnzyme-9.so\"","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"On your machine ${JULIA_PKG_DEVDIR} most likely corresponds to ~/.julia/dev. Now we can inspect \"${JULIA_PKG_DEVDIR}/Enzyme_jll/override/lib and see that there is a copy of LLVMEnzyme-9.so, which we can replace with a symbolic link or a copy of a version of Enzyme.","category":"page"},{"location":"dev_docs/#Building-Enzyme-against-Julia's-LLVM.","page":"For developers","title":"Building Enzyme against Julia's LLVM.","text":"","category":"section"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"Depending on how you installed Julia the LLVM Julia is using will be different.","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"Download from julialang.org (Recommended)\nManual build on your machine\nUses a pre-built Julia from your system vendor (Not recommended)","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"To check what LLVM Julia is using use:","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"julia> Base.libllvm_version_string\n\"9.0.1jl\"","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"If the LLVM version ends in a jl you a likely using the private LLVM.","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"In your source checkout of Enzyme:","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"mkdir build-jl\ncd build-jl","category":"page"},{"location":"dev_docs/#Prebuilt-binary-from-julialang.org","page":"For developers","title":"Prebuilt binary from julialang.org","text":"","category":"section"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"LLVM_MAJOR_VER=`julia -e \"print(Base.libllvm_version.major)\"`\njulia -e \"using Pkg; pkg\\\"add LLVM_full_jll@${LLVM_MAJOR_VER}\\\"\"\nLLVM_DIR=`julia -e \"using LLVM_full_jll; print(LLVM_full_jll.artifact_dir)\"`\necho \"LLVM_DIR=$LLVM_DIR\"\ncmake ../enzyme/ -G Ninja -DENZYME_EXTERNAL_SHARED_LIB=ON -DLLVM_DIR=${LLVM_DIR} -DLLVM_EXTERNAL_LIT=${LLVM_DIR}/tools/lit/lit.py","category":"page"},{"location":"dev_docs/#Manual-build-of-Julia","page":"For developers","title":"Manual build of Julia","text":"","category":"section"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"cmake ../enzyme/ -G Ninja -DENZYME_EXTERNAL_SHARED_LIB=ON -DLLVM_DIR=${PATH_TO_BUILDDIR_OF_JULIA}/usr/lib/cmake/llvm/","category":"page"},{"location":"internal_api/#Internal-API","page":"Internal API","title":"Internal API","text":"","category":"section"},{"location":"internal_api/","page":"Internal API","title":"Internal API","text":"note: Note\nThis is the documentation of Enzymes's internal API. The internal API is not subject to semantic versioning and may change at any time and without deprecation.","category":"page"},{"location":"internal_api/","page":"Internal API","title":"Internal API","text":"Modules = [Enzyme.Compiler]\nOrder = [:module, :type, :constant, :macro, :function]","category":"page"},{"location":"internal_api/#Enzyme.Compiler.fspec-Tuple{Any, Any, Integer}","page":"Internal API","title":"Enzyme.Compiler.fspec","text":"Create the methodinstance pair, and lookup the primal return type.\n\n\n\n\n\n","category":"method"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"EditURL = \"../../../examples/box.jl\"","category":"page"},{"location":"generated/box/#Enzyme-for-adjoint-tutorial:-Stommel-three-box-ocean-model","page":"Box model","title":"Enzyme for adjoint tutorial: Stommel three-box ocean model","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"The goal of this tutorial is to teach about a specific usage of Enzyme's automatic differentiation capabilities, and will be centered around the Stommel ocean model. This is a nice example to see how powerful Enzyme is, and the ability of it to take a derivative of a complicated function (namely one that has many parts and parameters). This tutorial will focus first on the computations and getting Enzyme running, for those interested a mathematical explanation of the model and what an adjoint variable is will be provided at the end.","category":"page"},{"location":"generated/box/#Brief-model-overview","page":"Box model","title":"Brief model overview","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"The Stommel box model can be viewed as a watered down full ocean model. In our example, we have three boxes (Box One, Box Two, and Box Three) and we model the transport of fluid between them. The full equations of our system are given by:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"beginaligned\n U = u_0 left rho_2 - left rho_1 + (1 - delta) rho_3 right right \n rho_i = -alpha T_i + beta S_i i = 1 2 3\nendaligned","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"for the transport U and densities rho, and then the time derivatives","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"beginaligned\n dotT_1 = U(T_3 - T_1)V_1 + gamma (T_1^* - T_1 ) dotS_1 = U(S_3 - S_1)V_1 + FW_1V_1 \n dotT_2 = U(T_1 - T_2)V_2 + gamma (T_2^* - T_2 ) dotS_2 = U(S_1 - S_2)V_2 + FW_2V_2 \n dotT_3 = U(T_2 - T_3)V_3 dotS_3 = U(S_2 - S_3)V_3\nendaligned","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"for positive transport, U 0, and","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"beginaligned\n dotT_1 = U(T_2 - T_1)V_1 + gamma (T_1^* - T_1) dotS_1 = U(S_2 - S_1)V_1 + FW_1V_1 \n dotT_2 = U(T_3 - T_2)V_2 + gamma (T_2^* - T_2 ) dotS_2 = U(S_3 - S_2)V_2 + FW_2V_2 \n dotT_3 = U(T_1 - T_3)V_3 dotS_3 = U(S_1 - S_3)V_3\nendaligned","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"for U leq 0. The only force driving our system is a density gradient generated via temperature and salinity differences between the boxes. This makes it a really easy model to play around with! With this in mind, the model is run forward with the steps:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Compute densities\nCompute transport\nCompute time derivatives of the box temperatures and salinities\nUpdate the state vector","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"We'll start by going through the model setup step by step, then providing a few test cases with Enzyme.","category":"page"},{"location":"generated/box/#Model-setup","page":"Box model","title":"Model setup","text":"","category":"section"},{"location":"generated/box/#Model-dependencies","page":"Box model","title":"Model dependencies","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Let's first add the necessary packages to run everything","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"using Enzyme","category":"page"},{"location":"generated/box/#Initialize-constants","page":"Box model","title":"Initialize constants","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"The system equations have quite a few constants that appear, here we initialize them for later use. We'll do this in a Julia way: we have an empty structure that will hold all the parameters, and a function (we'll call this setup) that initializes them. This means that, so long as we don't need to change parameters, we only need to run setup once.","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"struct ModelParameters\n\n # handy to have constants\n day::Float64\n year::Float64\n\n # Information related to the boxes\n boxlength::Vector{Float64} ## Vector with north-south size of each box [cm]\n boxdepth::Vector{Float64} ## \" \" the depth of each box [cm]\n boxwidth::Float64 ## \" \" the width of each box [cm]\n boxarea::Vector{Float64} ## \" \" the area of each box [cm^2]\n boxvol::Vector{Float64} ## \" \" the volume of each box [cm^3]\n\n delta::Float64 ## Constant ratio depth(box1) / (depth(box1) + depth(box3))\n\n # Parameters that appear in the box model equations\n u0::Float64\n alpha::Float64\n beta::Float64\n gamma::Float64\n\n # Coefficient for the Robert filter smoother\n rf_coeff::Float64\n\n # Freshwater forcing\n FW::Vector{Float64}\n\n # Restoring atmospheric temperatures and salinities\n Tstar::Vector{Float64}\n Sstar::Vector{Float64}\n\nend\n\nfunction setup()\n\n blength = [5000.0e5; 1000.0e5; 5000.0e5]\n bdepth = [1.0e5; 5.0e5; 4.0e5]\n\n delta = bdepth[1]/(bdepth[1] + bdepth[3])\n\n bwidth = 4000.0*1e5 ## box width, centimeters\n\n # box areas\n barea = [blength[1]*bwidth;\n blength[2]*bwidth;\n blength[3]*bwidth]\n\n # box volumes\n bvolume = [barea[1]*bdepth[1];\n barea[2]*bdepth[2];\n barea[3]*bdepth[3]]\n\n # parameters that are used to ensure units are in CGS (cent-gram-sec)\n\n day = 3600.0*24.0\n year = day*365.0\n Sv = 1e12 ## one Sverdrup (a unit of ocean transport), 1e6 meters^3/second\n\n # parameters that appear in box model equations\n u0 = 16.0*Sv/0.0004\n alpha = 1668e-7\n beta = 0.7811e-3\n\n gamma = 1/(300*day)\n\n # robert filter coefficient for the smoother part of the timestep\n robert_filter_coeff = 0.25\n\n # freshwater forcing\n FW = [(100/year) * 35.0 * barea[1]; -(100/year) * 35.0 * barea[1]]\n\n # restoring atmospheric temperatures\n Tstar = [22.0; 0.0]\n Sstar = [36.0; 34.0]\n\n structure_with_parameters = ModelParameters(day,\n year,\n blength,\n bdepth,\n bwidth,\n barea,\n bvolume,\n delta,\n u0,\n alpha,\n beta,\n gamma,\n robert_filter_coeff,\n FW,\n Tstar,\n Sstar\n )\n\n return structure_with_parameters\n\nend","category":"page"},{"location":"generated/box/#Define-model-functions","page":"Box model","title":"Define model functions","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Here we define functions that will calculate quantities used in the forward steps.","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"# function to compute transport\n# Input: rho - the density vector\n# Output: U - transport value\n\nfunction compute_transport(rho, params)\n\n U = params.u0 * (rho[2] - (params.delta * rho[1] + (1 - params.delta)*rho[3]))\n return U\n\nend\n\n# function to compute density\n# Input: state = [T1; T2; T3; S1; S2; S3]\n# Output: rho\n\nfunction compute_density(state, params)\n\n rho = -params.alpha * state[1:3] + params.beta * state[4:6]\n return rho\n\nend\n\n# lastly, a function that takes one step forward\n# Input: state_now = [T1(t), T2(t), ..., S3(t)]\n# state_old = [T1(t-dt), ..., S3(t-dt)]\n# u = transport(t)\n# dt = time step\n# Output: state_new = [T1(t+dt), ..., S3(t+dt)]\n\nfunction compute_update(state_now, state_old, u, params, dt)\n\n dstate_now_dt = zeros(6)\n state_new = zeros(6)\n\n # first computing the time derivatives of the various temperatures and salinities\n if u > 0\n\n dstate_now_dt[1] = u * (state_now[3] - state_now[1]) / params.boxvol[1] + params.gamma * (params.Tstar[1] - state_now[1])\n dstate_now_dt[2] = u * (state_now[1] - state_now[2]) / params.boxvol[2] + params.gamma * (params.Tstar[2] - state_now[2])\n dstate_now_dt[3] = u * (state_now[2] - state_now[3]) / params.boxvol[3]\n\n dstate_now_dt[4] = u * (state_now[6] - state_now[4]) / params.boxvol[1] + params.FW[1] / params.boxvol[1]\n dstate_now_dt[5] = u * (state_now[4] - state_now[5]) / params.boxvol[2] + params.FW[2] / params.boxvol[2]\n dstate_now_dt[6] = u * (state_now[5] - state_now[6]) / params.boxvol[3]\n\n elseif u <= 0\n\n dstate_now_dt[1] = u * (state_now[2] - state_now[1]) / params.boxvol[1] + params.gamma * (params.Tstar[1] - state_now[1])\n dstate_now_dt[2] = u * (state_now[3] - state_now[2]) / params.boxvol[2] + params.gamma * (params.Tstar[2] - state_now[2])\n dstate_now_dt[3] = u * (state_now[1] - state_now[3]) / params.boxvol[3]\n\n dstate_now_dt[4] = u * (state_now[5] - state_now[4]) / params.boxvol[1] + params.FW[1] / params.boxvol[1]\n dstate_now_dt[5] = u * (state_now[6] - state_now[5]) / params.boxvol[2] + params.FW[2] / params.boxvol[2]\n dstate_now_dt[6] = u * (state_now[4] - state_now[6]) / params.boxvol[3]\n\n end\n\n # update fldnew using a version of Euler's method\n state_new .= state_old + 2.0 * dt * dstate_now_dt\n\n return state_new\nend","category":"page"},{"location":"generated/box/#Define-forward-functions","page":"Box model","title":"Define forward functions","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Finally, we create two functions, the first of which computes and stores all the states of the system, and the second will take just a single step forward.","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Let's start with the standard forward function. This is just going to be used to store the states at every timestep:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"function integrate(state_now, state_old, dt, M, parameters)\n\n # Because of the adjoint problem we're setting up, we need to store both the states before\n # and after the Robert filter smoother has been applied\n states_before = [state_old]\n states_after = [state_old]\n\n for t = 1:M\n\n rho = compute_density(state_now, parameters)\n u = compute_transport(rho, parameters)\n state_new = compute_update(state_now, state_old, u, parameters, dt)\n\n # Applying the Robert filter smoother (needed for stability)\n state_new_smoothed = state_now + parameters.rf_coeff * (state_new - 2.0 * state_now + state_old)\n\n push!(states_after, state_new_smoothed)\n push!(states_before, state_new)\n\n # cycle the \"now, new, old\" states\n state_old = state_new_smoothed\n state_now = state_new\n\n end\n\n return states_after, states_before\nend","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Now, for the purposes of Enzyme, it would be convenient for us to have a function that runs a single step of the model forward rather than the whole integration. This would allow us to save as many of the adjoint variables as we wish when running the adjoint method, although for the example we'll discuss later we technically only need one of them","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"function one_step_forward(state_now, state_old, out_now, out_old, parameters, dt)\n\n state_new_smoothed = zeros(6)\n rho = compute_density(state_now, parameters) ## compute density\n u = compute_transport(rho, parameters) ## compute transport\n state_new = compute_update(state_now, state_old, u, parameters, dt) ## compute new state values\n\n # Robert filter smoother\n state_new_smoothed[:] = state_now + parameters.rf_coeff * (state_new - 2.0 * state_now + state_old)\n\n out_old[:] = state_new_smoothed\n out_now[:] = state_new\n\n return nothing\n\nend","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"One difference to note is that one_step_forward now returns nothing, but is rather a function of both its input and output. Since the output of the function is a vector, we need to have this return nothing for Enzyme to work. Now we can move on to some examples using Enzyme.","category":"page"},{"location":"generated/box/#Example-1:-Simply-using-Enzyme","page":"Box model","title":"Example 1: Simply using Enzyme","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"For the first example let's just compute the gradient of our forward function and examine the output. We'll just run the model for one step, and take a dt of ten days. The initial conditions of the system are given as Tbar and Sbar. We run setup once here, and never have to run it again! (Unless we decide to change a parameter)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"parameters = setup()\n\nTbar = [20.0; 1.0; 1.0] ## initial temperatures\nSbar = [35.5; 34.5; 34.5] ## initial salinities\n\n# Running the model one step forward\nstates_after_smoother, states_before_smoother = integrate(\n copy([Tbar; Sbar]),\n copy([Tbar; Sbar]),\n 10*parameters.day,\n 1,\n parameters\n)\n\n# Run Enzyme one time on `one_step_forward``\ndstate_now = zeros(6)\ndstate_old = zeros(6)\nout_now = zeros(6); dout_now = ones(6)\nout_old = zeros(6); dout_old = ones(6)\n\nautodiff(Reverse,\n one_step_forward,\n Duplicated([Tbar; Sbar], dstate_now),\n Duplicated([Tbar; Sbar], dstate_old),\n Duplicated(out_now, dout_now),\n Duplicated(out_old, dout_old),\n parameters,\n Const(10*parameters.day)\n)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"In order to run Enzyme on one_step_forward, we've needed to provide quite a few placeholders, and wrap everything in Duplicated as all components of our function are vectors, not scalars. Let's go through and see what Enzyme did with all of those placeholders.","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"First we can look at what happened to the zero vectors out_now and out_old:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show out_now, out_old","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Comparing to the results of forward func:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show states_before_smoother[2], states_after_smoother[2]","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"we see that Enzyme has computed and stored exactly the output of the forward step. Next, let's look at dstate_now:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show dstate_now","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Just a few numbers, but this is what makes AD so nice: Enzyme has exactly computed the derivative of all outputs with respect to the input state_now, evaluated at state_now, and acted with this gradient on what we gave as dout_now (in our case, all ones). Using AD notation for reverse mode, this is","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"overlinetextstate_now = fracpartial textout_nowpartial textstate_nowright_textstate_now overlinetextout_now + fracpartial textout_oldpartial textstate_nowright_textstate_now overlinetextout_old","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"We note here that had we initialized dstate_now and dstate_old as something else, our results will change. Let's multiply them by two and see what happens.","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"dstate_now_new = zeros(6)\ndstate_old_new = zeros(6)\nout_now = zeros(6); dout_now = 2*ones(6)\nout_old = zeros(6); dout_old = 2*ones(6)\nautodiff(Reverse,\n one_step_forward,\n Duplicated([Tbar; Sbar], dstate_now_new),\n Duplicated([Tbar; Sbar], dstate_old_new),\n Duplicated(out_now, dout_now),\n Duplicated(out_old, dout_old),\n parameters,\n Const(10*parameters.day)\n)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Now checking dstate_now and dstate_old we see","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show dstate_now_new","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"What happened? Enzyme is actually taking the computed gradient and acting on what we give as input to dout_now and dout_old. Checking this, we see","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show 2*dstate_now","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"and they match the new results. This exactly matches what we'd expect to happen since we scaled dout_now by two.","category":"page"},{"location":"generated/box/#Example-2:-Full-sensitivity-calculations","page":"Box model","title":"Example 2: Full sensitivity calculations","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Now we want to use Enzyme for a bit more than just a single derivative. Let's say we'd like to understand how sensitive the final temperature of Box One is to the initial salinity of Box Two. That is, given the function","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"J = (100000)^T cdot mathbfx(t_f)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"we want Enzyme to calculate the derivative","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"fracpartial Jpartial mathbfx(0)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"where x(t) is the state of the model at time t. If we think about x(t_f) as solely depending on the initial condition, then this derivative is really","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"fracpartial Jpartial mathbfx(0) = fracpartialpartial mathbfx(0) left( (100000)^T cdot L(ldots(L(mathbfx(0)))) right)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"with L(x(t)) = x(t + dt), i.e. one forward step. One could expand this derivative with the chain rule (and it would be very complicated), but really this is where Enzyme comes in. Each run of autodiff on our forward function is one piece of this big chain rule done for us! We also note that the chain rule goes from the outside in, so we start with the derivative of the forward function at the final state, and work backwards until the initial state. To get Enzyme to do this, we complete the following steps:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Run the forward model and store outputs (in a real ocean model this wouldn't be feasible and we'd need to use checkpointing)\nCompute the initial derivative from the final state\nUse Enzyme to work backwards until we reach the desired derivative.","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"For simplicity we define a function that takes completes our AD steps","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"function compute_adjoint_values(states_before_smoother, states_after_smoother, M, parameters)\n\n dout_now = [0.0;0.0;0.0;0.0;0.0;0.0]\n dout_old = [1.0;0.0;0.0;0.0;0.0;0.0]\n\n for j = M:-1:1\n\n dstate_now = zeros(6)\n dstate_old = zeros(6)\n\n autodiff(Reverse,\n one_step_forward,\n Duplicated(states_before_smoother[j], dstate_now),\n Duplicated(states_after_smoother[j], dstate_old),\n Duplicated(zeros(6), dout_now),\n Duplicated(zeros(6), dout_old),\n parameters,\n Const(10*parameters.day)\n )\n\n if j == 1\n return dstate_now, dstate_old\n end\n\n dout_now = copy(dstate_now)\n dout_old = copy(dstate_old)\n\n end\n\nend","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"First we integrate the model forward:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"M = 10000 ## Total number of forward steps to take\nTbar = [20.0; 1.0; 1.0] ## initial temperatures\nSbar = [35.5; 34.5; 34.5] ## initial salinities\n\nstates_after_smoother, states_before_smoother = integrate(\n copy([Tbar; Sbar]),\n copy([Tbar; Sbar]),\n 10*parameters.day,\n M,\n parameters\n)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Next, we pass all of our states to the AD function to get back to the desired derivative:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"dstate_now, dstate_old = compute_adjoint_values(\n states_before_smoother,\n states_after_smoother,\n M,\n parameters\n)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"And we're done! We were interested in sensitivity to the initial salinity of box two, which will live in what we've called dstate_old. Checking this value we see","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show dstate_old[5]","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"As it stands this is just a number, but a good check that Enzyme has computed what we want is to approximate the derivative with a Taylor series. Specifically,","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"J(mathbfx(0) + varepsilon) approx J(mathbfx(0)) +\nvarepsilon fracpartial Jpartial mathbfx(0)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"and a simple rearrangement yields","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"fracpartial Jpartial mathbfx(0) approx\nfracJ(mathbfx(0) + varepsilon) - J(mathbfx(0))varepsilon","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Hopefully we see that the analytical values converge close to the one we found with Enzyme:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"# unperturbed final state\nuse_to_check = states_after_smoother[M+1]\n\n# a loop to compute the perturbed final states\ndiffs = []\nstep_sizes = [1e-1, 1e-2, 1e-3, 1e-4, 1e-5, 1e-6, 1e-7, 1e-8, 1e-9, 1e-10]\nfor eps in step_sizes\n\n state_new_smoothed = zeros(6)\n\n initial_temperature = [20.0; 1.0; 1.0]\n perturbed_initial_salinity = [35.5; 34.5; 34.5] + [0.0; eps; 0.0]\n\n state_old = [initial_temperature; perturbed_initial_salinity]\n state_now = [20.0; 1.0; 1.0; 35.5; 34.5; 34.5]\n\n for t = 1:M\n\n rho = compute_density(state_now, parameters)\n u = compute_transport(rho, parameters)\n state_new = compute_update(state_now, state_old, u, parameters, 10*parameters.day)\n\n state_new_smoothed[:] = state_now + parameters.rf_coeff * (state_new - 2.0 * state_now + state_old)\n\n state_old = state_new_smoothed\n state_now = state_new\n\n end\n\n push!