Learning to Solve Multiple Partial Differential Equations Using Physics-informed Neural Networks

Abstract

Lately, researchers have used neural networks to solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering unseen data points and a high nonlinearity domain. To improve the generalizability, we introduce the novel approach of employing the multi-task learning technique, called the uncertainty-weighting loss, in the context of learning PDE solutions. The multi-task scheme exploits the benefits of learning shared representations, controlled by cross-stitch modules, between multiple related PDEs. An auxiliary PDE is obtainable by varying the PDE parameterization coefficient, to generalize better on the original PDE. Letting the network pay closer attention to the high nonlinearity domain regions that are more challenging to learn, we also propose adversarial training for generating supplementary high-loss samples. In the experiment, our proposed method is found to be effective and reduce the error on the unseen data points as compared to the previous approaches.