DEQGAN: Learning the Loss Function for PINNs with Generative Adversarial Networks

TL;DR

DEQGAN introduces a generative adversarial network approach to learn the optimal loss function for solving differential equations, significantly improving accuracy over traditional PINNs and matching numerical methods.

Contribution

This work presents a novel GAN-based framework for learning loss functions in PINNs, providing a theoretical basis and enhanced accuracy for differential equation solutions.

Findings

DEQGAN achieves lower mean squared errors than traditional PINNs.

DEQGAN's solutions are competitive with numerical methods.

Two methods improve DEQGAN's robustness to hyperparameters.

Abstract

Solutions to differential equations are of significant scientific and engineering relevance. Physics-Informed Neural Networks (PINNs) have emerged as a promising method for solving differential equations, but they lack a theoretical justification for the use of any particular loss function. This work presents Differential Equation GAN (DEQGAN), a novel method for solving differential equations using generative adversarial networks to "learn the loss function" for optimizing the neural network. Presenting results on a suite of twelve ordinary and partial differential equations, including the nonlinear Burgers', Allen-Cahn, Hamilton, and modified Einstein's gravity equations, we show that DEQGAN can obtain multiple orders of magnitude lower mean squared errors than PINNs that use $L_2$, $L_1$, and Huber loss functions. We also show that DEQGAN achieves solution accuracies that are competitive with popular numerical methods. Finally, we present two methods to improve the robustness of DEQGAN to different hyperparameter settings.