(diffs, (state_old[1] - use_to_check[1])/eps)\n\nend","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Then checking what we found the derivative to be analytically:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show diffs","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"which comes very close to our calculated value. We can go further and check the percent difference to see","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show abs.(diffs .- dstate_old[5])./dstate_old[5]","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"and we get down to a percent difference on the order of 1e^-5, showing Enzyme calculated the correct derivative. Success!","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"EditURL = \"../../../examples/custom_rule.jl\"","category":"page"},{"location":"generated/custom_rule/#Enzyme-custom-rules-tutorial","page":"Custom rules","title":"Enzyme custom rules tutorial","text":"","category":"section"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"note: More Examples\nThe tutorial below focuses on a simple setting to illustrate the basic concepts of writing custom rules. For more complex custom rules beyond the scope of this tutorial, you may take inspiration from the following in-the-wild examples:Enzyme internal rules\nKernelAbstractions.jl\nLinearSolve.jl\nNNlib.jl","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"The goal of this tutorial is to give a simple example of defining a custom rule with Enzyme. Specifically, our goal will be to write custom rules for the following function f:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"function f(y, x)\n y .= x.^2\n return sum(y)\nend","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Our function f populates its first input y with the element-wise square of x. In addition, it returns sum(y) as output. What a sneaky function!","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"In this case, Enzyme can differentiate through f automatically. For example, using forward mode:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"using Enzyme\nx = [3.0, 1.0]\ndx = [1.0, 0.0]\ny = [0.0, 0.0]\ndy = [0.0, 0.0]\n\ng(y, x) = f(y, x)^2 # function to differentiate\n\n@show autodiff(Forward, g, Duplicated(y, dy), Duplicated(x, dx)) # derivative of g w.r.t. x[1]\n@show dy; # derivative of y w.r.t. x[1] when g is run\nnothing #hide","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"(See the AutoDiff API tutorial for more information on using autodiff.)","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"But there may be special cases where we need to write a custom rule to help Enzyme out. Let's see how to write a custom rule for f!","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"warning: Don't use custom rules unnecessarily!\nEnzyme can efficiently handle a wide range of constructs, and so a custom rule should only be required in certain special cases. For example, a function may make a foreign call that Enzyme cannot differentiate, or we may have higher-level mathematical knowledge that enables us to write a more efficient rule. Even in these cases, try to make your custom rule encapsulate the minimum possible construct that Enzyme cannot differentiate, rather than expanding the scope of the rule unnecessarily. For pedagogical purposes, we will disregard this principle here and go ahead and write a custom rule for f :)","category":"page"},{"location":"generated/custom_rule/#Defining-our-first-rule","page":"Custom rules","title":"Defining our first rule","text":"","category":"section"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"First, we import the functions EnzymeRules.forward, EnzymeRules.augmented_primal, and EnzymeRules.reverse. We need to overload forward in order to define a custom forward rule, and we need to overload augmented_primal and reverse in order to define a custom reverse rule.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"import .EnzymeRules: forward, reverse, augmented_primal\nusing .EnzymeRules","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"In this section, we write a simple forward rule to start out:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"function forward(func::Const{typeof(f)}, ::Type{<:Duplicated}, y::Duplicated, x::Duplicated)\n println(\"Using custom rule!\")\n ret = func.val(y.val, x.val)\n y.dval .= 2 .* x.val .* x.dval\n return Duplicated(ret, sum(y.dval))\nend","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"In the signature of our rule, we have made use of Enzyme's activity annotations. Let's break down each one:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"the Const annotation on f indicates that we accept a function f that does not have a derivative component, which makes sense since f is not a closure with data that could be differentiated.\nthe Duplicated annotation given in the second argument annotates the return value of f. This means that our forward function should return an output of type Duplicated, containing the original output sum(y) and its derivative.\nthe Duplicated annotations for x and y mean that our forward function handles inputs x and y which have been marked as Duplicated. We should update their shadows with their derivative contributions.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"In the logic of our forward function, we run the original function, populate y.dval (the shadow of y), and finally return a Duplicated for the output as promised. Let's see our rule in action! With the same setup as before:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"x = [3.0, 1.0]\ndx = [1.0, 0.0]\ny = [0.0, 0.0]\ndy = [0.0, 0.0]\n\ng(y, x) = f(y, x)^2 # function to differentiate\n\n@show autodiff(Forward, g, Duplicated(y, dy), Duplicated(x, dx)) # derivative of g w.r.t. x[1]\n@show dy; # derivative of y w.r.t. x[1] when g is run\nnothing #hide","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"We see that our custom forward rule has been triggered and gives the same answer as before.","category":"page"},{"location":"generated/custom_rule/#Handling-more-activities","page":"Custom rules","title":"Handling more activities","text":"","category":"section"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Our custom rule applies for the specific set of activities that are annotated for f in the above autodiff call. However, Enzyme has a number of other annotations. Let us consider a particular example, where the output has a DuplicatedNoNeed annotation. This means we are only interested in its derivative, not its value. To squeeze out the last drop of performance, the below rule avoids computing the output of the original function and just computes its derivative.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"function forward(func::Const{typeof(f)}, ::Type{<:DuplicatedNoNeed}, y::Duplicated, x::Duplicated)\n println(\"Using custom rule with DuplicatedNoNeed output.\")\n y.val .= x.val.^2\n y.dval .= 2 .* x.val .* x.dval\n return sum(y.dval)\nend","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Our rule is triggered, for example, when we call autodiff directly on f, as the return value's derivative isn't needed:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"x = [3.0, 1.0]\ndx = [1.0, 0.0]\ny = [0.0, 0.0]\ndy = [0.0, 0.0]\n\n@show autodiff(Forward, f, Duplicated(y, dy), Duplicated(x, dx)) # derivative of f w.r.t. x[1]\n@show dy; # derivative of y w.r.t. x[1] when f is run\nnothing #hide","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"note: Custom rule dispatch\nWhen multiple custom rules for a function are defined, the correct rule is chosen using Julia's multiple dispatch. In particular, it is important to understand that the custom rule does not determine the activities of the inputs and the return value: rather, Enzyme decides the activity annotations independently, and then dispatches to the custom rule handling the activities, if one exists. If a custom rule is specified for the correct function/argument types, but not the correct activity annotation, a runtime error will be thrown alerting the user to the missing activity rule rather than silently ignoring the rule.\"","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Finally, it may be that either x, y, or the return value are marked as Const. We can in fact handle this case, along with the previous two cases, all together in a single rule:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Base.delete_method.(methods(forward, (Const{typeof(f)}, Vararg{Any}))) # delete our old rules\n\nfunction forward(func::Const{typeof(f)}, RT::Type{<:Union{Const, DuplicatedNoNeed, Duplicated}},\n y::Union{Const, Duplicated}, x::Union{Const, Duplicated})\n println(\"Using our general custom rule!\")\n y.val .= x.val.^2\n if !(x isa Const) && !(y isa Const)\n y.dval .= 2 .* x.val .* x.dval\n elseif !(y isa Const)\n y.dval .= 0\n end\n dret = !(y isa Const) ? sum(y.dval) : zero(eltype(y.val))\n if RT <: Const\n return sum(y.val)\n elseif RT <: DuplicatedNoNeed\n return dret\n else\n return Duplicated(sum(y.val), dret)\n end\nend","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Let's try out our rule:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"x = [3.0, 1.0]\ndx = [1.0, 0.0]\ny = [0.0, 0.0]\ndy = [0.0, 0.0]\n\ng(y, x) = f(y, x)^2 # function to differentiate\n\n@show autodiff(Forward, g, Duplicated(y, dy), Duplicated(x, dx)) # derivative of g w.r.t. x[1]\n@show autodiff(Forward, g, Const(y), Duplicated(x, dx)) # derivative of g w.r.t. x[1], with y annotated Const\n@show autodiff(Forward, g, Const(y), Const(x)); # derivative of g w.r.t. x[1], with x and y annotated Const\nnothing #hide","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Note that there are also exist batched duplicated annotations for forward mode, namely BatchDuplicated and BatchDuplicatedNoNeed, which are not covered in this tutorial.","category":"page"},{"location":"generated/custom_rule/#Defining-a-reverse-mode-rule","page":"Custom rules","title":"Defining a reverse-mode rule","text":"","category":"section"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Let's look at how to write a simple reverse-mode rule! First, we write a method for EnzymeRules.augmented_primal:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"function augmented_primal(config::ConfigWidth{1}, func::Const{typeof(f)}, ::Type{<:Active},\n y::Duplicated, x::Duplicated)\n println(\"In custom augmented primal rule.\")\n # Compute primal\n if needs_primal(config)\n primal = func.val(y.val, x.val)\n else\n y.val .= x.val.^2 # y still needs to be mutated even if primal not needed!\n primal = nothing\n end\n # Save x in tape if x will be overwritten\n if overwritten(config)[3]\n tape = copy(x.val)\n else\n tape = nothing\n end\n # Return an AugmentedReturn object with shadow = nothing\n return AugmentedReturn(primal, nothing, tape)\nend","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Let's unpack our signature for augmented_primal :","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"We accepted a EnzymeRules.Config object with a specified width of 1, which means that our rule does not support batched reverse mode.\nWe annotated f with Const as usual.\nWe dispatched on an Active annotation for the return value. This is a special annotation for scalar values, such as our return value, that indicates that that we care about the value's derivative but we need not explicitly allocate a mutable shadow since it is a scalar value.\nWe annotated x and y with Duplicated, similar to our first simple forward rule.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Now, let's unpack the body of our augmented_primal rule:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"We checked if the config requires the primal. If not, we need not compute the return value, but we make sure to mutate y in all cases.\nWe checked if x could possibly be overwritten using the Overwritten attribute of EnzymeRules.Config. If so, we save the elements of x on the tape of the returned EnzymeRules.AugmentedReturn object.\nWe return a shadow of nothing since the return value is Active and hence does not need a shadow.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Now, we write a method for EnzymeRules.reverse:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"function reverse(config::ConfigWidth{1}, func::Const{typeof(f)}, dret::Active, tape,\n y::Duplicated, x::Duplicated)\n println(\"In custom reverse rule.\")\n # retrieve x value, either from original x or from tape if x may have been overwritten.\n xval = overwritten(config)[3] ? tape : x.val\n # accumulate dret into x's shadow. don't assign!\n x.dval .+= 2 .* xval .* dret.val\n # also accumulate any derivative in y's shadow into x's shadow.\n x.dval .+= 2 .* xval .* y.dval\n y.dval .= 0\n return (nothing, nothing)\nend","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Let's make a few observations about our reverse rule:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"The activities used in the signature correspond to what we used for augmented_primal.\nHowever, for Active return types such as in this case, we now receive an instance dret of Active for the return type, not just a type annotation, which stores the derivative value for ret (not the original return value!). For the other annotations (e.g. Duplicated), we still receive only the type. In that case, if necessary a reference to the shadow of the output should be placed on the tape in augmented_primal.\nUsing dret.val and y.dval, we accumulate the backpropagated derivatives for x into its shadow x.dval. Note that we have to accumulate from both y.dval and dret.val. This is because in reverse-mode AD we have to sum up the derivatives from all uses: if y was read after our function, we need to consider derivatives from that use as well.\nWe zero-out y's shadow. This is because y is overwritten within f, so there is no derivative w.r.t. to the y that was originally inputted.\nFinally, since all derivatives are accumulated in place (in the shadows of the Duplicated arguments), these derivatives must not be communicated via the return value. Hence, we return (nothing, nothing). If, instead, one of our arguments was annotated as Active, we would have to provide its derivative at the corresponding index in the tuple returned.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Finally, let's see our reverse rule in action!","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"x = [3.0, 1.0]\ndx = [0.0, 0.0]\ny = [0.0, 0.0]\ndy = [0.0, 0.0]\n\ng(y, x) = f(y, x)^2\n\nautodiff(Reverse, g, Duplicated(y, dy), Duplicated(x, dx))\n@show dx # derivative of g w.r.t. x\n@show dy; # derivative of g w.r.t. y\nnothing #hide","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Let's also try a function which mutates x after running f, and also uses y directly rather than only ret after running f (but ultimately gives the same result as above):","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"function h(y, x)\n ret = f(y, x)\n x .= x.^2\n return ret * sum(y)\nend\n\nx = [3.0, 1.0]\ny = [0.0, 0.0]\ndx .= 0\ndy .= 0\n\nautodiff(Reverse, h, Duplicated(y, dy), Duplicated(x, dx))\n@show dx # derivative of h w.r.t. x\n@show dy; # derivative of h w.r.t. y\nnothing #hide","category":"page"},{"location":"generated/custom_rule/#Marking-functions-inactive","page":"Custom rules","title":"Marking functions inactive","text":"","category":"section"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"If we want to tell Enzyme that the function call does not affect the differentiation result in any form (i.e. not by side effects or through its return values), we can simply use EnzymeRules.inactive. So long as there exists a matching dispatch to EnzymeRules.inactive, the function will be considered inactive. For example:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"printhi() = println(\"Hi!\")\nEnzymeRules.inactive(::typeof(printhi), args...) = nothing\n\nfunction k(x)\n printhi()\n return x^2\nend\n\nautodiff(Forward, k, Duplicated(2.0, 1.0))","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Or for a case where we incorrectly mark a function inactive:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"double(x) = 2*x\nEnzymeRules.inactive(::typeof(double), args...) = nothing\n\nautodiff(Forward, x -> x + double(x), Duplicated(2.0, 1.0)) # mathematically should be 3.0, inactive rule causes it to be 1.0","category":"page"},{"location":"generated/custom_rule/#Testing-our-rules","page":"Custom rules","title":"Testing our rules","text":"","category":"section"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"We can test our rules using finite differences using EnzymeTestUtils.test_forward and EnzymeTestUtils.test_reverse.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"using EnzymeTestUtils, Test\n\n@testset \"f rules\" begin\n @testset \"forward\" begin\n @testset for RT in (Const, DuplicatedNoNeed, Duplicated),\n Tx in (Const, Duplicated),\n Ty in (Const, Duplicated)\n\n x = [3.0, 1.0]\n y = [0.0, 0.0]\n test_forward(g, RT, (x, Tx), (y, Ty))\n end\n end\n @testset \"reverse\" begin\n @testset for RT in (Active,),\n Tx in (Duplicated,),\n Ty in (Duplicated,),\n fun in (g, h)\n\n x = [3.0, 1.0]\n y = [0.0, 0.0]\n test_reverse(fun, RT, (x, Tx), (y, Ty))\n end\n end\nend","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"In any package that implements Enzyme rules using EnzymeRules, it is recommended to add EnzymeTestUtils as a test dependency to test the rules.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"This page was generated using Literate.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = Enzyme\nDocTestSetup = quote\n using Enzyme\nend","category":"page"},{"location":"#Enzyme","page":"Home","title":"Enzyme","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Documentation for Enzyme.jl, the Julia bindings for Enzyme.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Enzyme performs automatic differentiation (AD) of statically analyzable LLVM. It is highly-efficient and its ability to perform AD on optimized code allows Enzyme to meet or exceed the performance of state-of-the-art AD tools.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Enzyme.jl can be installed in the usual way Julia packages are installed:","category":"page"},{"location":"","page":"Home","title":"Home","text":"] add Enzyme","category":"page"},{"location":"","page":"Home","title":"Home","text":"The Enzyme binary dependencies will be installed automatically via Julia's binary artifact system.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The Enzyme.jl API revolves around the function autodiff. For some common operations, Enzyme additionally wraps autodiff in several convenience functions; e.g., gradient and jacobian.