Solving partial differential equations based on preconditioning-pretraining physics-informed neural network

Abstract

Physics-Informed Neural Network (PINN) is a deep learning framework that has been widely employed to solve spatial-temporal partial differential equations (PDEs) across various fields. However, recent numerical experiments indicate that the vanilla-PINN often struggles with PDEs featuring high-frequency solutions or strong nonlinearity. To enhance PINN's performance, we propose a novel strategy called the Preconditioning-Pretraining Physics-Informed Neural Network (PP-PINN). This approach involves transforming the original task into a new system characterized by low frequency and weak nonlinearity over an extended time scale. The transformed PDEs are then solved using a pretraining approach. Additionally, we introduce a new constraint termed “fixed point”, which is beneficial for scenarios with extremely high frequency or strong nonlinearity. To demonstrate the efficacy of our method, we apply the newly developed strategy to three different equations, achieving improved accuracy and reduced computational costs compared to previous approaches which incorporate the pretraining technique. The effectiveness and interpretability of our PP-PINN are also discussed, emphasizing its advantages in tackling high-frequency solutions and strong nonlinearity, thereby offering insights into its broader applicability in complex mathematical modeling.