Pre-training strategy for solving evolution equations based on physics-informed neural networks

The physics informed neural network (PINN) is a promising method for solving time-evolution partial differential equations (PDEs). However, the standard PINN method may fail to solve the PDEs with strongly nonlinear characteristics or those with high-frequency solutions. To address this problem, we propose a novel method named pre-training PINN (PT-PINN) which can improve the convergence and accuracy of the standard PINN method by combining with the resampling strategy and the existing optimizer combination technique. The PT-PINN method transforms the difficult problem on the entire time domain to relatively simple problems defined on small subdomains. The neural network trained on small subdomains provides the neural network initialization and extra supervised learning data for the problems on larger subdomains or on the entire time-domain. Numerical experiments show that the PT-PINN successfully solves the evolution PDEs with strong non-linearity and/or high frequency solutions, including the strongly nonlinear heat equation, the Allen-Cahn equation, the convection equation with high-frequency solutions and so on. The PT-PINN method is a reliable and competitive method for solving the time-evolution PDEs.