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The tutorial below covers the basic usage of these functions. For a complete overview of Enzyme's functionality, see the API documentation. Also see Implementing pullbacks on how to implement back-propagation for functions with non-scalar results.","category":"page"},{"location":"#Getting-started","page":"Home","title":"Getting started","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"julia> rosenbrock(x, y) = (1.0 - x)^2 + 100.0 * (y - x^2)^2\nrosenbrock (generic function with 1 method)\n\njulia> rosenbrock_inp(x) = (1.0 - x[1])^2 + 100.0 * (x[2] - x[1]^2)^2\nrosenbrock_inp (generic function with 1 method)","category":"page"},{"location":"#Reverse-mode","page":"Home","title":"Reverse mode","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"The return value of reverse mode autodiff is a tuple that contains as a first value the derivative value of the active inputs and optionally the primal return value.","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> autodiff(Reverse, rosenbrock, Active, Active(1.0), Active(2.0))\n((-400.0, 200.0),)\n\njulia> autodiff(ReverseWithPrimal, rosenbrock, Active, Active(1.0), Active(2.0))\n((-400.0, 200.0), 100.0)","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> x = [1.0, 2.0]\n2-element Vector{Float64}:\n 1.0\n 2.0\n\njulia> dx = [0.0, 0.0]\n2-element Vector{Float64}:\n 0.0\n 0.0\n\njulia> autodiff(Reverse, rosenbrock_inp, Active, Duplicated(x, dx))\n((nothing,),)\n\njulia> dx\n2-element Vector{Float64}:\n -400.0\n 200.0","category":"page"},{"location":"","page":"Home","title":"Home","text":"Both the inplace and \"normal\" variant return the gradient. The difference is that with Active the gradient is returned and with Duplicated the gradient is accumulated in place.","category":"page"},{"location":"#Forward-mode","page":"Home","title":"Forward mode","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"The return value of forward mode with a Duplicated return is a tuple containing as the first value the primal return value and as the second value the derivative.","category":"page"},{"location":"","page":"Home","title":"Home","text":"In forward mode Duplicated(x, 0.0) is equivalent to Const(x), except that we can perform more optimizations for Const.","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> autodiff(Forward, rosenbrock, Duplicated, Const(1.0), Duplicated(3.0, 1.0))\n(400.0, 400.0)\n\njulia> autodiff(Forward, rosenbrock, Duplicated, Duplicated(1.0, 1.0), Const(3.0))\n(400.0, -800.0)","category":"page"},{"location":"","page":"Home","title":"Home","text":"Of note, when we seed both arguments at once the tangent return is the sum of both.","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> autodiff(Forward, rosenbrock, Duplicated, Duplicated(1.0, 1.0), Duplicated(3.0, 1.0))\n(400.0, -400.0)","category":"page"},{"location":"","page":"Home","title":"Home","text":"We can also use forward mode with our inplace method.","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> x = [1.0, 3.0]\n2-element Vector{Float64}:\n 1.0\n 3.0\n\njulia> dx = [1.0, 1.0]\n2-element Vector{Float64}:\n 1.0\n 1.0\n\njulia> autodiff(Forward, rosenbrock_inp, Duplicated, Duplicated(x, dx))\n(400.0, -400.0)","category":"page"},{"location":"","page":"Home","title":"Home","text":"Note the seeding through dx.","category":"page"},{"location":"#Vector-forward-mode","page":"Home","title":"Vector forward mode","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"We can also use vector mode to calculate both derivatives at once.","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> autodiff(Forward, rosenbrock, BatchDuplicated, BatchDuplicated(1.0, (1.0, 0.0)), BatchDuplicated(3.0, (0.0, 1.0)))\n(400.0, (var\"1\" = -800.0, var\"2\" = 400.0))\n\njulia> x = [1.0, 3.0]\n2-element Vector{Float64}:\n 1.0\n 3.0\n\njulia> dx_1 = [1.0, 0.0]; dx_2 = [0.0, 1.0];\n\njulia> autodiff(Forward, rosenbrock_inp, BatchDuplicated, BatchDuplicated(x, (dx_1, dx_2)))\n(400.0, (var\"1\" = -800.0, var\"2\" = 400.0))","category":"page"},{"location":"#Convenience-functions","page":"Home","title":"Convenience functions","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"note: Note\nWhile the convenience functions discussed below use autodiff internally, they are generally more limited in their functionality. Beyond that, these convenience functions may also come with performance penalties; especially if one makes a closure of a multi-argument function instead of calling the appropriate multi-argument autodiff function directly.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Key convenience functions for common derivative computations are gradient (and its inplace variant gradient!) and jacobian. Like autodiff, the mode (forward or reverse) is determined by the first argument.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The functions gradient and gradient! compute the gradient of function with vector input and scalar return.","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> gradient(Reverse, rosenbrock_inp, [1.0, 2.0])\n2-element Vector{Float64}:\n -400.0\n 200.0\n\njulia> # inplace variant\n dx = [0.0, 0.0];\n gradient!(Reverse, dx, rosenbrock_inp, [1.0, 2.0])\n2-element Vector{Float64}:\n -400.0\n 200.0\n\njulia> dx\n2-element Vector{Float64}:\n -400.0\n 200.0\n\njulia> gradient(Forward, rosenbrock_inp, [1.0, 2.0])\n(-400.0, 200.0)\n\njulia> # in forward mode, we can also optionally pass a chunk size\n # to specify the number of derivatives computed simulateneously\n # using vector forward mode\n chunk_size = Val(2)\n gradient(Forward, rosenbrock_inp, [1.0, 2.0], chunk_size)\n(-400.0, 200.0)","category":"page"},{"location":"","page":"Home","title":"Home","text":"The function jacobian computes the Jacobian of a function vector input and vector return.","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> foo(x) = [rosenbrock_inp(x), prod(x)];\n\njulia> output_size = Val(2) # here we have to provide the output size of `foo` since it cannot be statically inferred\n jacobian(Reverse, foo, [1.0, 2.0], output_size) \n2×2 Matrix{Float64}:\n -400.0 200.0\n 2.0 1.0\n\njulia> chunk_size = Val(2) # By specifying the optional chunk size argument, we can use vector inverse mode to propogate derivatives of multiple outputs at once.\n jacobian(Reverse, foo, [1.0, 2.0], output_size, chunk_size)\n2×2 Matrix{Float64}:\n -400.0 200.0\n 2.0 1.0\n\njulia> jacobian(Forward, foo, [1.0, 2.0])\n2×2 Matrix{Float64}:\n -400.0 200.0\n 2.0 1.0\n\njulia> # Again, the optinal chunk size argument allows us to use vector forward mode\n jacobian(Forward, foo, [1.0, 2.0], chunk_size)\n2×2 Matrix{Float64}:\n -400.0 200.0\n 2.0 1.0","category":"page"},{"location":"#Caveats-/-Known-issues","page":"Home","title":"Caveats / Known-issues","text":"","category":"section"},{"location":"#Activity-of-temporary-storage","page":"Home","title":"Activity of temporary storage","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"If you pass in any temporary storage which may be involved in an active computation to a function you want to differentiate, you must also pass in a duplicated temporary storage for use in computing the derivatives. ","category":"page"},{"location":"","page":"Home","title":"Home","text":"function f(x, tmp, k, n)\n tmp[1] = 1.0\n for i in 1:n\n tmp[k] *= x\n end\n tmp[1]\nend\n\n# output\n\nf (generic function with 1 method)","category":"page"},{"location":"","page":"Home","title":"Home","text":"Enzyme.autodiff(Reverse, f, Active(1.2), Const(Vector{Float64}(undef, 1)), Const(1), Const(5)) # Incorrect\n\n# output\n\n((0.0, nothing, nothing, nothing),)","category":"page"},{"location":"","page":"Home","title":"Home","text":"Enzyme.autodiff(Reverse, f, Active(1.2), Duplicated(Vector{Float64}(undef, 1), Vector{Float64}(undef, 1)), Const(1), Const(5)) # Correct (returns 10.367999999999999 == 1.2^4 * 5)\n\n# output\n\n((10.367999999999999, nothing, nothing, nothing),)","category":"page"},{"location":"#CUDA.jl-support","page":"Home","title":"CUDA.jl support","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"CUDA.jl is only supported on Julia v1.7.0 and onwards. On v1.6, attempting to differentiate CUDA kernel functions will not use device overloads correctly and thus returns fundamentally wrong results.","category":"page"},{"location":"#Sparse-Arrays","page":"Home","title":"Sparse Arrays","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"At the moment there is limited support for sparse linear algebra operations. Sparse arrays may be used, but care must be taken because backing arrays drop zeros in Julia (unless told not to).","category":"page"},{"location":"","page":"Home","title":"Home","text":"using SparseArrays\na = sparse([2.0])\nda1 = sparse([0.0]) # Incorrect: SparseMatrixCSC drops explicit zeros\nEnzyme.autodiff(Reverse, sum, Active, Duplicated(a, da1))\nda1\n\n# output\n\n1-element SparseVector{Float64, Int64} with 0 stored entries","category":"page"},{"location":"","page":"Home","title":"Home","text":"da2 = sparsevec([1], [0.0]) # Correct: Prevent SparseMatrixCSC from dropping zeros\nEnzyme.autodiff(Reverse, sum, Active, Duplicated(a, da2))\nda2\n\n# output\n\n1-element SparseVector{Float64, Int64} with 1 stored entry:\n [1] = 1.0","category":"page"},{"location":"","page":"Home","title":"Home","text":"Sometimes, determining how to perform this zeroing can be complicated. That is why Enzyme provides a helper function Enzyme.make_zero that does this automatically.","category":"page"}] +[{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"EditURL = \"../../../examples/autodiff.jl\"","category":"page"},{"location":"generated/autodiff/#AutoDiff-API","page":"AutoDiff API","title":"AutoDiff API","text":"","category":"section"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"The goal of this tutorial is to give users already familiar with automatic differentiation (AD) an overview of the Enzyme differentiation API for the following differentiation modes","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"Reverse mode\nForward mode\nForward over reverse mode\nVector Forward over reverse mode","category":"page"},{"location":"generated/autodiff/#Defining-a-function","page":"AutoDiff API","title":"Defining a function","text":"","category":"section"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"Enzyme differentiates arbitrary multivariate vector functions as the most general case in automatic differentiation","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"f mathbbR^n rightarrow mathbbR^m y = f(x)","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"For simplicity we define a vector function with m=1. However, this tutorial can easily be applied to arbitrary m in mathbbN.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"using Enzyme\n\nfunction f(x::Array{Float64}, y::Array{Float64})\n y[1] = x[1] * x[1] + x[2] * x[1]\n return nothing\nend;\nnothing #hide","category":"page"},{"location":"generated/autodiff/#Reverse-mode","page":"AutoDiff API","title":"Reverse mode","text":"","category":"section"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"The reverse model in AD is defined as","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"beginaligned\ny = f(x) \nbarx = bary cdot nabla f(x)\nendaligned","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"bar denotes an adjoint variable. Note that executing an AD in reverse mode computes both y and the adjoint barx.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"x = [2.0, 2.0]\nbx = [0.0, 0.0]\ny = [0.0]\nby = [1.0];\nnothing #hide","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"Enzyme stores the value and adjoint of a variable in an object of type Duplicated where the first element represent the value and the second the adjoint. Evaluating the reverse model using Enzyme is done via the following call.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"Enzyme.autodiff(Reverse, f, Duplicated(x, bx), Duplicated(y, by));\nnothing #hide","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"This yields the gradient of f in bx at point x = [2.0, 2.0]. by is called the seed and has to be set to 10 in order to compute the gradient. Let's save the gradient for later.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"g = copy(bx)","category":"page"},{"location":"generated/autodiff/#Forward-mode","page":"AutoDiff API","title":"Forward mode","text":"","category":"section"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"The forward model in AD is defined as","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"beginaligned\ny = f(x) \ndoty = nabla f(x) cdot dotx\nendaligned","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"To obtain the first element of the gradient using the forward model we have to seed dotx with dotx = 1000","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"x = [2.0, 2.0]\ndx = [1.0, 0.0]\ny = [0.0]\ndy = [0.0];\nnothing #hide","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"In the forward mode the second element of Duplicated stores the tangent.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"Enzyme.autodiff(Forward, f, Duplicated(x, dx), Duplicated(y, dy));\nnothing #hide","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"We can now verify that indeed the reverse mode and forward mode yield the same result for the first component of the gradient. Note that to acquire the full gradient one needs to execute the forward model a second time with the seed dx set to [0.0,1.0].","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"Let's verify whether the reverse and forward model agree.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"g[1] == dy[1]","category":"page"},{"location":"generated/autodiff/#Forward-over-reverse","page":"AutoDiff API","title":"Forward over reverse","text":"","category":"section"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"The forward over reverse (FoR) model is obtained by applying the forward model to the reverse model using the chain rule for the product in the adjoint statement.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"beginaligned\ny = f(x) \ndoty = nabla f(x) cdot dotx \nbarx = bary cdot nabla f(x) \ndotbarx = bary cdot nabla^2 f(x) cdot dotx + dotbary cdot nabla f(x)\nendaligned","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"To obtain the first column/row of the Hessian nabla^2 f(x) we have to seed dotbary with 00, bary with 10 and dotx with 10 00.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"y = [0.0]\nx = [2.0, 2.0]\n\ndy = [0.0]\ndx = [1.0, 0.0]\n\nbx = [0.0, 0.0]\nby = [1.0]\ndbx = [0.0, 0.0]\ndby = [0.0]\n\nEnzyme.autodiff(\n Forward,\n (x,y) -> Enzyme.autodiff_deferred(Reverse, f, x, y),\n Duplicated(Duplicated(x, bx), Duplicated(dx, dbx)),\n Duplicated(Duplicated(y, by), Duplicated(dy, dby)),\n)","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"The FoR model also computes the forward model from before, giving us again the first component of the gradient.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"g[1] == dy[1]","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"In addition we now have the first row/column of the Hessian.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"dbx[1] == 2.0\ndbx[2] == 1.0","category":"page"},{"location":"generated/autodiff/#Vector-forward-over-reverse","page":"AutoDiff API","title":"Vector forward over reverse","text":"","category":"section"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"The vector FoR allows us to propagate several tangents at once through the second-order model by computing the derivative of the gradient at multiple points at once. We begin by defining a helper function for the gradient. Since we will not need the original results (stored in y), we can mark it DuplicatedNoNeed. Specifically, this will perform the following:","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"beginaligned\nbarx = barx + bary cdot nabla f(x) \nbary = 0\nendaligned","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"function grad(x, dx, y, dy)\n Enzyme.autodiff_deferred(Reverse, f, Duplicated(x, dx), DuplicatedNoNeed(y, dy))\n nothing\nend","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"To compute the conventional gradient, we would call this function with our given inputs, dy = [1.0], and dx = [0.0, 0.0]. Since y is not needed, we can just set it to an undef vector.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"x = [2.0, 2.0]\ny = Vector{Float64}(undef, 1)\ndx = [0.0, 0.0]\ndy = [1.0]\n\ngrad(x, dx, y, dy)","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"dx now contains the gradient","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"@show dx","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"To compute the hessian, we need to take the dervative of this gradient function at every input. Following the same seeding strategy as before, we now seed both in the vx[1]=[1.0, 0.0] and vx[2]=[0.0, 1.0] direction. These tuples have to be put into a BatchDuplicated type. We then compute the forward mode derivative at all these points.","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"vx = ([1.0, 0.0], [0.0, 1.0])\nhess = ([0.0, 0.0], [0.0, 0.0])\ndx = [0.0, 0.0]\ndy = [1.0]\n\nEnzyme.autodiff(Enzyme.Forward, grad,\n Enzyme.BatchDuplicated(x, vx),\n Enzyme.BatchDuplicated(dx, hess),\n Const(y),\n Const(dy))","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"Again we obtain the first-order gradient. If we did not want to compute the gradient again, we could instead have used Enzyme.BatchDuplicatedNoNeed(dx, hess)","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"g[1] == dx[1]","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"We have now the first row/column of the Hessian","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"hess[1][1] == 2.0\n\nhess[1][2] == 1.0","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"as well as the second row/column","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"hess[2][1] == 1.0\n\nhess[2][2] == 0.0","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"","category":"page"},{"location":"generated/autodiff/","page":"AutoDiff API","title":"AutoDiff API","text":"This page was generated using Literate.jl.","category":"page"},{"location":"api/#API","page":"API","title":"API","text":"","category":"section"},{"location":"api/#Types-and-constants","page":"API","title":"Types and constants","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"Modules = [Enzyme, EnzymeCore, EnzymeCore.EnzymeRules, EnzymeTestUtils, Enzyme.API]\nOrder = [:type, :constant]","category":"page"},{"location":"api/#Functions-and-macros","page":"API","title":"Functions and macros","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"Modules = [Enzyme, EnzymeCore, EnzymeCore.EnzymeRules, EnzymeTestUtils, Enzyme.API]\nOrder = [:macro, :function]","category":"page"},{"location":"api/#Documentation","page":"API","title":"Documentation","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"Modules = [Enzyme, EnzymeCore, EnzymeCore.EnzymeRules, EnzymeTestUtils, Enzyme.API]\nOrder = [:module, :type, :constant, :macro, :function]","category":"page"},{"location":"api/#Enzyme.gradient!-Tuple{ReverseMode, Any, Any, Any}","page":"API","title":"Enzyme.gradient!","text":"gradient!(::ReverseMode, dx, f, x)\n\nCompute the gradient of an array-input function f using reverse mode, storing the derivative result in an existing array dx.\n\nExample:\n\nf(x) = x[1]*x[2]\n\ndx = [0.0, 0.0]\ngradient!(Reverse, dx, f, [2.0, 3.0])\n\n# output\n\n2-element Vector{Float64}:\n 3.0\n 2.0\n\n\n\n\n\n","category":"method"},{"location":"api/#Enzyme.gradient-Tuple{ForwardMode, Any, Any}","page":"API","title":"Enzyme.gradient","text":"gradient(::ForwardMode, f, x; shadow=onehot(x))\n\nCompute the gradient of an array-input function f using forward mode. The optional keyword argument shadow is a vector of one-hot vectors of type x which are used to forward-propagate into the return. For performance reasons, this should be computed once, outside the call to gradient, rather than within this call.\n\nExample:\n\nf(x) = x[1]*x[2]\n\ngrad = gradient(Forward, f, [2.0, 3.0])\n\n# output\n\n(3.0, 2.0)\n\n\n\n\n\n","category":"method"},{"location":"api/#Enzyme.gradient-Tuple{ReverseMode, Any, Any}","page":"API","title":"Enzyme.gradient","text":"gradient(::ReverseMode, f, x)\n\nCompute the gradient of an array-input function f using reverse mode. This will allocate and return new array with the gradient result.\n\nExample:\n\nf(x) = x[1]*x[2]\n\ngrad = gradient(Reverse, f, [2.0, 3.0])\n\n# output\n\n2-element Vector{Float64}:\n 3.0\n 2.0\n\n\n\n\n\n","category":"method"},{"location":"api/#Enzyme.gradient-Union{Tuple{chunk}, Tuple{X}, Tuple{F}, Tuple{ForwardMode, F, X, Val{chunk}}} where {F, X, chunk}","page":"API","title":"Enzyme.gradient","text":"gradient(::ForwardMode, f, x, ::Val{chunk}; shadow=onehot(x))\n\nCompute the gradient of an array-input function f using vector forward mode. Like gradient, except it uses a chunk size of chunk to compute chunk derivatives in a single call.\n\nExample:\n\nf(x) = x[1]*x[2]\n\ngrad = gradient(Forward, f, [2.0, 3.0], Val(2))\n\n# output\n\n(3.0, 2.0)\n\n\n\n\n\n","category":"method"},{"location":"api/#Enzyme.jacobian-Tuple{ForwardMode, Any, Any}","page":"API","title":"Enzyme.jacobian","text":"jacobian(::ForwardMode, f, x; shadow=onehot(x))\njacobian(::ForwardMode, f, x, ::Val{chunk}; shadow=onehot(x))\n\nCompute the jacobian of an array-input function f using (potentially vector) forward mode. This is a simple rename of the gradient function, and all relevant arguments apply here.