Cross-time causal physics-informed neural networks for dynamic time-dependent problems

Physics-informed neural networks (PINNs) provide a new paradigm for solving partial differential equations (PDEs) and have made significant progress in scientific computing. By incorporating the physical constraints of the governing equations into the loss function, PINNs enable neural networks to learn and approximate the behavior of physical systems during the optimization process. Despite the considerable potential of the standard formulation, it shows limitations in accurately predicting the dynamic behavior of time-dependent problems. To address this challenge, previous research has proposed dividing the time domain into multiple segments and using different neural networks for each segment; however, this method does not guarantee causality within individual temporal subdomains. Inspired by previous work on causal relationships in PINNs, we propose a Cross-Time Causal PINNs algorithm — a overlapping domain decomposition method that integrates both soft and hard causality constraints. Specifically, within each time subdomain, we employ an exponentially decaying soft causal weighting scheme that prioritizes learning physical patterns at earlier time points. Between subdomains, we establish a hard causality constraint using a parameter transfer operator, which takes the output of the previous network as the initial condition for the next subdomain, thereby ensuring continuous differentiability of the solution across subdomains. Additionally, we introduce a composite loss term over the overlapping regions and use an adaptive penalty coefficient to balance the accuracy and generalizability of the solution, while expanding the receptive field of causality. The proposed method demonstrates superior performance and higher computational efficiency compared to existing methods when solving the Allen–Cahn and Korteweg–de Vries equations.