Unsupervised Learning of Solutions to Differential Equations with Generative Adversarial Networks

TL;DR

This paper introduces DEQGAN, an unsupervised neural network method using GANs to solve differential equations, achieving lower errors and competitive accuracy compared to traditional methods.

Contribution

It develops a novel GAN-based approach for solving differential equations that learns the loss function, improving accuracy over existing neural network methods.

Findings

DEQGAN achieves significantly lower mean squared errors than alternative neural methods.

DEQGAN attains solution accuracy comparable to traditional numerical methods.

The approach's stability is sensitive to hyperparameter choices.

Abstract

Solutions to differential equations are of significant scientific and engineering relevance. Recently, there has been a growing interest in solving differential equations with neural networks. This work develops a novel method for solving differential equations with unsupervised neural networks that applies Generative Adversarial Networks (GANs) to \emph{learn the loss function} for optimizing the neural network. We present empirical results showing that our method, which we call Differential Equation GAN (DEQGAN), can obtain multiple orders of magnitude lower mean squared errors than an alternative unsupervised neural network method based on (squared) $L_2$, $L_1$, and Huber loss functions. Moreover, we show that DEQGAN achieves solution accuracy that is competitive with traditional numerical methods. Finally, we analyze the stability of our approach and find it to be sensitive to the selection of hyperparameters, which we provide in the appendix. Code available at https://github.com/dylanrandle/denn. Please address any electronic correspondence to dylanrandle@alumni.harvard.edu.