\n\nExample:\n\nf(x) = [x[1]*x[2], x[2]]\n\ngrad = jacobian(Forward, f, [2.0, 3.0])\n\n# output\n\n2×2 Matrix{Float64}:\n 3.0 2.0\n 0.0 1.0\n\n\n\n\n\n","category":"method"},{"location":"api/#Enzyme.jacobian-Union{Tuple{RABI}, Tuple{ReturnPrimal}, Tuple{n_out_val}, Tuple{chunk}, Tuple{X}, Tuple{F}, Tuple{ReverseMode{ReturnPrimal, RABI}, F, X, Val{n_out_val}, Val{chunk}}} where {F, X, chunk, n_out_val, ReturnPrimal, RABI<:EnzymeCore.ABI}","page":"API","title":"Enzyme.jacobian","text":"jacobian(::ReverseMode, f, x, ::Val{num_outs}, ::Val{chunk})\n\nCompute the jacobian of an array-input function f using (potentially vector) reverse mode. The chunk argument denotes the chunk size to use and num_outs denotes the number of outputs f will return in an array.\n\nExample:\n\nf(x) = [x[1]*x[2], x[2]]\n\ngrad = jacobian(Reverse, f, [2.0, 3.0], Val(2))\n\n# output\n\n2×2 Matrix{Float64}:\n 3.0 2.0\n 0.0 1.0\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff-Union{Tuple{A}, Tuple{FA}, Tuple{Nargs}, Tuple{RABI}, Tuple{ForwardMode{RABI}, FA, Type{A}, Vararg{Annotation, Nargs}}} where {RABI<:EnzymeCore.ABI, Nargs, FA<:Annotation, A<:Annotation}","page":"API","title":"EnzymeCore.autodiff","text":"autodiff(::ForwardMode, f, Activity, args::Vararg{Annotation, Nargs})\n\nAuto-differentiate function f at arguments args using forward mode.\n\nargs may be numbers, arrays, structs of numbers, structs of arrays and so on. Enzyme will only differentiate in respect to arguments that are wrapped in a Duplicated or similar argument. Unlike reverse mode in autodiff, Active arguments are not allowed here, since all derivative results of immutable objects will be returned and should instead use Duplicated or variants like DuplicatedNoNeed.\n\nActivity is the Activity of the return value, it may be:\n\nConst if the return is not to be differentiated with respect to\nDuplicated, if the return is being differentiated with respect to and both the original value and the derivative return are desired\nDuplicatedNoNeed, if the return is being differentiated with respect to and only the derivative return is desired.\nBatchDuplicated, like Duplicated, but computing multiple derivatives at once. All batch sizes must be the same for all arguments.\nBatchDuplicatedNoNeed, like DuplicatedNoNeed, but computing multiple derivatives at one. All batch sizes must be the same for all arguments.\n\nExample returning both original return and derivative:\n\nf(x) = x*x\nres, ∂f_∂x = autodiff(Forward, f, Duplicated, Duplicated(3.14, 1.0))\n\n# output\n\n(9.8596, 6.28)\n\nExample returning just the derivative:\n\nf(x) = x*x\n∂f_∂x = autodiff(Forward, f, DuplicatedNoNeed, Duplicated(3.14, 1.0))\n\n# output\n\n(6.28,)\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff-Union{Tuple{Holomorphic}, Tuple{Nargs}, Tuple{RABI}, Tuple{ReturnPrimal}, Tuple{A}, Tuple{FA}, Tuple{ReverseMode{ReturnPrimal, RABI, Holomorphic}, FA, Type{A}, Vararg{Annotation, Nargs}}} where {FA<:Annotation, A<:Annotation, ReturnPrimal, RABI<:EnzymeCore.ABI, Nargs, Holomorphic}","page":"API","title":"EnzymeCore.autodiff","text":"autodiff(::ReverseMode, f, Activity, args::Vararg{Annotation, Nargs})\n\nAuto-differentiate function f at arguments args using reverse mode.\n\nLimitations:\n\nf may only return a Real (of a built-in/primitive type) or nothing, not an array, struct, BigFloat, etc. To handle vector-valued return types, use a mutating f! that returns nothing and stores it's return value in one of the arguments, which must be wrapped in a Duplicated.\n\nargs may be numbers, arrays, structs of numbers, structs of arrays and so on. Enzyme will only differentiate in respect to arguments that are wrapped in an Active (for arguments whose derivative result must be returned rather than mutated in place, such as primitive types and structs thereof) or Duplicated (for mutable arguments like arrays, Refs and structs thereof).\n\nActivity is the Activity of the return value, it may be Const or Active.\n\nExample:\n\na = 4.2\nb = [2.2, 3.3]; ∂f_∂b = zero(b)\nc = 55; d = 9\n\nf(a, b, c, d) = a * √(b[1]^2 + b[2]^2) + c^2 * d^2\n∂f_∂a, _, _, ∂f_∂d = autodiff(Reverse, f, Active, Active(a), Duplicated(b, ∂f_∂b), Const(c), Active(d))[1]\n\n# output\n\n(3.966106403010388, nothing, nothing, 54450.0)\n\nhere, autodiff returns a tuple (partial fpartial a partial fpartial d), while partial fpartial b will be added to ∂f_∂b (but not returned). c will be treated as Const(c).\n\nOne can also request the original returned value of the computation.\n\nExample:\n\nEnzyme.autodiff(ReverseWithPrimal, x->x*x, Active(3.0))\n\n# output\n\n((6.0,), 9.0)\n\nnote: Note\nEnzyme gradients with respect to integer values are zero. Active will automatically convert plain integers to floating point values, but cannot do so for integer values in tuples and structs.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff-Union{Tuple{Nargs}, Tuple{CMode}, Tuple{FA}, Tuple{CMode, FA, Vararg{Annotation, Nargs}}} where {FA<:Annotation, CMode<:EnzymeCore.Mode, Nargs}","page":"API","title":"EnzymeCore.autodiff","text":"autodiff(mode::Mode, f, args::Vararg{Annotation, Nargs})\n\nLike autodiff but will try to guess the activity of the return value.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff-Union{Tuple{Nargs}, Tuple{CMode}, Tuple{F}, Tuple{CMode, F, Vararg{Annotation, Nargs}}} where {F, CMode<:EnzymeCore.Mode, Nargs}","page":"API","title":"EnzymeCore.autodiff","text":"autodiff(mode::Mode, f, ::Type{A}, args::Vararg{Annotation, Nargs})\n\nLike autodiff but will try to extend f to an annotation, if needed.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff_deferred-Union{Tuple{Nargs}, Tuple{A}, Tuple{FA}, Tuple{ForwardMode, FA, Type{A}, Vararg{Annotation, Nargs}}} where {FA<:Annotation, A<:Annotation, Nargs}","page":"API","title":"EnzymeCore.autodiff_deferred","text":"autodiff_deferred(::ForwardMode, f, Activity, args::Vararg{Annotation, Nargs})\n\nSame as autodiff(::ForwardMode, f, Activity, args) but uses deferred compilation to support usage in GPU code, as well as high-order differentiation.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff_deferred-Union{Tuple{Nargs}, Tuple{CMode}, Tuple{F}, Tuple{CMode, F, Vararg{Annotation, Nargs}}} where {F, CMode<:EnzymeCore.Mode, Nargs}","page":"API","title":"EnzymeCore.autodiff_deferred","text":"autodiff_deferred(mode::Mode, f, ::Type{A}, args::Vararg{Annotation, Nargs})\n\nLike autodiff_deferred but will try to extend f to an annotation, if needed.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff_deferred-Union{Tuple{Nargs}, Tuple{ReturnPrimal}, Tuple{A}, Tuple{FA}, Tuple{ReverseMode{ReturnPrimal}, FA, Type{A}, Vararg{Annotation, Nargs}}} where {FA<:Annotation, A<:Annotation, ReturnPrimal, Nargs}","page":"API","title":"EnzymeCore.autodiff_deferred","text":"autodiff_deferred(::ReverseMode, f, Activity, args::Vararg{Annotation, Nargs})\n\nSame as autodiff but uses deferred compilation to support usage in GPU code, as well as high-order differentiation.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff_deferred_thunk-Union{Tuple{Nargs}, Tuple{RABI}, Tuple{ModifiedBetweenT}, Tuple{Width}, Tuple{ReturnShadow}, Tuple{ReturnPrimal}, Tuple{TapeType}, Tuple{A2}, Tuple{A}, Tuple{FA}, Tuple{EnzymeCore.ReverseModeSplit{ReturnPrimal, ReturnShadow, Width, ModifiedBetweenT, RABI}, Type{TapeType}, Type{FA}, Type{A}, Type{A2}, Vararg{Type{<:Annotation}, Nargs}}} where {FA<:Annotation, A<:Annotation, A2, TapeType, ReturnPrimal, ReturnShadow, Width, ModifiedBetweenT, RABI<:EnzymeCore.ABI, Nargs}","page":"API","title":"EnzymeCore.autodiff_deferred_thunk","text":"autodiff_deferred_thunk(::ReverseModeSplit, ftype, Activity, argtypes::Vararg{Type{<:Annotation}, Nargs})\n\nProvide the split forward and reverse pass functions for annotated function type ftype when called with args of type argtypes when using reverse mode.\n\nActivity is the Activity of the return value, it may be Const, Active, or Duplicated (or its variants DuplicatedNoNeed, BatchDuplicated, and BatchDuplicatedNoNeed).\n\nThe forward function will return a tape, the primal (or nothing if not requested), and the shadow (or nothing if not a Duplicated variant), and tapes the corresponding type arguements provided.\n\nThe reverse function will return the derivative of Active arguments, updating the Duplicated arguments in place. The same arguments to the forward pass should be provided, followed by the adjoint of the return (if the return is active), and finally the tape from the forward pass.\n\nExample:\n\n\nA = [2.2]; ∂A = zero(A)\nv = 3.3\n\nfunction f(A, v)\n res = A[1] * v\n A[1] = 0\n res\nend\n\nTapeType = tape_type(ReverseSplitWithPrimal, Const{typeof(f)}, Active, Duplicated{typeof(A)}, Active{typeof(v)})\nforward, reverse = autodiff_deferred_thunk(ReverseSplitWithPrimal, TapeType, Const{typeof(f)}, Active, Active{Float64}, Duplicated{typeof(A)}, Active{typeof(v)})\n\ntape, result, shadow_result = forward(Const(f), Duplicated(A, ∂A), Active(v))\n_, ∂v = reverse(Const(f), Duplicated(A, ∂A), Active(v), 1.0, tape)[1]\n\nresult, ∂v, ∂A \n\n# output\n\n(7.26, 2.2, [3.3])\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff_thunk-Union{Tuple{Nargs}, Tuple{RABI}, Tuple{A}, Tuple{FA}, Tuple{ForwardMode{RABI}, Type{FA}, Type{A}, Vararg{Type{<:Annotation}, Nargs}}} where {FA<:Annotation, A<:Annotation, RABI<:EnzymeCore.ABI, Nargs}","page":"API","title":"EnzymeCore.autodiff_thunk","text":"autodiff_thunk(::ForwardMode, ftype, Activity, argtypes::Vararg{Type{<:Annotation}, Nargs})\n\nProvide the thunk forward mode function for annotated function type ftype when called with args of type argtypes.\n\nActivity is the Activity of the return value, it may be Const or Duplicated (or its variants DuplicatedNoNeed, BatchDuplicated, andBatchDuplicatedNoNeed).\n\nThe forward function will return the primal (if requested) and the shadow (or nothing if not a Duplicated variant).\n\nExample returning both original return and derivative:\n\na = 4.2\nb = [2.2, 3.3]; ∂f_∂b = zero(b)\nc = 55; d = 9\n\nf(x) = x*x\nforward = autodiff_thunk(Forward, Const{typeof(f)}, Duplicated, Duplicated{Float64})\nres, ∂f_∂x = forward(Const(f), Duplicated(3.14, 1.0))\n\n# output\n\n(9.8596, 6.28)\n\nExample returning just the derivative:\n\na = 4.2\nb = [2.2, 3.3]; ∂f_∂b = zero(b)\nc = 55; d = 9\n\nf(x) = x*x\nforward = autodiff_thunk(Forward, Const{typeof(f)}, DuplicatedNoNeed, Duplicated{Float64})\n∂f_∂x = forward(Const(f), Duplicated(3.14, 1.0))\n\n# output\n\n(6.28,)\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.autodiff_thunk-Union{Tuple{Nargs}, Tuple{RABI}, Tuple{ModifiedBetweenT}, Tuple{Width}, Tuple{ReturnShadow}, Tuple{ReturnPrimal}, Tuple{A}, Tuple{FA}, Tuple{EnzymeCore.ReverseModeSplit{ReturnPrimal, ReturnShadow, Width, ModifiedBetweenT, RABI}, Type{FA}, Type{A}, Vararg{Type{<:Annotation}, Nargs}}} where {FA<:Annotation, A<:Annotation, ReturnPrimal, ReturnShadow, Width, ModifiedBetweenT, RABI<:EnzymeCore.ABI, Nargs}","page":"API","title":"EnzymeCore.autodiff_thunk","text":"autodiff_thunk(::ReverseModeSplit, ftype, Activity, argtypes::Vararg{Type{<:Annotation}, Nargs})\n\nProvide the split forward and reverse pass functions for annotated function type ftype when called with args of type argtypes when using reverse mode.\n\nActivity is the Activity of the return value, it may be Const, Active, or Duplicated (or its variants DuplicatedNoNeed, BatchDuplicated, and BatchDuplicatedNoNeed).\n\nThe forward function will return a tape, the primal (or nothing if not requested), and the shadow (or nothing if not a Duplicated variant), and tapes the corresponding type arguements provided.\n\nThe reverse function will return the derivative of Active arguments, updating the Duplicated arguments in place. The same arguments to the forward pass should be provided, followed by the adjoint of the return (if the return is active), and finally the tape from the forward pass.\n\nExample:\n\n\nA = [2.2]; ∂A = zero(A)\nv = 3.3\n\nfunction f(A, v)\n res = A[1] * v\n A[1] = 0\n res\nend\n\nforward, reverse = autodiff_thunk(ReverseSplitWithPrimal, Const{typeof(f)}, Active, Duplicated{typeof(A)}, Active{typeof(v)})\n\ntape, result, shadow_result = forward(Const(f), Duplicated(A, ∂A), Active(v))\n_, ∂v = reverse(Const(f), Duplicated(A, ∂A), Active(v), 1.0, tape)[1]\n\nresult, ∂v, ∂A \n\n# output\n\n(7.26, 2.2, [3.3])\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.ABI","page":"API","title":"EnzymeCore.ABI","text":"abstract type ABI\n\nAbstract type for what ABI will be used.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.Active","page":"API","title":"EnzymeCore.Active","text":"Active(x)\n\nMark a function argument x of autodiff as active, Enzyme will auto-differentiate in respect Active arguments.\n\nnote: Note\nEnzyme gradients with respect to integer values are zero. Active will automatically convert plain integers to floating point values, but cannot do so for integer values in tuples and structs.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.Annotation","page":"API","title":"EnzymeCore.Annotation","text":"abstract type Annotation{T}\n\nAbstract type for autodiff function argument wrappers like Const, Active and Duplicated.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.BatchDuplicated","page":"API","title":"EnzymeCore.BatchDuplicated","text":"BatchDuplicated(x, ∂f_∂xs)\n\nLike Duplicated, except contains several shadows to compute derivatives for all at once. Argument ∂f_∂xs should be a tuple of the several values of type x.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.BatchDuplicatedNoNeed","page":"API","title":"EnzymeCore.BatchDuplicatedNoNeed","text":"BatchDuplicatedNoNeed(x, ∂f_∂xs)\n\nLike DuplicatedNoNeed, except contains several shadows to compute derivatives for all at once. Argument ∂f_∂xs should be a tuple of the several values of type x.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.Const","page":"API","title":"EnzymeCore.Const","text":"Const(x)\n\nMark a function argument x of autodiff as constant, Enzyme will not auto-differentiate in respect Const arguments.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.Duplicated","page":"API","title":"EnzymeCore.Duplicated","text":"Duplicated(x, ∂f_∂x)\n\nMark a function argument x of autodiff as duplicated, Enzyme will auto-differentiate in respect to such arguments, with dx acting as an accumulator for gradients (so partial f partial x will be added to) ∂f_∂x.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.DuplicatedNoNeed","page":"API","title":"EnzymeCore.DuplicatedNoNeed","text":"DuplicatedNoNeed(x, ∂f_∂x)\n\nLike Duplicated, except also specifies that Enzyme may avoid computing the original result and only compute the derivative values.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.FFIABI","page":"API","title":"EnzymeCore.FFIABI","text":"struct FFIABI <: ABI\n\nForeign function call ABI. JIT the differentiated function, then inttoptr call the address.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.ForwardMode","page":"API","title":"EnzymeCore.ForwardMode","text":"struct Forward <: Mode\n\nForward mode differentiation\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.InlineABI","page":"API","title":"EnzymeCore.InlineABI","text":"struct InlineABI <: ABI\n\nInlining function call ABI. \n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.Mode","page":"API","title":"EnzymeCore.Mode","text":"abstract type Mode\n\nAbstract type for what differentiation mode will be used.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.ReverseMode","page":"API","title":"EnzymeCore.ReverseMode","text":"struct ReverseMode{ReturnPrimal,ABI,Holomorphic} <: Mode{ABI}\n\nReverse mode differentiation.\n\nReturnPrimal: Should Enzyme return the primal return value from the augmented-forward.\nABI: What runtime ABI to use\nHolomorphic: Whether the complex result function is holomorphic and we should compute d/dz\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.ReverseModeSplit","page":"API","title":"EnzymeCore.ReverseModeSplit","text":"struct ReverseModeSplit{ReturnPrimal,ReturnShadow,Width,ModifiedBetween,ABI} <: Mode{ABI}\n\nReverse mode differentiation.\n\nReturnPrimal: Should Enzyme return the primal return value from the augmented-forward.\nReturnShadow: Should Enzyme return the shadow return value from the augmented-forward.\nWidth: Batch Size (0 if to be automatically derived)\nModifiedBetween: Tuple of each argument's modified between state (true if to be automatically derived).\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.compiler_job_from_backend","page":"API","title":"EnzymeCore.compiler_job_from_backend","text":"compiler_job_from_backend(::KernelAbstractions.Backend, F::Type, TT:Type)::GPUCompiler.CompilerJob\n\nReturns a GPUCompiler CompilerJob from a backend as specified by the first argument to the function.\n\nFor example, in CUDA one would do:\n\nfunction EnzymeCore.compiler_job_from_backend(::CUDABackend, @nospecialize(F::Type), @nospecialize(TT::Type))\n mi = GPUCompiler.methodinstance(F, TT)\n return GPUCompiler.CompilerJob(mi, CUDA.compiler_config(CUDA.device()))\nend\n\n\n\n\n\n","category":"function"},{"location":"api/#EnzymeCore.make_zero","page":"API","title":"EnzymeCore.make_zero","text":"make_zero(::Type{T}, seen::IdDict, prev::T, ::Val{copy_if_inactive}=Val(false))::T\n\nRecursively make a zero'd copy of the value `prev` of type `T`. The argument `copy_if_inactive` specifies\nwhat to do if the type `T` is guaranteed to be inactive, use the primal (the default) or still copy the value.\n\n\n\n\n\n","category":"function"},{"location":"api/#EnzymeCore.make_zero-Union{Tuple{copy_if_inactive}, Tuple{T}, Tuple{T, Val{copy_if_inactive}}} where {T, copy_if_inactive}","page":"API","title":"EnzymeCore.make_zero","text":"make_zero(prev::T)\n\nHelper function to recursively make zero.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.EnzymeRules.AugmentedReturn","page":"API","title":"EnzymeCore.EnzymeRules.AugmentedReturn","text":"AugmentedReturn(primal, shadow, tape)\n\nAugment the primal return value of a function with its shadow, as well as any additional information needed to correctly compute the reverse pass, stored in tape.\n\nUnless specified by the config that a variable is not overwritten, rules must assume any arrays/data structures/etc are overwritten between the forward and the reverse pass. Any floats or variables passed by value are always preserved as is (as are the arrays themselves, just not necessarily the values in the array).\n\nSee also augmented_primal.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.EnzymeRules.Config","page":"API","title":"EnzymeCore.EnzymeRules.Config","text":"Config{NeedsPrimal, NeedsShadow, Width, Overwritten}\nConfigWidth{Width} = Config{<:Any,<:Any, Width}\n\nConfiguration type to dispatch on in custom reverse rules (see augmented_primal and reverse).\n\nNeedsPrimal and NeedsShadow: boolean values specifying whether the primal and shadow (resp.) should be returned. \nWidth: an integer that specifies the number of adjoints/shadows simultaneously being propagated.\nOverwritten: a tuple of booleans of whether each argument (including the function itself) is modified between the forward and reverse pass (true if potentially modified between).\n\nGetters for the four type parameters are provided by needs_primal, needs_shadow, width, and overwritten.\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeCore.EnzymeRules.augmented_primal","page":"API","title":"EnzymeCore.EnzymeRules.augmented_primal","text":"augmented_primal(::Config, func::Annotation{typeof(f)}, RT::Type{<:Annotation}, args::Annotation...)\n\nMust return an AugmentedReturn type.\n\nThe primal must be the same type of the original return if needs_primal(config), otherwise nothing.\nThe shadow must be nothing if needs_shadow(config) is false. If width is 1, the shadow should be the same type of the original return. If the width is greater than 1, the shadow should be NTuple{original return, width}.\nThe tape can be any type (including Nothing) and is preserved for the reverse call.\n\n\n\n\n\n","category":"function"},{"location":"api/#EnzymeCore.EnzymeRules.forward","page":"API","title":"EnzymeCore.EnzymeRules.forward","text":"forward(func::Annotation{typeof(f)}, RT::Type{<:Annotation}, args::Annotation...)\n\nCalculate the forward derivative. The first argument func is the callable for which the rule applies to. Either wrapped in a Const), or a Duplicated if it is a closure. The second argument is the return type annotation, and all other arguments are the annotated function arguments.\n\n\n\n\n\n","category":"function"},{"location":"api/#EnzymeCore.EnzymeRules.inactive","page":"API","title":"EnzymeCore.EnzymeRules.inactive","text":"inactive(func::typeof(f), args...)\n\nMark a particular function as always being inactive in both its return result and the function call itself.\n\n\n\n\n\n","category":"function"},{"location":"api/#EnzymeCore.EnzymeRules.inactive_noinl","page":"API","title":"EnzymeCore.EnzymeRules.inactive_noinl","text":"inactive_noinl(func::typeof(f), args...)\n\nMark a particular function as always being inactive in both its return result and the function call itself, but do not prevent inlining of the function.