DP-PINN+: A Dual-Phase PINN learning with automated phase division

Abstract

Physics-Informed Neural Networks (PINNs) are a promising application of deep neural networks for the numerical solution of nonlinear partial differential equations (PDEs). However, it has been observed that standard PINNs may not be able to accurately fit all types of PDEs, leading to poor predictions for specific regions in the domain. A common solution is to partition the domain by time and train each time interval separately. However, this approach leads to the prediction errors being accumulated over time, which is especially the case when solving “stiff” PDEs. To address these issues, we propose a new PINN training scheme, called DP-PINN+ (Dual-Phase PINN+). DP-PINN+ divides the training into two phases based on a carefully chosen time point $t_s$. The phase-1 training aims to generate the accurate solution at $t_s$, which will serve as the additional intermediate condition for the phase-2 training. The method for determining the optimized value of $t_s$ is proposed in this paper. Further,new sampling strategies are proposed to enhance the training process. These design considerations improve the prediction accuracy significantly. We have conducted the experiments to evaluate DP-PINN+ with both “stiff” and non-stiff PDEs, including 1D Burger’s Equation, 1D Allen–Cahn Equation, 2D and 3D Navier–Stokes Equations (i.e., 2D cylinder wake and 3D unsteady Beltrami flow). We compared DP-PINN+ with the state-of-the-art PINNs in literature, including Time Adaptive PINN, SA-PINN, bc-PINN and NSFNets.The results show that the solutions predicted by DP-PINN+ exhibit significantly higher accuracy. This paper is extended from our conference paper published in ICCS2024 Yan and He (2024) [1].