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Solutions to differential equations are of significant scientific andengineering relevance. Physics-Informed Neural Networks (PINNs) have emerged asa promising method for solving differential equations, but they lack atheoretical justification for the use of any particular loss function. Thiswork presents Differential Equation GAN (DEQGAN), a novel method for solvingdifferential equations using generative adversarial networks to "learn the lossfunction" for optimizing the neural network. Presenting results on a suite oftwelve ordinary and partial differential equations, including the nonlinearBurgers', Allen-Cahn, Hamilton, and modified Einstein's gravity equations, weshow that DEQGAN can obtain multiple orders of magnitude lower mean squarederrors than PINNs that use $L_2$, $L_1$, and Huber loss functions. We also showthat DEQGAN achieves solution accuracies that are competitive with popularnumerical methods. Finally, we present two methods to improve the robustness ofDEQGAN to different hyperparameter settings.

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