\n\n\n\n\n\n","category":"function"},{"location":"api/#EnzymeCore.EnzymeRules.inactive_type-Tuple{Type}","page":"API","title":"EnzymeCore.EnzymeRules.inactive_type","text":"inactive_type(::Type{Ty})\n\nMark a particular type Ty as always being inactive.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeCore.EnzymeRules.reverse","page":"API","title":"EnzymeCore.EnzymeRules.reverse","text":"reverse(::Config, func::Annotation{typeof(f)}, dret::Active, tape, args::Annotation...)\nreverse(::Config, func::Annotation{typeof(f)}, ::Type{<:Annotation), tape, args::Annotation...)\n\nTakes gradient of derivative, activity annotation, and tape. If there is an active return dret is passed as Active{T} with the derivative of the active return val. Otherwise dret is passed as Type{Duplicated{T}}, etc.\n\n\n\n\n\n","category":"function"},{"location":"api/#EnzymeTestUtils.ExprAndMsg","page":"API","title":"EnzymeTestUtils.ExprAndMsg","text":"A cunning hack to carry extra message along with the original expression in a test\n\n\n\n\n\n","category":"type"},{"location":"api/#EnzymeTestUtils.@test_msg-Tuple{Any, Any, Vararg{Any}}","page":"API","title":"EnzymeTestUtils.@test_msg","text":"@test_msg msg condion kws...\n\nThis is per Test.@test condion kws... except that if it fails it also prints the msg. If msg==\"\" then this is just like @test, nothing is printed\n\nExamles\n\njulia> @test_msg \"It is required that the total is under 10\" sum(1:1000) < 10;\nTest Failed at REPL[1]:1\n Expression: sum(1:1000) < 10\n Problem: It is required that the total is under 10\n Evaluated: 500500 < 10\nERROR: There was an error during testing\n\n\njulia> @test_msg \"It is required that the total is under 10\" error(\"not working at all\");\nError During Test at REPL[2]:1\n Test threw exception\n Expression: error(\"not working at all\")\n Problem: It is required that the total is under 10\n \"not working at all\"\n Stacktrace:\n\njulia> a = \"\";\n\njulia> @test_msg a sum(1:1000) < 10;\n Test Failed at REPL[153]:1\n Expression: sum(1:1000) < 10\n Evaluated: 500500 < 10\n ERROR: There was an error during testing\n\n\n\n\n\n","category":"macro"},{"location":"api/#EnzymeTestUtils.are_activities_compatible-Tuple{Any, Vararg{Any}}","page":"API","title":"EnzymeTestUtils.are_activities_compatible","text":"are_activities_compatible(Tret, activities...) -> Bool\n\nReturn true if return activity type Tret and activity types activities are compatible.\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeTestUtils.test_forward-Tuple{Any, Any, Vararg{Any}}","page":"API","title":"EnzymeTestUtils.test_forward","text":"test_forward(f, Activity, args...; kwargs...)\n\nTest Enzyme.autodiff of f in Forward-mode against finite differences.\n\nf has all constraints of the same argument passed to Enzyme.autodiff, with several additional constraints:\n\nIf it mutates one of its arguments, it must return that argument.\n\nArguments\n\nActivity: the activity of the return value of f\nargs: Each entry is either an argument to f, an activity type accepted by autodiff, or a tuple of the form (arg, Activity), where Activity is the activity type of arg. If the activity type specified requires a tangent, a random tangent will be automatically generated.\n\nKeywords\n\nfdm=FiniteDifferences.central_fdm(5, 1): The finite differences method to use.\nfkwargs: Keyword arguments to pass to f.\nrtol: Relative tolerance for isapprox.\natol: Absolute tolerance for isapprox.\ntestset_name: Name to use for a testset in which all tests are evaluated.\n\nExamples\n\nHere we test a rule for a function of scalars. Because we don't provide an activity annotation for y, it is assumed to be Const.\n\nusing Enzyme, EnzymeTestUtils\n\nx, y = randn(2)\nfor Tret in (Const, Duplicated, DuplicatedNoNeed), Tx in (Const, Duplicated)\n test_forward(*, Tret, (x, Tx), y)\nend\n\nHere we test a rule for a function of an array in batch forward-mode:\n\nx = randn(3)\ny = randn()\nfor Tret in (Const, BatchDuplicated, BatchDuplicatedNoNeed),\n Tx in (Const, BatchDuplicated),\n Ty in (Const, BatchDuplicated)\n\n test_forward(*, Tret, (x, Tx), (y, Ty))\nend\n\n\n\n\n\n","category":"method"},{"location":"api/#EnzymeTestUtils.test_reverse-Tuple{Any, Any, Vararg{Any}}","page":"API","title":"EnzymeTestUtils.test_reverse","text":"test_reverse(f, Activity, args...; kwargs...)\n\nTest Enzyme.autodiff_thunk of f in ReverseSplitWithPrimal-mode against finite differences.\n\nf has all constraints of the same argument passed to Enzyme.autodiff_thunk, with several additional constraints:\n\nIf it mutates one of its arguments, it must not also return that argument.\nIf the return value is a struct, then all floating point numbers contained in the struct or its fields must be in arrays.\n\nArguments\n\nActivity: the activity of the return value of f.\nargs: Each entry is either an argument to f, an activity type accepted by autodiff, or a tuple of the form (arg, Activity), where Activity is the activity type of arg. If the activity type specified requires a shadow, one will be automatically generated.\n\nKeywords\n\nfdm=FiniteDifferences.central_fdm(5, 1): The finite differences method to use.\nfkwargs: Keyword arguments to pass to f.\nrtol: Relative tolerance for isapprox.\natol: Absolute tolerance for isapprox.\ntestset_name: Name to use for a testset in which all tests are evaluated.\n\nExamples\n\nHere we test a rule for a function of scalars. Because we don't provide an activity annotation for y, it is assumed to be Const.\n\nusing Enzyme, EnzymeTestUtils\n\nx = randn()\ny = randn()\nfor Tret in (Const, Active), Tx in (Const, Active)\n test_reverse(*, Tret, (x, Tx), y)\nend\n\nHere we test a rule for a function of an array in batch reverse-mode:\n\nx = randn(3)\nfor Tret in (Const, Active), Tx in (Const, BatchDuplicated)\n test_reverse(prod, Tret, (x, Tx))\nend\n\n\n\n\n\n","category":"method"},{"location":"api/#Enzyme.API.runtimeActivity!-Tuple{Bool}","page":"API","title":"Enzyme.API.runtimeActivity!","text":"runtimeActivity!(val::Bool)\n\nEnzyme runs an activity analysis which deduces which values, instructions, etc are necessary to be differentiated and therefore involved in the differentiation procedure. This runs at compile time. However, there may be implementation flaws in this analysis that means that Enzyme cannot deduce that an inactive (const) value is actually const. Alternatively, there may be some data which is conditionally active, depending on which runtime branch is taken. In these cases Enzyme conservatively presumes the value is active.\n\nHowever, in certain cases, an insufficiently aggressive activity analysis may result in derivative errors – for example by mistakenly using the primal (const) argument and mistaking it for the duplicated shadow. As a result this may result in incorrect results, or accidental updates to the primal.\n\nThis flag enables runntime activity which tells all load/stores to check at runtime whether the value they are updating is indeed active (in addition to the compile-time activity analysis). This will remedy these such errors, but at a performance penalty of performing such checks.\n\nIt is on the Enzyme roadmap to add a PotentiallyDuplicated style activity, in addition to the current Const and Duplicated styles that will disable the need for this, which does not require the check when a value is guaranteed active, but still supports runtime-based activity information.\n\nThis function takes an argument to set the runtime activity value, true means it is on, and false means off. By default it is off.\n\n\n\n\n\n","category":"method"},{"location":"api/#Enzyme.API.runtimeActivity-Tuple{}","page":"API","title":"Enzyme.API.runtimeActivity","text":"runtimeActivity()\n\nGets the current value of the runtime activity. See runtimeActivity! for more information.\n\n\n\n\n\n","category":"method"},{"location":"pullbacks/#Implementing-pullbacks","page":"Implementing pullbacks","title":"Implementing pullbacks","text":"","category":"section"},{"location":"pullbacks/","page":"Implementing pullbacks","title":"Implementing pullbacks","text":"Enzyme's autodiff function can only handle functions with scalar output. To implement pullbacks (back-propagation of gradients/tangents) for array-valued functions, use a mutating function that returns nothing and stores it's result in one of the arguments, which must be passed wrapped in a Duplicated.","category":"page"},{"location":"pullbacks/#Example","page":"Implementing pullbacks","title":"Example","text":"","category":"section"},{"location":"pullbacks/","page":"Implementing pullbacks","title":"Implementing pullbacks","text":"Given a function mymul! that performs the equivalent of R = A * B for matrices A and B, and given a gradient (tangent) ∂z_∂R, we can compute ∂z_∂A and ∂z_∂B like this:","category":"page"},{"location":"pullbacks/","page":"Implementing pullbacks","title":"Implementing pullbacks","text":"using Enzyme, Random\n\nfunction mymul!(R, A, B)\n @assert axes(A,2) == axes(B,1)\n @inbounds @simd for i in eachindex(R)\n R[i] = 0\n end\n @inbounds for j in axes(B, 2), i in axes(A, 1)\n @inbounds @simd for k in axes(A,2)\n R[i,j] += A[i,k] * B[k,j]\n end\n end\n nothing\nend\n\nRandom.seed!(1234)\nA = rand(5, 3)\nB = rand(3, 7)\n\nR = zeros(size(A,1), size(B,2))\n∂z_∂R = rand(size(R)...) # Some gradient/tangent passed to us\n∂z_∂R0 = copyto!(similar(∂z_∂R), ∂z_∂R) # exact copy for comparison\n\n∂z_∂A = zero(A)\n∂z_∂B = zero(B)\n\nEnzyme.autodiff(Reverse, mymul!, Const, Duplicated(R, ∂z_∂R), Duplicated(A, ∂z_∂A), Duplicated(B, ∂z_∂B))","category":"page"},{"location":"pullbacks/","page":"Implementing pullbacks","title":"Implementing pullbacks","text":"Now we have:","category":"page"},{"location":"pullbacks/","page":"Implementing pullbacks","title":"Implementing pullbacks","text":"R ≈ A * B &&\n∂z_∂A ≈ ∂z_∂R0 * B' && # equivalent to Zygote.pullback(*, A, B)[2](∂z_∂R)[1]\n∂z_∂B ≈ A' * ∂z_∂R0 # equivalent to Zygote.pullback(*, A, B)[2](∂z_∂R)[2]","category":"page"},{"location":"pullbacks/","page":"Implementing pullbacks","title":"Implementing pullbacks","text":"Note that the result of the backpropagation is added to ∂z_∂A and ∂z_∂B, they act as accumulators for gradient information.","category":"page"},{"location":"dev_docs/#Enzyme-developer-documentation","page":"For developers","title":"Enzyme developer documentation","text":"","category":"section"},{"location":"dev_docs/#Development-of-Enzyme-and-Enzyme.jl-together","page":"For developers","title":"Development of Enzyme and Enzyme.jl together","text":"","category":"section"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"Normally Enzyme.jl downloads and install Enzyme for the user automatically since Enzyme needs to be built against Julia bundeled LLVM. In case that you are making updates to Enzyme and want to test them against Enzyme.jl the instructions below should help you get started.","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"Start Julia in your development copy of Enzyme.jl","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"~/s/Enzyme (master)> julia --project=.","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"Then create a development copy of Enzyme_jll and activate it within.","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"julia> using Enzyme_jll\njulia> Enzyme_jll.dev_jll()\n[ Info: Enzyme_jll dev'ed out to ${JULIA_PKG_DEVDIR}/Enzyme_jll with pre-populated override directory\n(Enzyme) pkg> dev Enzyme_jll\nPath `${JULIA_PKG_DEVDIR}/Enzyme_jll` exists and looks like the correct package. Using existing path.","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"After restarting Julia:","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"julia> Enzyme_jll.dev_jll()\njulia> Enzyme_jll.libEnzyme_path\n\"${JULIA_PKG_DEVDIR}/Enzyme_jll/override/lib/LLVMEnzyme-9.so\"","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"On your machine ${JULIA_PKG_DEVDIR} most likely corresponds to ~/.julia/dev. Now we can inspect \"${JULIA_PKG_DEVDIR}/Enzyme_jll/override/lib and see that there is a copy of LLVMEnzyme-9.so, which we can replace with a symbolic link or a copy of a version of Enzyme.","category":"page"},{"location":"dev_docs/#Building-Enzyme-against-Julia's-LLVM.","page":"For developers","title":"Building Enzyme against Julia's LLVM.","text":"","category":"section"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"Depending on how you installed Julia the LLVM Julia is using will be different.","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"Download from julialang.org (Recommended)\nManual build on your machine\nUses a pre-built Julia from your system vendor (Not recommended)","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"To check what LLVM Julia is using use:","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"julia> Base.libllvm_version_string\n\"9.0.1jl\"","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"If the LLVM version ends in a jl you a likely using the private LLVM.","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"In your source checkout of Enzyme:","category":"page"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"mkdir build-jl\ncd build-jl","category":"page"},{"location":"dev_docs/#Prebuilt-binary-from-julialang.org","page":"For developers","title":"Prebuilt binary from julialang.org","text":"","category":"section"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"LLVM_MAJOR_VER=`julia -e \"print(Base.libllvm_version.major)\"`\njulia -e \"using Pkg; pkg\\\"add LLVM_full_jll@${LLVM_MAJOR_VER}\\\"\"\nLLVM_DIR=`julia -e \"using LLVM_full_jll; print(LLVM_full_jll.artifact_dir)\"`\necho \"LLVM_DIR=$LLVM_DIR\"\ncmake ../enzyme/ -G Ninja -DENZYME_EXTERNAL_SHARED_LIB=ON -DLLVM_DIR=${LLVM_DIR} -DLLVM_EXTERNAL_LIT=${LLVM_DIR}/tools/lit/lit.py","category":"page"},{"location":"dev_docs/#Manual-build-of-Julia","page":"For developers","title":"Manual build of Julia","text":"","category":"section"},{"location":"dev_docs/","page":"For developers","title":"For developers","text":"cmake ../enzyme/ -G Ninja -DENZYME_EXTERNAL_SHARED_LIB=ON -DLLVM_DIR=${PATH_TO_BUILDDIR_OF_JULIA}/usr/lib/cmake/llvm/","category":"page"},{"location":"internal_api/#Internal-API","page":"Internal API","title":"Internal API","text":"","category":"section"},{"location":"internal_api/","page":"Internal API","title":"Internal API","text":"note: Note\nThis is the documentation of Enzymes's internal API. The internal API is not subject to semantic versioning and may change at any time and without deprecation.","category":"page"},{"location":"internal_api/","page":"Internal API","title":"Internal API","text":"Modules = [Enzyme.Compiler]\nOrder = [:module, :type, :constant, :macro, :function]","category":"page"},{"location":"internal_api/#Enzyme.Compiler.fspec-Tuple{Any, Any, Integer}","page":"Internal API","title":"Enzyme.Compiler.fspec","text":"Create the methodinstance pair, and lookup the primal return type.\n\n\n\n\n\n","category":"method"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"EditURL = \"../../../examples/box.jl\"","category":"page"},{"location":"generated/box/#Enzyme-for-adjoint-tutorial:-Stommel-three-box-ocean-model","page":"Box model","title":"Enzyme for adjoint tutorial: Stommel three-box ocean model","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"The goal of this tutorial is to teach about a specific usage of Enzyme's automatic differentiation capabilities, and will be centered around the Stommel ocean model. This is a nice example to see how powerful Enzyme is, and the ability of it to take a derivative of a complicated function (namely one that has many parts and parameters). This tutorial will focus first on the computations and getting Enzyme running, for those interested a mathematical explanation of the model and what an adjoint variable is will be provided at the end.","category":"page"},{"location":"generated/box/#Brief-model-overview","page":"Box model","title":"Brief model overview","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"The Stommel box model can be viewed as a watered down full ocean model. In our example, we have three boxes (Box One, Box Two, and Box Three) and we model the transport of fluid between them. The full equations of our system are given by:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"beginaligned\n U = u_0 left rho_2 - left rho_1 + (1 - delta) rho_3 right right \n rho_i = -alpha T_i + beta S_i i = 1 2 3\nendaligned","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"for the transport U and densities rho, and then the time derivatives","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"beginaligned\n dotT_1 = U(T_3 - T_1)V_1 + gamma (T_1^* - T_1 ) dotS_1 = U(S_3 - S_1)V_1 + FW_1V_1 \n dotT_2 = U(T_1 - T_2)V_2 + gamma (T_2^* - T_2 ) dotS_2 = U(S_1 - S_2)V_2 + FW_2V_2 \n dotT_3 = U(T_2 - T_3)V_3 dotS_3 = U(S_2 - S_3)V_3\nendaligned","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"for positive transport, U 0, and","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"beginaligned\n dotT_1 = U(T_2 - T_1)V_1 + gamma (T_1^* - T_1) dotS_1 = U(S_2 - S_1)V_1 + FW_1V_1 \n dotT_2 = U(T_3 - T_2)V_2 + gamma (T_2^* - T_2 ) dotS_2 = U(S_3 - S_2)V_2 + FW_2V_2 \n dotT_3 = U(T_1 - T_3)V_3 dotS_3 = U(S_1 - S_3)V_3\nendaligned","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"for U leq 0. The only force driving our system is a density gradient generated via temperature and salinity differences between the boxes. This makes it a really easy model to play around with! With this in mind, the model is run forward with the steps:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Compute densities\nCompute transport\nCompute time derivatives of the box temperatures and salinities\nUpdate the state vector","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"We'll start by going through the model setup step by step, then providing a few test cases with Enzyme.","category":"page"},{"location":"generated/box/#Model-setup","page":"Box model","title":"Model setup","text":"","category":"section"},{"location":"generated/box/#Model-dependencies","page":"Box model","title":"Model dependencies","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Let's first add the necessary packages to run everything","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"using Enzyme","category":"page"},{"location":"generated/box/#Initialize-constants","page":"Box model","title":"Initialize constants","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"The system equations have quite a few constants that appear, here we initialize them for later use. We'll do this in a Julia way: we have an empty structure that will hold all the parameters, and a function (we'll call this setup) that initializes them. This means that, so long as we don't need to change parameters, we only need to run setup once.","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"struct ModelParameters\n\n # handy to have constants\n day::Float64\n year::Float64\n\n # Information related to the boxes\n boxlength::Vector{Float64} ## Vector with north-south size of each box [cm]\n boxdepth::Vector{Float64} ## \" \" the depth of each box [cm]\n boxwidth::Float64 ## \" \" the width of each box [cm]\n boxarea::Vector{Float64} ## \" \" the area of each box [cm^2]\n boxvol::Vector{Float64} ## \" \" the volume of each box [cm^3]\n\n delta::Float64 ## Constant ratio depth(box1) / (depth(box1) + depth(box3))\n\n # Parameters that appear in the box model equations\n u0::Float64\n alpha::Float64\n beta::Float64\n gamma::Float64\n\n # Coefficient for the Robert filter smoother\n rf_coeff::Float64\n\n # Freshwater forcing\n FW::Vector{Float64}\n\n # Restoring atmospheric temperatures and salinities\n Tstar::Vector{Float64}\n Sstar::Vector{Float64}\n\nend\n\nfunction setup()\n\n blength = [5000.0e5; 1000.0e5; 5000.0e5]\n bdepth = [1.0e5; 5.0e5; 4.0e5]\n\n delta = bdepth[1]/(bdepth[1] + bdepth[3])\n\n bwidth = 4000.0*1e5 ## box width, centimeters\n\n # box areas\n barea = [blength[1]*bwidth;\n blength[2]*bwidth;\n blength[3]*bwidth]\n\n # box volumes\n bvolume = [barea[1]*bdepth[1];\n barea[2]*bdepth[2];\n barea[3]*bdepth[3]]\n\n # parameters that are used to ensure units are in CGS (cent-gram-sec)\n\n day = 3600.0*24.0\n year = day*365.0\n Sv = 1e12 ## one Sverdrup (a unit of ocean transport), 1e6 meters^3/second\n\n # parameters that appear in box model equations\n u0 = 16.0*Sv/0.0004\n alpha = 1668e-7\n beta = 0.7811e-3\n\n gamma = 1/(300*day)\n\n # robert filter coefficient for the smoother part of the timestep\n robert_filter_coeff = 0.25\n\n # freshwater forcing\n FW = [(100/year) * 35.0 * barea[1]; -(100/year) * 35.0 * barea[1]]\n\n # restoring atmospheric temperatures\n Tstar = [22.0; 0.0]\n Sstar = [36.0; 34.0]\n\n structure_with_parameters = ModelParameters(day,\n year,\n blength,\n bdepth,\n bwidth,\n barea,\n bvolume,\n delta,\n u0,\n alpha,\n beta,\n gamma,\n robert_filter_coeff,\n FW,\n Tstar,\n Sstar\n )\n\n return structure_with_parameters\n\nend","category":"page"},{"location":"generated/box/#Define-model-functions","page":"Box model","title":"Define model functions","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Here we define functions that will calculate quantities used in the forward steps.","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"# function to compute transport\n# Input: rho - the density vector\n# Output: U - transport value\n\nfunction compute_transport(rho, params)\n\n U = params.u0 * (rho[2] - (params.delta * rho[1] + (1 - params.delta)*rho[3]))\n return U\n\nend\n\n# function to compute density\n# Input: state = [T1; T2; T3; S1; S2; S3]\n# Output: rho\n\nfunction compute_density(state, params)\n\n rho = -params.alpha * state[1:3] + params.beta * state[4:6]\n return rho\n\nend\n\n# lastly, a function that takes one step forward\n# Input: state_now = [T1(t), T2(t), ..., S3(t)]\n# state_old = [T1(t-dt), ..., S3(t-dt)]\n# u = transport(t)\n# dt = time step\n# Output: state_new = [T1(t+dt), ..., S3(t+dt)]\n\nfunction compute_update(state_now, state_old, u, params, dt)\n\n dstate_now_dt = zeros(6)\n state_new = zeros(6)\n\n # first computing the time derivatives of the various temperatures and salinities\n if u > 0\n\n dstate_now_dt[1] = u * (state_now[3] - state_now[1]) / params.boxvol[1] + params.gamma * (params.Tstar[1] - state_now[1])\n dstate_now_dt[2] = u * (state_now[1] - state_now[2]) / params.boxvol[2] + params.gamma * (params.Tstar[2] - state_now[2])\n dstate_now_dt[3] = u * (state_now[2] - state_now[3]) / params.boxvol[3]\n\n dstate_now_dt[4] = u * (state_now[6] - state_now[4]) / params.boxvol[1] + params.FW[1] / params.boxvol[1]\n dstate_now_dt[5] = u * (state_now[4] - state_now[5]) / params.boxvol[2] + params.FW[2] / params.boxvol[2]\n dstate_now_dt[6] = u * (state_now[5] - state_now[6]) / params.boxvol[3]\n\n elseif u <= 0\n\n dstate_now_dt[1] = u * (state_now[2] - state_now[1]) / params.boxvol[1] + params.gamma * (params.Tstar[1] - state_now[1])\n dstate_now_dt[2] = u * (state_now[3] - state_now[2]) / params.boxvol[2] + params.gamma * (params.Tstar[2] - state_now[2])\n dstate_now_dt[3] = u * (state_now[1] - state_now[3]) / params.boxvol[3]\n\n dstate_now_dt[4] = u * (state_now[5] - state_now[4]) / params.boxvol[1] + params.FW[1] / params.boxvol[1]\n dstate_now_dt[5] = u * (state_now[6] - state_now[5]) / params.boxvol[2] + params.FW[2] / params.boxvol[2]\n dstate_now_dt[6] = u * (state_now[4] - state_now[6]) / params.boxvol[3]\n\n end\n\n # update fldnew using a version of Euler's method\n state_new .