Long-time Integration of Linear Partial Differential Equations Based on Discrete Physics-informed Neural Network

In response to the potential failure of continuous physics-informed neural networks (PINNs) in long-time integration of partial differential equations (PDEs) due to violations of temporal causality, a discrete physics-informed neural network method combining warm-start and double-precision computation was proposed. A time-discretization strategy was introduced into the physics-informed neural network to better adhere to temporal causality. Simultaneously, a warm-start scheme was implemented, where the model parameters trained at the previous time step were used as the initial parameters for the next time step, thereby reducing computational overhead. On the other hand, double-precision floating-point computation was employed to enhance the accuracy of solving linear PDEs. The effectiveness of the proposed method was validated through numerical experiments on the convection equation and the wave equation. The results demonstrate that the introduction of the warm-start scheme improves computational efficiency by 2-3 times and can enhance computational accuracy by approximately 20%. In long-time integration tasks, double-precision computation improves computational accuracy by about 10 times, albeit at the cost of increased computation time (7-8 times). Additionally, the introduction of extra physical constraints can improve computational accuracy by 5-6 times with almost no increase in computational overhead. By combining the warm-start scheme and double-precision computation, the discrete physics-informed neural network achieves high-precision long-time integration at an acceptable computational cost, providing a new solution for long-time simulations of complex systems.