A symmetry group based supervised learning method for solving partial differential equations

Abstract

Deep neural network provides an effective way to solve partial differential equations (PDEs) where physics-informed neural network (PINN) is a typical representative of which the outputs are constrained to approximate the solution of PDEs. However, the PINN method still fails to learn some solutions of particular properties and the reasons of failure are still unclear. In this paper, we employ both the loss landscape of the final optimization parameters of network and the imbalance of back-propagation gradient during the training to analyze why the PINN method fails to learn large amplitude solution and high-frequency solution of the Klein–Gordon equation. Consequently, we find that the PDE-based soft constraints in the loss function take a major responsibility for the failures of the PINN method in predicting such types of solutions. To remedy the pathologies we remove the PDE-based soft constraints in the loss function and calibrate a supervised learning model for predicting such types of particular solutions for a class of PDEs which cannot be learned by the PINN method and the two weight-related improved PINN methods. Specifically, if the governing PDEs under study admit a Lie symmetry group and the unique solution of the corresponding initial boundary problem is obtained via the symmetry group, we utilize the symmetry group to generate labeled data in the interior domain of PDEs from the discrete points on the known initial and boundary conditions. Then, with the labeled data we leverage the supervised learning model to predict the large amplitude solution and the high-frequency solution of the Klein–Gordon equation on a rectangular domain by means of a space–time translation group and a singular solution of the heat equation on a moving boundary domain with a physical symmetry group. Numerical experiments confirm the good performances of the proposed supervised learning method.