= state_old + 2.0 * dt * dstate_now_dt\n\n return state_new\nend","category":"page"},{"location":"generated/box/#Define-forward-functions","page":"Box model","title":"Define forward functions","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Finally, we create two functions, the first of which computes and stores all the states of the system, and the second will take just a single step forward.","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Let's start with the standard forward function. This is just going to be used to store the states at every timestep:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"function integrate(state_now, state_old, dt, M, parameters)\n\n # Because of the adjoint problem we're setting up, we need to store both the states before\n # and after the Robert filter smoother has been applied\n states_before = [state_old]\n states_after = [state_old]\n\n for t = 1:M\n\n rho = compute_density(state_now, parameters)\n u = compute_transport(rho, parameters)\n state_new = compute_update(state_now, state_old, u, parameters, dt)\n\n # Applying the Robert filter smoother (needed for stability)\n state_new_smoothed = state_now + parameters.rf_coeff * (state_new - 2.0 * state_now + state_old)\n\n push!(states_after, state_new_smoothed)\n push!(states_before, state_new)\n\n # cycle the \"now, new, old\" states\n state_old = state_new_smoothed\n state_now = state_new\n\n end\n\n return states_after, states_before\nend","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Now, for the purposes of Enzyme, it would be convenient for us to have a function that runs a single step of the model forward rather than the whole integration. This would allow us to save as many of the adjoint variables as we wish when running the adjoint method, although for the example we'll discuss later we technically only need one of them","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"function one_step_forward(state_now, state_old, out_now, out_old, parameters, dt)\n\n state_new_smoothed = zeros(6)\n rho = compute_density(state_now, parameters) ## compute density\n u = compute_transport(rho, parameters) ## compute transport\n state_new = compute_update(state_now, state_old, u, parameters, dt) ## compute new state values\n\n # Robert filter smoother\n state_new_smoothed[:] = state_now + parameters.rf_coeff * (state_new - 2.0 * state_now + state_old)\n\n out_old[:] = state_new_smoothed\n out_now[:] = state_new\n\n return nothing\n\nend","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"One difference to note is that one_step_forward now returns nothing, but is rather a function of both its input and output. Since the output of the function is a vector, we need to have this return nothing for Enzyme to work. Now we can move on to some examples using Enzyme.","category":"page"},{"location":"generated/box/#Example-1:-Simply-using-Enzyme","page":"Box model","title":"Example 1: Simply using Enzyme","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"For the first example let's just compute the gradient of our forward function and examine the output. We'll just run the model for one step, and take a dt of ten days. The initial conditions of the system are given as Tbar and Sbar. We run setup once here, and never have to run it again! (Unless we decide to change a parameter)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"parameters = setup()\n\nTbar = [20.0; 1.0; 1.0] ## initial temperatures\nSbar = [35.5; 34.5; 34.5] ## initial salinities\n\n# Running the model one step forward\nstates_after_smoother, states_before_smoother = integrate(\n copy([Tbar; Sbar]),\n copy([Tbar; Sbar]),\n 10*parameters.day,\n 1,\n parameters\n)\n\n# Run Enzyme one time on `one_step_forward``\ndstate_now = zeros(6)\ndstate_old = zeros(6)\nout_now = zeros(6); dout_now = ones(6)\nout_old = zeros(6); dout_old = ones(6)\n\nautodiff(Reverse,\n one_step_forward,\n Duplicated([Tbar; Sbar], dstate_now),\n Duplicated([Tbar; Sbar], dstate_old),\n Duplicated(out_now, dout_now),\n Duplicated(out_old, dout_old),\n Const(parameters),\n Const(10*parameters.day)\n)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"In order to run Enzyme on one_step_forward, we've needed to provide quite a few placeholders, and wrap everything in Duplicated as all components of our function are vectors, not scalars. Let's go through and see what Enzyme did with all of those placeholders.","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"First we can look at what happened to the zero vectors out_now and out_old:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show out_now, out_old","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Comparing to the results of forward func:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show states_before_smoother[2], states_after_smoother[2]","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"we see that Enzyme has computed and stored exactly the output of the forward step. Next, let's look at dstate_now:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show dstate_now","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Just a few numbers, but this is what makes AD so nice: Enzyme has exactly computed the derivative of all outputs with respect to the input state_now, evaluated at state_now, and acted with this gradient on what we gave as dout_now (in our case, all ones). Using AD notation for reverse mode, this is","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"overlinetextstate_now = fracpartial textout_nowpartial textstate_nowright_textstate_now overlinetextout_now + fracpartial textout_oldpartial textstate_nowright_textstate_now overlinetextout_old","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"We note here that had we initialized dstate_now and dstate_old as something else, our results will change. Let's multiply them by two and see what happens.","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"dstate_now_new = zeros(6)\ndstate_old_new = zeros(6)\nout_now = zeros(6); dout_now = 2*ones(6)\nout_old = zeros(6); dout_old = 2*ones(6)\nautodiff(Reverse,\n one_step_forward,\n Duplicated([Tbar; Sbar], dstate_now_new),\n Duplicated([Tbar; Sbar], dstate_old_new),\n Duplicated(out_now, dout_now),\n Duplicated(out_old, dout_old),\n Const(parameters),\n Const(10*parameters.day)\n)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Now checking dstate_now and dstate_old we see","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show dstate_now_new","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"What happened? Enzyme is actually taking the computed gradient and acting on what we give as input to dout_now and dout_old. Checking this, we see","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show 2*dstate_now","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"and they match the new results. This exactly matches what we'd expect to happen since we scaled dout_now by two.","category":"page"},{"location":"generated/box/#Example-2:-Full-sensitivity-calculations","page":"Box model","title":"Example 2: Full sensitivity calculations","text":"","category":"section"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Now we want to use Enzyme for a bit more than just a single derivative. Let's say we'd like to understand how sensitive the final temperature of Box One is to the initial salinity of Box Two. That is, given the function","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"J = (100000)^T cdot mathbfx(t_f)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"we want Enzyme to calculate the derivative","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"fracpartial Jpartial mathbfx(0)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"where x(t) is the state of the model at time t. If we think about x(t_f) as solely depending on the initial condition, then this derivative is really","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"fracpartial Jpartial mathbfx(0) = fracpartialpartial mathbfx(0) left( (100000)^T cdot L(ldots(L(mathbfx(0)))) right)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"with L(x(t)) = x(t + dt), i.e. one forward step. One could expand this derivative with the chain rule (and it would be very complicated), but really this is where Enzyme comes in. Each run of autodiff on our forward function is one piece of this big chain rule done for us! We also note that the chain rule goes from the outside in, so we start with the derivative of the forward function at the final state, and work backwards until the initial state. To get Enzyme to do this, we complete the following steps:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Run the forward model and store outputs (in a real ocean model this wouldn't be feasible and we'd need to use checkpointing)\nCompute the initial derivative from the final state\nUse Enzyme to work backwards until we reach the desired derivative.","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"For simplicity we define a function that takes completes our AD steps","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"function compute_adjoint_values(states_before_smoother, states_after_smoother, M, parameters)\n\n dout_now = [0.0;0.0;0.0;0.0;0.0;0.0]\n dout_old = [1.0;0.0;0.0;0.0;0.0;0.0]\n\n for j = M:-1:1\n\n dstate_now = zeros(6)\n dstate_old = zeros(6)\n\n autodiff(Reverse,\n one_step_forward,\n Duplicated(states_before_smoother[j], dstate_now),\n Duplicated(states_after_smoother[j], dstate_old),\n Duplicated(zeros(6), dout_now),\n Duplicated(zeros(6), dout_old),\n Const(parameters),\n Const(10*parameters.day)\n )\n\n if j == 1\n return dstate_now, dstate_old\n end\n\n dout_now = copy(dstate_now)\n dout_old = copy(dstate_old)\n\n end\n\nend","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"First we integrate the model forward:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"M = 10000 ## Total number of forward steps to take\nTbar = [20.0; 1.0; 1.0] ## initial temperatures\nSbar = [35.5; 34.5; 34.5] ## initial salinities\n\nstates_after_smoother, states_before_smoother = integrate(\n copy([Tbar; Sbar]),\n copy([Tbar; Sbar]),\n 10*parameters.day,\n M,\n parameters\n)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Next, we pass all of our states to the AD function to get back to the desired derivative:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"dstate_now, dstate_old = compute_adjoint_values(\n states_before_smoother,\n states_after_smoother,\n M,\n parameters\n)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"And we're done! We were interested in sensitivity to the initial salinity of box two, which will live in what we've called dstate_old. Checking this value we see","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show dstate_old[5]","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"As it stands this is just a number, but a good check that Enzyme has computed what we want is to approximate the derivative with a Taylor series. Specifically,","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"J(mathbfx(0) + varepsilon) approx J(mathbfx(0)) +\nvarepsilon fracpartial Jpartial mathbfx(0)","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"and a simple rearrangement yields","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"fracpartial Jpartial mathbfx(0) approx\nfracJ(mathbfx(0) + varepsilon) - J(mathbfx(0))varepsilon","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Hopefully we see that the analytical values converge close to the one we found with Enzyme:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"# unperturbed final state\nuse_to_check = states_after_smoother[M+1]\n\n# a loop to compute the perturbed final states\ndiffs = []\nstep_sizes = [1e-1, 1e-2, 1e-3, 1e-4, 1e-5, 1e-6, 1e-7, 1e-8, 1e-9, 1e-10]\nfor eps in step_sizes\n\n state_new_smoothed = zeros(6)\n\n initial_temperature = [20.0; 1.0; 1.0]\n perturbed_initial_salinity = [35.5; 34.5; 34.5] + [0.0; eps; 0.0]\n\n state_old = [initial_temperature; perturbed_initial_salinity]\n state_now = [20.0; 1.0; 1.0; 35.5; 34.5; 34.5]\n\n for t = 1:M\n\n rho = compute_density(state_now, parameters)\n u = compute_transport(rho, parameters)\n state_new = compute_update(state_now, state_old, u, parameters, 10*parameters.day)\n\n state_new_smoothed[:] = state_now + parameters.rf_coeff * (state_new - 2.0 * state_now + state_old)\n\n state_old = state_new_smoothed\n state_now = state_new\n\n end\n\n push!(diffs, (state_old[1] - use_to_check[1])/eps)\n\nend","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"Then checking what we found the derivative to be analytically:","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show diffs","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"which comes very close to our calculated value. We can go further and check the percent difference to see","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"@show abs.(diffs .- dstate_old[5])./dstate_old[5]","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"and we get down to a percent difference on the order of 1e^-5, showing Enzyme calculated the correct derivative. Success!","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"","category":"page"},{"location":"generated/box/","page":"Box model","title":"Box model","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"EditURL = \"../../../examples/custom_rule.jl\"","category":"page"},{"location":"generated/custom_rule/#Enzyme-custom-rules-tutorial","page":"Custom rules","title":"Enzyme custom rules tutorial","text":"","category":"section"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"note: More Examples\nThe tutorial below focuses on a simple setting to illustrate the basic concepts of writing custom rules. For more complex custom rules beyond the scope of this tutorial, you may take inspiration from the following in-the-wild examples:Enzyme internal rules\nKernelAbstractions.jl\nLinearSolve.jl\nNNlib.jl","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"The goal of this tutorial is to give a simple example of defining a custom rule with Enzyme. Specifically, our goal will be to write custom rules for the following function f:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"function f(y, x)\n y .= x.^2\n return sum(y)\nend","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Our function f populates its first input y with the element-wise square of x. In addition, it returns sum(y) as output. What a sneaky function!","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"In this case, Enzyme can differentiate through f automatically. For example, using forward mode:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"using Enzyme\nx = [3.0, 1.0]\ndx = [1.0, 0.0]\ny = [0.0, 0.0]\ndy = [0.0, 0.0]\n\ng(y, x) = f(y, x)^2 # function to differentiate\n\n@show autodiff(Forward, g, Duplicated(y, dy), Duplicated(x, dx)) # derivative of g w.r.t. x[1]\n@show dy; # derivative of y w.r.t. x[1] when g is run\nnothing #hide","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"(See the AutoDiff API tutorial for more information on using autodiff.)","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"But there may be special cases where we need to write a custom rule to help Enzyme out. Let's see how to write a custom rule for f!","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"warning: Don't use custom rules unnecessarily!\nEnzyme can efficiently handle a wide range of constructs, and so a custom rule should only be required in certain special cases. For example, a function may make a foreign call that Enzyme cannot differentiate, or we may have higher-level mathematical knowledge that enables us to write a more efficient rule. Even in these cases, try to make your custom rule encapsulate the minimum possible construct that Enzyme cannot differentiate, rather than expanding the scope of the rule unnecessarily. For pedagogical purposes, we will disregard this principle here and go ahead and write a custom rule for f :)","category":"page"},{"location":"generated/custom_rule/#Defining-our-first-rule","page":"Custom rules","title":"Defining our first rule","text":"","category":"section"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"First, we import the functions EnzymeRules.forward, EnzymeRules.augmented_primal, and EnzymeRules.reverse. We need to overload forward in order to define a custom forward rule, and we need to overload augmented_primal and reverse in order to define a custom reverse rule.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"import .EnzymeRules: forward, reverse, augmented_primal\nusing .EnzymeRules","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"In this section, we write a simple forward rule to start out:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"function forward(func::Const{typeof(f)}, ::Type{<:Duplicated}, y::Duplicated, x::Duplicated)\n println(\"Using custom rule!\")\n ret = func.val(y.val, x.val)\n y.dval .= 2 .* x.val .* x.dval\n return Duplicated(ret, sum(y.dval))\nend","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"In the signature of our rule, we have made use of Enzyme's activity annotations. Let's break down each one:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"the Const annotation on f indicates that we accept a function f that does not have a derivative component, which makes sense since f is not a closure with data that could be differentiated.\nthe Duplicated annotation given in the second argument annotates the return value of f. This means that our forward function should return an output of type Duplicated, containing the original output sum(y) and its derivative.\nthe Duplicated annotations for x and y mean that our forward function handles inputs x and y which have been marked as Duplicated. We should update their shadows with their derivative contributions.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"In the logic of our forward function, we run the original function, populate y.dval (the shadow of y), and finally return a Duplicated for the output as promised. Let's see our rule in action! With the same setup as before:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"x = [3.0, 1.0]\ndx = [1.0, 0.0]\ny = [0.0, 0.0]\ndy = [0.0, 0.0]\n\ng(y, x) = f(y, x)^2 # function to differentiate\n\n@show autodiff(Forward, g, Duplicated(y, dy), Duplicated(x, dx)) # derivative of g w.r.t. x[1]\n@show dy; # derivative of y w.r.t. x[1] when g is run\nnothing #hide","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"We see that our custom forward rule has been triggered and gives the same answer as before.","category":"page"},{"location":"generated/custom_rule/#Handling-more-activities","page":"Custom rules","title":"Handling more activities","text":"","category":"section"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Our custom rule applies for the specific set of activities that are annotated for f in the above autodiff call. However, Enzyme has a number of other annotations. Let us consider a particular example, where the output has a DuplicatedNoNeed annotation. This means we are only interested in its derivative, not its value. To squeeze out the last drop of performance, the below rule avoids computing the output of the original function and just computes its derivative.