A Staged Training Approach for Physics-Informed Neural Networks in Solving Partial Differential Equations

Partial differential equations (PDEs) serve as fundamental mathematical frameworks for modeling natural phenomena and engineering challenges, playing an irreplaceable role in scientific research and engineering practice. Since exact solutions are rarely obtainable for most PDEs, numerical approachs have traditionally been used to compute approximate solutions. With recent advances in deep learning, physics-informed neural networks (PINNs) have emerged as a promising approach for numerically solving PDEs. In this paper, we propose a novel staged training strategy for PINNs to improve the performance in solving PDEs. The approach implements a two-stage process: first, an initial phase pre-trains model on a short time domain, then a formal phase extends the initially trained model to the full time domain. The initial phase provides better parameter initialization and additional training data for the formal phase training, thereby improving the accuracy and efficiency of standard PINNs approach. Furthermore, we incorporate an uncertainty-guided sampling strategy to optimize model's ability to generalize beyond training samples. Numerous numerical simulations confirm the efficacy and robustness of the proposed approach in solving time-dependent PDEs.