PIAT: Physics Informed Adversarial Training for Solving Partial Differential Equations

TL;DR

PIAT introduces a physics-informed adversarial training method that enhances neural network solutions for nonlinear differential equations by promoting smoothness and incorporating physical laws, outperforming traditional PINN approaches.

Contribution

The paper presents a novel physics-informed adversarial training framework for neural networks to solve nonlinear differential equations more effectively and smoothly.

Findings

PIAT outperforms PINN in solving NDEs up to 10 dimensions.

Adversarial training improves the smoothness of neural network solutions.

Weight decay and Gaussian smoothing further enhance PIAT's performance.

Abstract

In this paper, we propose the physics informed adversarial training (PIAT) of neural networks for solving nonlinear differential equations (NDE). It is well-known that the standard training of neural networks results in non-smooth functions. Adversarial training (AT) is an established defense mechanism against adversarial attacks, which could also help in making the solution smooth. AT include augmenting the training mini-batch with a perturbation that makes the network output mismatch the desired output adversarially. Unlike formal AT, which relies only on the training data, here we encode the governing physical laws in the form of nonlinear differential equations using automatic differentiation in the adversarial network architecture. We compare PIAT with PINN to indicate the effectiveness of our method in solving NDEs for up to 10 dimensions. Moreover, we propose weight decay and Gaussian smoothing to demonstrate the PIAT advantages. The code repository is available at https://github.com/rohban-lab/PIAT.