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"function forward(func::Const{typeof(f)}, ::Type{<:DuplicatedNoNeed}, y::Duplicated, x::Duplicated)\n println(\"Using custom rule with DuplicatedNoNeed output.\")\n y.val .= x.val.^2\n y.dval .= 2 .* x.val .* x.dval\n return sum(y.dval)\nend","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Our rule is triggered, for example, when we call autodiff directly on f, as the return value's derivative isn't needed:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"x = [3.0, 1.0]\ndx = [1.0, 0.0]\ny = [0.0, 0.0]\ndy = [0.0, 0.0]\n\n@show autodiff(Forward, f, Duplicated(y, dy), Duplicated(x, dx)) # derivative of f w.r.t. x[1]\n@show dy; # derivative of y w.r.t. x[1] when f is run\nnothing #hide","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"note: Custom rule dispatch\nWhen multiple custom rules for a function are defined, the correct rule is chosen using Julia's multiple dispatch. In particular, it is important to understand that the custom rule does not determine the activities of the inputs and the return value: rather, Enzyme decides the activity annotations independently, and then dispatches to the custom rule handling the activities, if one exists. If a custom rule is specified for the correct function/argument types, but not the correct activity annotation, a runtime error will be thrown alerting the user to the missing activity rule rather than silently ignoring the rule.\"","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Finally, it may be that either x, y, or the return value are marked as Const. We can in fact handle this case, along with the previous two cases, all together in a single rule:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Base.delete_method.(methods(forward, (Const{typeof(f)}, Vararg{Any}))) # delete our old rules\n\nfunction forward(func::Const{typeof(f)}, RT::Type{<:Union{Const, DuplicatedNoNeed, Duplicated}},\n y::Union{Const, Duplicated}, x::Union{Const, Duplicated})\n println(\"Using our general custom rule!\")\n y.val .= x.val.^2\n if !(x isa Const) && !(y isa Const)\n y.dval .= 2 .* x.val .* x.dval\n elseif !(y isa Const)\n y.dval .= 0\n end\n dret = !(y isa Const) ? sum(y.dval) : zero(eltype(y.val))\n if RT <: Const\n return sum(y.val)\n elseif RT <: DuplicatedNoNeed\n return dret\n else\n return Duplicated(sum(y.val), dret)\n end\nend","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Let's try out our rule:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"x = [3.0, 1.0]\ndx = [1.0, 0.0]\ny = [0.0, 0.0]\ndy = [0.0, 0.0]\n\ng(y, x) = f(y, x)^2 # function to differentiate\n\n@show autodiff(Forward, g, Duplicated(y, dy), Duplicated(x, dx)) # derivative of g w.r.t. x[1]\n@show autodiff(Forward, g, Const(y), Duplicated(x, dx)) # derivative of g w.r.t. x[1], with y annotated Const\n@show autodiff(Forward, g, Const(y), Const(x)); # derivative of g w.r.t. x[1], with x and y annotated Const\nnothing #hide","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Note that there are also exist batched duplicated annotations for forward mode, namely BatchDuplicated and BatchDuplicatedNoNeed, which are not covered in this tutorial.","category":"page"},{"location":"generated/custom_rule/#Defining-a-reverse-mode-rule","page":"Custom rules","title":"Defining a reverse-mode rule","text":"","category":"section"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Let's look at how to write a simple reverse-mode rule! First, we write a method for EnzymeRules.augmented_primal:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"function augmented_primal(config::ConfigWidth{1}, func::Const{typeof(f)}, ::Type{<:Active},\n y::Duplicated, x::Duplicated)\n println(\"In custom augmented primal rule.\")\n # Compute primal\n if needs_primal(config)\n primal = func.val(y.val, x.val)\n else\n y.val .= x.val.^2 # y still needs to be mutated even if primal not needed!\n primal = nothing\n end\n # Save x in tape if x will be overwritten\n if overwritten(config)[3]\n tape = copy(x.val)\n else\n tape = nothing\n end\n # Return an AugmentedReturn object with shadow = nothing\n return AugmentedReturn(primal, nothing, tape)\nend","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Let's unpack our signature for augmented_primal :","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"We accepted a EnzymeRules.Config object with a specified width of 1, which means that our rule does not support batched reverse mode.\nWe annotated f with Const as usual.\nWe dispatched on an Active annotation for the return value. This is a special annotation for scalar values, such as our return value, that indicates that that we care about the value's derivative but we need not explicitly allocate a mutable shadow since it is a scalar value.\nWe annotated x and y with Duplicated, similar to our first simple forward rule.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Now, let's unpack the body of our augmented_primal rule:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"We checked if the config requires the primal. If not, we need not compute the return value, but we make sure to mutate y in all cases.\nWe checked if x could possibly be overwritten using the Overwritten attribute of EnzymeRules.Config. If so, we save the elements of x on the tape of the returned EnzymeRules.AugmentedReturn object.\nWe return a shadow of nothing since the return value is Active and hence does not need a shadow.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Now, we write a method for EnzymeRules.reverse:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"function reverse(config::ConfigWidth{1}, func::Const{typeof(f)}, dret::Active, tape,\n y::Duplicated, x::Duplicated)\n println(\"In custom reverse rule.\")\n # retrieve x value, either from original x or from tape if x may have been overwritten.\n xval = overwritten(config)[3] ? tape : x.val\n # accumulate dret into x's shadow. don't assign!\n x.dval .+= 2 .* xval .* dret.val\n # also accumulate any derivative in y's shadow into x's shadow.\n x.dval .+= 2 .* xval .* y.dval\n y.dval .= 0\n return (nothing, nothing)\nend","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Let's make a few observations about our reverse rule:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"The activities used in the signature correspond to what we used for augmented_primal.\nHowever, for Active return types such as in this case, we now receive an instance dret of Active for the return type, not just a type annotation, which stores the derivative value for ret (not the original return value!). For the other annotations (e.g. Duplicated), we still receive only the type. In that case, if necessary a reference to the shadow of the output should be placed on the tape in augmented_primal.\nUsing dret.val and y.dval, we accumulate the backpropagated derivatives for x into its shadow x.dval. Note that we have to accumulate from both y.dval and dret.val. This is because in reverse-mode AD we have to sum up the derivatives from all uses: if y was read after our function, we need to consider derivatives from that use as well.\nWe zero-out y's shadow. This is because y is overwritten within f, so there is no derivative w.r.t. to the y that was originally inputted.\nFinally, since all derivatives are accumulated in place (in the shadows of the Duplicated arguments), these derivatives must not be communicated via the return value. Hence, we return (nothing, nothing). If, instead, one of our arguments was annotated as Active, we would have to provide its derivative at the corresponding index in the tuple returned.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Finally, let's see our reverse rule in action!","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"x = [3.0, 1.0]\ndx = [0.0, 0.0]\ny = [0.0, 0.0]\ndy = [0.0, 0.0]\n\ng(y, x) = f(y, x)^2\n\nautodiff(Reverse, g, Duplicated(y, dy), Duplicated(x, dx))\n@show dx # derivative of g w.r.t. x\n@show dy; # derivative of g w.r.t. y\nnothing #hide","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Let's also try a function which mutates x after running f, and also uses y directly rather than only ret after running f (but ultimately gives the same result as above):","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"function h(y, x)\n ret = f(y, x)\n x .= x.^2\n return ret * sum(y)\nend\n\nx = [3.0, 1.0]\ny = [0.0, 0.0]\ndx .= 0\ndy .= 0\n\nautodiff(Reverse, h, Duplicated(y, dy), Duplicated(x, dx))\n@show dx # derivative of h w.r.t. x\n@show dy; # derivative of h w.r.t. y\nnothing #hide","category":"page"},{"location":"generated/custom_rule/#Marking-functions-inactive","page":"Custom rules","title":"Marking functions inactive","text":"","category":"section"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"If we want to tell Enzyme that the function call does not affect the differentiation result in any form (i.e. not by side effects or through its return values), we can simply use EnzymeRules.inactive. So long as there exists a matching dispatch to EnzymeRules.inactive, the function will be considered inactive. For example:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"printhi() = println(\"Hi!\")\nEnzymeRules.inactive(::typeof(printhi), args...) = nothing\n\nfunction k(x)\n printhi()\n return x^2\nend\n\nautodiff(Forward, k, Duplicated(2.0, 1.0))","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"Or for a case where we incorrectly mark a function inactive:","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"double(x) = 2*x\nEnzymeRules.inactive(::typeof(double), args...) = nothing\n\nautodiff(Forward, x -> x + double(x), Duplicated(2.0, 1.0)) # mathematically should be 3.0, inactive rule causes it to be 1.0","category":"page"},{"location":"generated/custom_rule/#Testing-our-rules","page":"Custom rules","title":"Testing our rules","text":"","category":"section"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"We can test our rules using finite differences using EnzymeTestUtils.test_forward and EnzymeTestUtils.test_reverse.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"using EnzymeTestUtils, Test\n\n@testset \"f rules\" begin\n @testset \"forward\" begin\n @testset for RT in (Const, DuplicatedNoNeed, Duplicated),\n Tx in (Const, Duplicated),\n Ty in (Const, Duplicated)\n\n x = [3.0, 1.0]\n y = [0.0, 0.0]\n test_forward(g, RT, (x, Tx), (y, Ty))\n end\n end\n @testset \"reverse\" begin\n @testset for RT in (Active,),\n Tx in (Duplicated,),\n Ty in (Duplicated,),\n fun in (g, h)\n\n x = [3.0, 1.0]\n y = [0.0, 0.0]\n test_reverse(fun, RT, (x, Tx), (y, Ty))\n end\n end\nend","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"In any package that implements Enzyme rules using EnzymeRules, it is recommended to add EnzymeTestUtils as a test dependency to test the rules.","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"","category":"page"},{"location":"generated/custom_rule/","page":"Custom rules","title":"Custom rules","text":"This page was generated using Literate.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = Enzyme\nDocTestSetup = quote\n using Enzyme\nend","category":"page"},{"location":"#Enzyme","page":"Home","title":"Enzyme","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Documentation for Enzyme.jl, the Julia bindings for Enzyme.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Enzyme performs automatic differentiation (AD) of statically analyzable LLVM. It is highly-efficient and its ability to perform AD on optimized code allows Enzyme to meet or exceed the performance of state-of-the-art AD tools.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Enzyme.jl can be installed in the usual way Julia packages are installed:","category":"page"},{"location":"","page":"Home","title":"Home","text":"] add Enzyme","category":"page"},{"location":"","page":"Home","title":"Home","text":"The Enzyme binary dependencies will be installed automatically via Julia's binary artifact system.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The Enzyme.jl API revolves around the function autodiff. For some common operations, Enzyme additionally wraps autodiff in several convenience functions; e.g., gradient and jacobian.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The tutorial below covers the basic usage of these functions. For a complete overview of Enzyme's functionality, see the API documentation. Also see Implementing pullbacks on how to implement back-propagation for functions with non-scalar results.","category":"page"},{"location":"#Getting-started","page":"Home","title":"Getting started","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"julia> rosenbrock(x, y) = (1.0 - x)^2 + 100.0 * (y - x^2)^2\nrosenbrock (generic function with 1 method)\n\njulia> rosenbrock_inp(x) = (1.0 - x[1])^2 + 100.0 * (x[2] - x[1]^2)^2\nrosenbrock_inp (generic function with 1 method)","category":"page"},{"location":"#Reverse-mode","page":"Home","title":"Reverse mode","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"The return value of reverse mode autodiff is a tuple that contains as a first value the derivative value of the active inputs and optionally the primal return value.","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> autodiff(Reverse, rosenbrock, Active, Active(1.0), Active(2.0))\n((-400.0, 200.0),)\n\njulia> autodiff(ReverseWithPrimal, rosenbrock, Active, Active(1.0), Active(2.0))\n((-400.0, 200.0), 100.0)","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> x = [1.0, 2.0]\n2-element Vector{Float64}:\n 1.0\n 2.0\n\njulia> dx = [0.0, 0.0]\n2-element Vector{Float64}:\n 0.0\n 0.0\n\njulia> autodiff(Reverse, rosenbrock_inp, Active, Duplicated(x, dx))\n((nothing,),)\n\njulia> dx\n2-element Vector{Float64}:\n -400.0\n 200.0","category":"page"},{"location":"","page":"Home","title":"Home","text":"Both the inplace and \"normal\" variant return the gradient. The difference is that with Active the gradient is returned and with Duplicated the gradient is accumulated in place.","category":"page"},{"location":"#Forward-mode","page":"Home","title":"Forward mode","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"The return value of forward mode with a Duplicated return is a tuple containing as the first value the primal return value and as the second value the derivative.","category":"page"},{"location":"","page":"Home","title":"Home","text":"In forward mode Duplicated(x, 0.0) is equivalent to Const(x), except that we can perform more optimizations for Const.","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> autodiff(Forward, rosenbrock, Duplicated, Const(1.0), Duplicated(3.0, 1.0))\n(400.0, 400.0)\n\njulia> autodiff(Forward, rosenbrock, Duplicated, Duplicated(1.0, 1.0), Const(3.0))\n(400.0, -800.0)","category":"page"},{"location":"","page":"Home","title":"Home","text":"Of note, when we seed both arguments at once the tangent return is the sum of both.","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> autodiff(Forward, rosenbrock, Duplicated, Duplicated(1.0, 1.0), Duplicated(3.0, 1.0))\n(400.0, -400.0)","category":"page"},{"location":"","page":"Home","title":"Home","text":"We can also use forward mode with our inplace method.","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> x = [1.0, 3.0]\n2-element Vector{Float64}:\n 1.0\n 3.0\n\njulia> dx = [1.0, 1.0]\n2-element Vector{Float64}:\n 1.0\n 1.0\n\njulia> autodiff(Forward, rosenbrock_inp, Duplicated, Duplicated(x, dx))\n(400.0, -400.0)","category":"page"},{"location":"","page":"Home","title":"Home","text":"Note the seeding through dx.","category":"page"},{"location":"#Vector-forward-mode","page":"Home","title":"Vector forward mode","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"We can also use vector mode to calculate both derivatives at once.","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> autodiff(Forward, rosenbrock, BatchDuplicated, BatchDuplicated(1.0, (1.0, 0.0)), BatchDuplicated(3.0, (0.0, 1.0)))\n(400.0, (var\"1\" = -800.0, var\"2\" = 400.0))\n\njulia> x = [1.0, 3.0]\n2-element Vector{Float64}:\n 1.0\n 3.0\n\njulia> dx_1 = [1.0, 0.0]; dx_2 = [0.0, 1.0];\n\njulia> autodiff(Forward, rosenbrock_inp, BatchDuplicated, BatchDuplicated(x, (dx_1, dx_2)))\n(400.0, (var\"1\" = -800.0, var\"2\" = 400.0))","category":"page"},{"location":"#Convenience-functions","page":"Home","title":"Convenience functions","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"note: Note\nWhile the convenience functions discussed below use autodiff internally, they are generally more limited in their functionality. Beyond that, these convenience functions may also come with performance penalties; especially if one makes a closure of a multi-argument function instead of calling the appropriate multi-argument autodiff function directly.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Key convenience functions for common derivative computations are gradient (and its inplace variant gradient!) and jacobian. Like autodiff, the mode (forward or reverse) is determined by the first argument.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The functions gradient and gradient! compute the gradient of function with vector input and scalar return.","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> gradient(Reverse, rosenbrock_inp, [1.0, 2.0])\n2-element Vector{Float64}:\n -400.0\n 200.0\n\njulia> # inplace variant\n dx = [0.0, 0.0];\n gradient!(Reverse, dx, rosenbrock_inp, [1.0, 2.0])\n2-element Vector{Float64}:\n -400.0\n 200.0\n\njulia> dx\n2-element Vector{Float64}:\n -400.0\n 200.0\n\njulia> gradient(Forward, rosenbrock_inp, [1.0, 2.0])\n(-400.0, 200.0)\n\njulia> # in forward mode, we can also optionally pass a chunk size\n # to specify the number of derivatives computed simulateneously\n # using vector forward mode\n chunk_size = Val(2)\n gradient(Forward, rosenbrock_inp, [1.0, 2.0], chunk_size)\n(-400.0, 200.0)","category":"page"},{"location":"","page":"Home","title":"Home","text":"The function jacobian computes the Jacobian of a function vector input and vector return.","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> foo(x) = [rosenbrock_inp(x), prod(x)];\n\njulia> output_size = Val(2) # here we have to provide the output size of `foo` since it cannot be statically inferred\n jacobian(Reverse, foo, [1.0, 2.0], output_size) \n2×2 Matrix{Float64}:\n -400.0 200.0\n 2.0 1.0\n\njulia> chunk_size = Val(2) # By specifying the optional chunk size argument, we can use vector inverse mode to propogate derivatives of multiple outputs at once.\n jacobian(Reverse, foo, [1.0, 2.0], output_size, chunk_size)\n2×2 Matrix{Float64}:\n -400.0 200.0\n 2.0 1.0\n\njulia> jacobian(Forward, foo, [1.0, 2.0])\n2×2 Matrix{Float64}:\n -400.0 200.0\n 2.0 1.0\n\njulia> # Again, the optinal chunk size argument allows us to use vector forward mode\n jacobian(Forward, foo, [1.0, 2.0], chunk_size)\n2×2 Matrix{Float64}:\n -400.0 200.0\n 2.0 1.0","category":"page"},{"location":"#Caveats-/-Known-issues","page":"Home","title":"Caveats / Known-issues","text":"","category":"section"},{"location":"#Activity-of-temporary-storage-/-Activity-Unstable-Code","page":"Home","title":"Activity of temporary storage / Activity Unstable Code","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"If you pass in any temporary storage which may be involved in an active computation to a function you want to differentiate, you must also pass in a duplicated temporary storage for use in computing the derivatives. For example, consider the following function which uses a temporary buffer to compute the result.","category":"page"},{"location":"","page":"Home","title":"Home","text":"function f(x, tmp, k, n)\n tmp[1] = 1.0\n for i in 1:n\n tmp[k] *= x\n end\n tmp[1]\nend\n\n# output\n\nf (generic function with 1 method)","category":"page"},{"location":"","page":"Home","title":"Home","text":"Marking the argument for tmp as Const (aka non-differentiable) means that Enzyme believes that all variables loaded from or stored into tmp must also be non-differentiable, since all values inside a non-differentiable variable must also by definition be non-differentiable.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Enzyme.autodiff(Reverse, f, Active(1.2), Const(Vector{Float64}(undef, 1)), Const(1), Const(5)) # Incorrect\n\n# output\n\n((0.0, nothing, nothing, nothing),)","category":"page"},{"location":"","page":"Home","title":"Home","text":"Passing in a dupliacted (e.g. differentiable) variable for tmp now leads to the correct answer.