Unsupervised Learning of Solutions to Differential Equations with Generative Adversarial Networks
This addresses the problem of efficiently solving differential equations for scientific and engineering applications, representing an incremental improvement with a novel method for a known bottleneck.
The paper tackles solving differential equations by developing a novel unsupervised method using Generative Adversarial Networks (GANs) to learn the loss function, achieving multiple orders of magnitude lower mean squared errors compared to alternative unsupervised neural network methods and competitive accuracy with traditional numerical methods.
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.