DEQGAN: Learning the Loss Function for PINNs with Generative Adversarial Networks
This addresses the lack of theoretical justification for loss functions in PINNs, offering a novel approach for scientific and engineering applications, though it appears incremental as it builds on existing PINN and GAN frameworks.
The paper tackles the problem of solving differential equations by proposing DEQGAN, a method that uses generative adversarial networks to learn the loss function for Physics-Informed Neural Networks (PINNs), achieving multiple orders of magnitude lower mean squared errors than PINNs with traditional loss functions on a suite of twelve equations.
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.