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Enzyme.autodiff(Reverse, f, Active(1.2), Duplicated(Vector{Float64}(undef, 1), Vector{Float64}(undef, 1)), Const(1), Const(5)) # Correct (returns 10.367999999999999 == 1.2^4 * 5)\n\n# output\n\n((10.367999999999999, nothing, nothing, nothing),)","category":"page"},{"location":"","page":"Home","title":"Home","text":"However, even if we ignore the semantic guarantee provided by marking tmp as constant, another issue arises. When computing the original function, intermediate computations (like in f above) can use tmp for temporary storage. When computing the derivative, Enzyme also needs additional temporary storage space for the corresponding derivative variables as well. If tmp is marked as Const, Enzyme does not have any temporary storage space for the derivatives!","category":"page"},{"location":"","page":"Home","title":"Home","text":"Recent versions of Enzyme will attempt to error when they detect these latter types of situations, which we will refer to as activity unstable. This term is chosen to mirror the Julia notion of type-unstable code (e.g. where a type is not known at compile time). If an expression is activity unstable, it could either be constant, or active, depending on data not known at compile time. For example, consider the following:","category":"page"},{"location":"","page":"Home","title":"Home","text":"function g(cond, active_var, constant_var)\n if cond\n return active_var\n else\n return constant_var\nend\n\nEnzyme.autodiff(Forward, g, Const(condition), Duplicated(x, dx), Const(y))","category":"page"},{"location":"","page":"Home","title":"Home","text":"The returned value here could either by constant or duplicated, depending on the runtime-defined value of cond. If cond is true, Enzyme simply returns the shadow of active_var as the derivative. However, if cond is false, there is no derivative shadow for constant_var and Enzyme will throw a \"Mismatched activity\" error. For some simple types, e.g. a float Enzyme can circumvent this issue, for example by returning the float 0. Similarly, for some types like the Symbol type, which are never differentiable, such a shadow value will never be used, and Enzyme can return the original \"primal\" value as its derivative. However, for arbitrary data structures, Enzyme presently has no generic mechanism to resolve this.","category":"page"},{"location":"","page":"Home","title":"Home","text":"For example consider a third function:","category":"page"},{"location":"","page":"Home","title":"Home","text":"function h(cond, active_var, constant_var)\n return [g(cond, active_var, constant_var), g(cond, active_var, constant_var)]\nend\n\nEnzyme.autodiff(Forward, h, Const(condition), Duplicated(x, dx), Const(y))","category":"page"},{"location":"","page":"Home","title":"Home","text":"Enzyme provides a nice utility Enzyme.make_zero which takes a data structure and constructs a deepcopy of the data structure with all of the floats set to zero and non-differentiable types like Symbols set to their primal value. If Enzyme gets into such a \"Mismatched activity\" situation where it needs to return a differentiable data structure from a constant variable, it could try to resolve this situation by constructing a new shadow data structure, such as with Enzyme.make_zero. However, this still can lead to incorrect results. In the case of h above, suppose that active_var and consant_var are both arrays, which are mutable (aka in-place) data types. This means that the return of h is going to either be result = [active_var, active_var] or result = [constant_var, constant_var]. Thus an update to result[1][1] would also change result[2][1] since result[1] and result[2] are the same array. ","category":"page"},{"location":"","page":"Home","title":"Home","text":"If one created a new zero'd copy of each return from g, this would mean that the derivative dresult would have one copy made for the first element, and a second copy made for the second element. This could lead to incorrect results, and is unfortunately not a general resolution. However, for non-mutable variables (e.g. like floats) or non-differrentiable types (e.g. like Symbols) this problem can never arise.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Instead, Enzyme has a special mode known as \"Runtime Activity\" which can handle these types of situations. It can come with a minor performance reduction, and is therefore off by default. It can be enabled with Enzyme.API.runtimeActivity!(true) right after importing Enzyme for the first time. ","category":"page"},{"location":"","page":"Home","title":"Home","text":"The way Enzyme's runtime activity resolves this issue is to return the original primal variable as the derivative whenever it needs to denote the fact that a variable is a constant. As this issue can only arise with mutable variables, they must be represented in memory via a pointer. All addtional loads and stores will now be modified to first check if the primal pointer is the same as the shadow pointer, and if so, treat it as a constant. Note that this check is not saying that the same arrays contain the same values, but rather the same backing memory represents both the primal and the shadow (e.g. a === b or equivalently pointer(a) == pointer(b)). ","category":"page"},{"location":"","page":"Home","title":"Home","text":"Enabling runtime activity does therefore, come with a sharp edge, which is that if the computed derivative of a function is mutable, one must also check to see if the primal and shadow represent the same pointer, and if so the true derivative of the function is actually zero.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Generally, the preferred solution to these type of activity unstable codes should be to make your variables all activity-stable (e.g. always containing differentiable memory or always containing non-differentiable memory). However, with care, Enzyme does support \"Runtime Activity\" as a way to differentiate these programs without having to modify your code.","category":"page"},{"location":"#Mixed-Activity","page":"Home","title":"Mixed Activity","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Sometimes in Reverse mode (but not forward mode), you may see an error Type T has mixed internal activity types for some type. This error arises when a variable in a computation cannot be fully represented as either a Duplicated or Active variable.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Active variables are used for immutable variables (like Float64), whereas Duplicated variables are used for mutable variables (like Vector{Float64}). Speciically, since Active variables are immutable, functions with Active inputs will return the adjoint of that variable. In contrast Duplicated variables will have their derivatives +='d in place.","category":"page"},{"location":"","page":"Home","title":"Home","text":"This error indicates that you have a type, like Tuple{Float, Vector{Float64}} that has immutable components and mutable components. Therefore neither Active nor Duplicated can be used for this type.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Internally, by virtue of working at the LLVM level, most Julia types are represented as pointers, and this issue does not tend to arise within code fully differentiated by Enzyme internally. However, when a program needs to interact with Julia API's (e.g. as arguments to a custom rule, a type unstable call, or the outermost function being differentiated), Enzyme must adhere to Julia's notion of immutability and will throw this error rather than risk an incorrect result.","category":"page"},{"location":"","page":"Home","title":"Home","text":"For example, consider the following code, which has a type unstable call to myfirst, passing in a mixed type Tuple{Float64, Vector{Float64}}.","category":"page"},{"location":"","page":"Home","title":"Home","text":"@noinline function myfirst(tup::T) where T\n return tup[1]\nend\n\nfunction f(x::Float64)\n vec = [x]\n tup = (x, vec)\n Base.inferencebarrier(myfirst)(tup)::Float64\nend\n\nEnzyme.autodiff(Reverse, f, Active, Active(3.1))","category":"page"},{"location":"","page":"Home","title":"Home","text":"When this situation arises, it is often easiest to resolve it by adding a level of indirection to ensure the entire variable is mutable. For example, one could enclose this variable in a reference, such as Ref{Tuple{Float, Vector{Float64}}}, like as follows.","category":"page"},{"location":"","page":"Home","title":"Home","text":"@noinline function myfirst_ref(tup_ref::T) where T\n tup = tup_ref[]\n return tup[1]\nend\n\nfunction f2(x::Float64)\n vec = [x]\n tup = (x, vec)\n tup_ref = Ref(tup)\n Base.inferencebarrier(myfirst_ref)(tup_ref)::Float64\nend\n\nEnzyme.autodiff(Reverse, f2, Active, Active(3.1))","category":"page"},{"location":"#CUDA.jl-support","page":"Home","title":"CUDA.jl support","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"CUDA.jl is only supported on Julia v1.7.0 and onwards. On v1.6, attempting to differentiate CUDA kernel functions will not use device overloads correctly and thus returns fundamentally wrong results.","category":"page"},{"location":"#Sparse-Arrays","page":"Home","title":"Sparse Arrays","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"At the moment there is limited support for sparse linear algebra operations. Sparse arrays may be used, but care must be taken because backing arrays drop zeros in Julia (unless told not to).","category":"page"},{"location":"","page":"Home","title":"Home","text":"using SparseArrays\na = sparse([2.0])\nda1 = sparse([0.0]) # Incorrect: SparseMatrixCSC drops explicit zeros\nEnzyme.autodiff(Reverse, sum, Active, Duplicated(a, da1))\nda1\n\n# output\n\n1-element SparseVector{Float64, Int64} with 0 stored entries","category":"page"},{"location":"","page":"Home","title":"Home","text":"da2 = sparsevec([1], [0.0]) # Correct: Prevent SparseMatrixCSC from dropping zeros\nEnzyme.autodiff(Reverse, sum, Active, Duplicated(a, da2))\nda2\n\n# output\n\n1-element SparseVector{Float64, Int64} with 1 stored entry:\n [1] = 1.0","category":"page"},{"location":"","page":"Home","title":"Home","text":"Sometimes, determining how to perform this zeroing can be complicated. That is why Enzyme provides a helper function Enzyme.make_zero that does this automatically.","category":"page"},{"location":"#Complex-Numbers","page":"Home","title":"Complex Numbers","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Differentiation of a function which returns a complex number is ambiguous, because there are several different gradients which may be desired. Rather than assume a specific of these conventions and potentially result in user error when the resulting derivative is not the desired one, Enzyme forces users to specify the desired convention by returning a real number instead.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Consider the function f(z) = z*z. If we were to differentiate this and have real inputs and outputs, the derivative f'(z) would be unambiguously 2*z. However, consider breaking down a complex number down into real and imaginary parts. Suppose now we were to call f with the explicit real and imaginary components, z = x + i y. This means that f is a function that takes an input of two values and returns two values f(x, y) = u(x, y) + i v(x, y). In the case of z*z this means that u(x,y) = x*x-y*y and v(x,y) = 2*x*y.","category":"page"},{"location":"","page":"Home","title":"Home","text":"If we were to look at all first-order derivatives in total, we would end up with a 2x2 matrix (i.e. Jacobian), the derivative of each output wrt each input. Let's try to compute this, first by hand, then with Enzyme.","category":"page"},{"location":"","page":"Home","title":"Home","text":"grad u(x, y) = [d/dx u, d/dy u] = [d/dx x*x-y*y, d/dy x*x-y*y] = [2*x, -2*y];\ngrad v(x, y) = [d/dx v, d/dy v] = [d/dx 2*x*y, d/dy 2*x*y] = [2*y, 2*x];","category":"page"},{"location":"","page":"Home","title":"Home","text":"Reverse mode differentiation computes the derivative of all inputs with respect to a single output by propagating the derivative of the return to its inputs. Here, we can explicitly differentiate with respect to the real and imaginary results, respectively, to find this matrix.","category":"page"},{"location":"","page":"Home","title":"Home","text":"f(z) = z * z\n\n# a fixed input to use for testing\nz = 3.1 + 2.7im\n\ngrad_u = Enzyme.autodiff(Reverse, z->real(f(z)), Active, Active(z))[1][1]\ngrad_v = Enzyme.autodiff(Reverse, z->imag(f(z)), Active, Active(z))[1][1]\n\n(grad_u, grad_v)\n# output\n(6.2 - 5.4im, 5.4 + 6.2im)","category":"page"},{"location":"","page":"Home","title":"Home","text":"This is somewhat inefficient, since we need to call the forward pass twice, once for the real part, once for the imaginary. We can solve this using batched derivatives in Enzyme, which computes several derivatives for the same function all in one go. To make it work, we're going to need to use split mode, which allows us to provide a custom derivative return value.","category":"page"},{"location":"","page":"Home","title":"Home","text":"fwd, rev = Enzyme.autodiff_thunk(ReverseSplitNoPrimal, Const{typeof(f)}, Active, Active{ComplexF64})\n\n# Compute the reverse pass seeded with a differntial return of 1.0 + 0.0im\ngrad_u = rev(Const(f), Active(z), 1.0 + 0.0im, fwd(Const(f), Active(z))[1])[1][1]\n# Compute the reverse pass seeded with a differntial return of 0.0 + 1.0im\ngrad_v = rev(Const(f), Active(z), 0.0 + 1.0im, fwd(Const(f), Active(z))[1])[1][1]\n\n(grad_u, grad_v)\n\n# output\n(6.2 - 5.4im, 5.4 + 6.2im)","category":"page"},{"location":"","page":"Home","title":"Home","text":"Now let's make this batched","category":"page"},{"location":"","page":"Home","title":"Home","text":"fwd, rev = Enzyme.autodiff_thunk(ReverseSplitWidth(ReverseSplitNoPrimal, Val(2)), Const{typeof(f)}, Active, Active{ComplexF64})\n\n# Compute the reverse pass seeded with a differential return of 1.0 + 0.0im and 0.0 + 1.0im in one go!\nrev(Const(f), Active(z), (1.0 + 0.0im, 0.0 + 1.0im), fwd(Const(f), Active(z))[1])[1][1]\n\n# output\n(6.2 - 5.4im, 5.4 + 6.2im)","category":"page"},{"location":"","page":"Home","title":"Home","text":"In contrast, Forward mode differentiation computes the derivative of all outputs with respect to a single input by providing a differential input. Thus we need to seed the shadow input with either 1.0 or 1.0im, respectively. This will compute the transpose of the matrix we found earlier.","category":"page"},{"location":"","page":"Home","title":"Home","text":"d/dx f(x, y) = d/dx [u(x,y), v(x,y)] = d/dx [x*x-y*y, 2*x*y] = [ 2*x, 2*y];\nd/dy f(x, y) = d/dy [u(x,y), v(x,y)] = d/dy [x*x-y*y, 2*x*y] = [-2*y, 2*x];","category":"page"},{"location":"","page":"Home","title":"Home","text":"d_dx = Enzyme.autodiff(Forward, f, Duplicated(z, 1.0+0.0im))[1]\nd_dy = Enzyme.autodiff(Forward, f, Duplicated(z, 0.0+1.0im))[1]\n\n(d_dx, d_dy)\n\n# output\n(6.2 + 5.4im, -5.4 + 6.2im)","category":"page"},{"location":"","page":"Home","title":"Home","text":"Again, we can go ahead and batch this.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Enzyme.autodiff(Forward, f, BatchDuplicated(z, (1.0+0.0im, 0.0+1.0im)))[1]\n\n# output\n(var\"1\" = 6.2 + 5.4im, var\"2\" = -5.4 + 6.2im)","category":"page"},{"location":"","page":"Home","title":"Home","text":"Taking Jacobians with respect to the real and imaginary results is fine, but for a complex scalar function it would be really nice to have a single complex derivative. More concretely, in this case when differentiating z*z, it would be nice to simply return 2*z. However, there are four independent variables in the 2x2 jacobian, but only two in a complex number. ","category":"page"},{"location":"","page":"Home","title":"Home","text":"Complex differentiation is often viewed in the lens of directional derivatives. For example, what is the derivative of the function as the real input increases, or as the imaginary input increases. Consider the derivative along the real axis, textttlim_Delta x rightarrow 0 fracf(x+Delta x y)-f(x y)Delta x. This simplifies to textttlim_Delta x rightarrow 0 fracu(x+Delta x y)-u(x y) + i left v(x+Delta x y)-v(x y)rightDelta x = fracpartialpartial x u(xy) + ifracpartialpartial x v(xy). This is exactly what we computed by seeding forward mode with a shadow of 1.0 + 0.0im.","category":"page"},{"location":"","page":"Home","title":"Home","text":"For completeness, we can also consider the derivative along the imaginary axis textttlim_Delta y rightarrow 0 fracf(x y+Delta y)-f(x y)iDelta y. Here this simplifies to textttlim_u(x y+Delta y)-u(x y) + i left v(x y+Delta y)-v(x y)rightiDelta y = -ifracpartialpartial y u(xy) + fracpartialpartial y v(xy). Except for the i in the denominator of the limit, this is the same as the result of Forward mode, when seeding x with a shadow of 0.0 + 1.0im. We can thus compute the derivative along the real axis by multiplying our second Forward mode call by -im.","category":"page"},{"location":"","page":"Home","title":"Home","text":"d_real = Enzyme.autodiff(Forward, f, Duplicated(z, 1.0+0.0im))[1]\nd_im = -im * Enzyme.autodiff(Forward, f, Duplicated(z, 0.0+1.0im))[1]\n\n(d_real, d_im)\n\n# output\n(6.2 + 5.4im, 6.2 + 5.4im)","category":"page"},{"location":"","page":"Home","title":"Home","text":"Interestingly, the derivative of z*z is the same when computed in either axis. That is because this function is part of a special class of functions that are invariant to the input direction, called holomorphic. ","category":"page"},{"location":"","page":"Home","title":"Home","text":"Thus, for holomorphic functions, we can simply seed Forward-mode AD with a shadow of one for whatever input we are differenitating. This is nice since seeding the shadow with an input of one is exactly what we'd do for real-valued funtions as well.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Reverse-mode AD, however, is more tricky. This is because holomorphic functions are invariant to the direction of differentiation (aka the derivative inputs), not the direction of the differential return.","category":"page"},{"location":"","page":"Home","title":"Home","text":"However, if a function is holomorphic, the two derivative functions we computed above must be the same. As a result, fracpartialpartial x u = fracpartialpartial y v and fracpartialpartial y u = -fracpartialpartial x v. ","category":"page"},{"location":"","page":"Home","title":"Home","text":"We saw earlier, that performing reverse-mode AD with a return seed of 1.0 + 0.0im yielded [d/dx u, d/dy u]. Thus, for a holomorphic function, a real-seeded Reverse-mode AD computes [d/dx u, -d/dx v], which is the complex conjugate of the derivative.","category":"page"},{"location":"","page":"Home","title":"Home","text":"conj(grad_u)\n\n# output\n\n6.2 + 5.4im","category":"page"},{"location":"","page":"Home","title":"Home","text":"In the case of a scalar-input scalar-output function, that's sufficient. However, most of the time one uses reverse mode, it involves either several inputs or outputs, perhaps via memory. This case requires additional handling to properly sum all the partial derivatives from the use of each input and apply the conjugate operator at only the ones relevant to the differential return.","category":"page"},{"location":"","page":"Home","title":"Home","text":"For simplicity, Enzyme provides a helper utlity ReverseHolomorphic which performs Reverse mode properly here, assuming that the function is indeed holomorphic and thus has a well-defined single derivative.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Enzyme.autodiff(ReverseHolomorphic, f, Active, Active(z))[1][1]\n\n# output\n\n6.2 + 5.4im","category":"page"},{"location":"","page":"Home","title":"Home","text":"For even non-holomorphic functions, complex analysis allows us to define fracpartialpartial z = frac12left(fracpartialpartial x - i fracpartialpartial y right). For non-holomorphic functions, this allows us to compute d/dz. Let's consider myabs2(z) = z * conj(z). We can compute the derivative wrt z of this in Forward mode as follows, which as one would expect results in a result of conj(z):","category":"page"},{"location":"","page":"Home","title":"Home","text":"myabs2(z) = z * conj(z)\n\ndabs2_dx, dabs2_dy = Enzyme.autodiff(Forward, myabs2, BatchDuplicated(z, (1.0 + 0.0im, 0.0 + 1.0im)))[1]\n(dabs2_dx - im * dabs2_dy) / 2\n\n# output\n\n3.1 - 2.7im","category":"page"},{"location":"","page":"Home","title":"Home","text":"Similarly, we can compute d/d conj(z) = d/dx + i d/dy.","category":"page"},{"location":"","page":"Home","title":"Home","text":"(dabs2_dx + im * dabs2_dy) / 2\n\n# output\n\n3.1 + 2.7im","category":"page"},{"location":"","page":"Home","title":"Home","text":"Computing this in Reverse mode is more tricky. Let's expand f in terms of u and v. fracpartialpartial z f = frac12 left( u_x + i v_x - i u_y + i v_y right) = frac12 left( u_x + v_y + i v_x - u_y right). Thus d/dz = (conj(grad_u) + im * conj(grad_v))/2.","category":"page"},{"location":"","page":"Home","title":"Home","text":"abs2_fwd, abs2_rev = Enzyme.autodiff_thunk(ReverseSplitWidth(ReverseSplitNoPrimal, Val(2)), Const{typeof(myabs2)}, Active, Active{ComplexF64})\n\n# Compute the reverse pass seeded with a differential return of 1.0 + 0.0im and 0.0 + 1.0im in one go!\ngradabs2_u, gradabs2_v = abs2_rev(Const(myabs2), Active(z), (1.0 + 0.0im, 0.0 + 1.0im), abs2_fwd(Const(myabs2), Active(z))[1])[1][1]\n\n(conj(gradabs2_u) + im * conj(gradabs2_v)) / 2\n\n# output\n\n3.1 - 2.7im","category":"page"},{"location":"","page":"Home","title":"Home","text":"For d/d conj(z), frac12 left( u_x + i v_x + i u_y + i v_y right) = frac12 left( u_x - v_y + i v_x + u_y right). Thus d/d conj(z) = (grad_u + im * grad_v)/2.","category":"page"},{"location":"","page":"Home","title":"Home","text":"(gradabs2_u + im * gradabs2_v) / 2\n\n# output\n\n3.1 + 2.7im","category":"page"},{"location":"","page":"Home","title":"Home","text":"Note: when writing rules for complex scalar functions, in reverse mode one needs to conjugate the differential return, and similarly the true result will be the conjugate of that value (in essence you can think of reverse-mode AD as working in the conjugate space).","category":"page"}] }