Studying Shallow and Deep Convolutional Neural Networks as Learned Numerical Schemes on the 1D Heat Equation and Burgers' Equation

TL;DR

This paper explores how shallow and deep convolutional neural networks can learn numerical schemes for solving 1D heat and Burgers' equations, showing convergence to traditional methods and comparable accuracy.

Contribution

It demonstrates that CNNs can learn traditional finite difference and finite volume schemes, bridging neural networks with classical numerical methods for PDEs.

Findings

Single-layer CNN converges to finite difference stencil for heat equation

Adversarial training does not find expected weights

Deep CNN achieves accuracy and stability similar to Godunov's method for Burgers' equation

Abstract

This paper examines the coincidence of neural networks with numerical methods for solving spatiotemporal physical problems. Neural networks are used to learn predictive numerical models from trajectory datasets from two well understood 1D problems: the heat equation and the inviscid Burgers' equation. Coincidence with established numerical methods is shown by demonstrating that a single layer convolutional neural network (CNN) converges to a traditional finite difference stencil for the heat equation. However, a discriminator-based adversarial training method, such as those used in generative adversarial networks (GANs), does not find the expected weights. A compact deep CNN is applied to nonlinear Burgers' equation, where the models' architecture is reminiscent of existing winding finite volume methods. By searching over architectures and using multiple recurrent steps in the training loss, a model is found that can integrate in time, recurring on its outputs, with similar accuracy and stability to Godunov's method.