Solving time-dependent partial differential equations via hard-constrained physics-informed neural networks with adaptive causal weighting

Physics-informed neural networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs). However, PINNs often struggle to accurately capture the temporal dynamics of complex time-dependent PDEs, with one major factor being their failure to respect the intrinsic temporal causality structure that governs the forward evolution of PDE systems. In this study, we conduct an error analysis of PINNs for a class of first-order in time PDEs and another class of second-order in time PDEs with a linear diffusion term. Our analysis demonstrates the necessity of respecting temporal causality in PINNs, and shows that the poor initial condition matching and the imbalanced PDE residual distribution in the causal sense can lead to the causality-related failure mode. To overcome these issues, we propose the temporal causality-enhanced PINN (TC-PINN), which integrates an adaptive causal weighting strategy and hard-constraint approaches for initial conditions. The former adaptively adjusts the parameterized loss weighting function based on the PDE residual distribution over time, while the latter eliminates the initial condition mismatch through a specific solution ansatz. Experiments on several challenging time-dependent PDE benchmarks demonstrate the effectiveness of the TC-PINN and its applicability to more general problems. Furthermore, we validate the importance of the two core components through an ablation study and confirm the robustness of the TC-PINN across different experimental settings.