Neural networks can be FLOP-efficient integrators of 1D oscillatory integrands

Authors: Anshuman Sinha, Spencer H. Bryngelson (Georgia Tech)

Published at TMLR (2024) ISSN 2835-8856, link: https://openreview.net/pdf?id=5psgQEHn6t

We demonstrate that neural networks can be FLOP-efficient integrators of one-dimensional oscillatory integrands. We train a feed-forward neural network to compute integrals of highly oscillatory 1D functions. The training set is a parametric combination of functions with varying characters and oscillatory behavior degrees. Numerical examples show that these networks are FLOP-efficient for sufficiently oscillatory integrands with an average FLOP gain of $10^3$ FLOPs. The network calculates oscillatory integrals better than traditional quadrature methods under the same computational budget or number of floating point operations. We find that feed-forward networks of 5 hidden layers are satisfactory for a relative accuracy of $10^{-3}$. The computational burden of inference of the neural network is relatively small, even compared to inner-product pattern quadrature rules. We postulate that our result follows from learning latent patterns in the oscillatory integrands that are otherwise opaque to traditional numerical integrators.

Dependencies

• DeepXDE (pip install deepxde)
• Tensorflow >= 3.8 (pip install tensorflow)

Install

To reproduce our results:

git clone https://github.com/comp-physics/deepOscillations.git
cd deepOscillations
bash run.sh

Example case

• Choose the desired function, for example func_str='Levin1'
• Set the desired n_array= (_) b_array=(_) s_array=(_) values in the run.sh script
• Execute

bash run.sh
python3 collect_results.py
python3 plot_result.py

Acknowledgement

The authors appreciate discussion with Dr. Ethan Pickering at an early stage of this work.

License

MIT