Causality-respecting adaptive refinement for PINNs: enabling precise interface evolution in phase field modeling

Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving physical systems described by
partial differential equations (PDEs). However, their accuracy in dynamical systems, particularly those involving sharp
moving boundaries with complex initial morphologies, remains a challenge. This study introduces an approach combining
residual-based adaptive refinement (RBAR) with causality-informed training to enhance the performance of PINNs in solving
spatio-temporal PDEs. Our method employs a three-step iterative process: initial causality-based training, RBAR-guided
domain refinement, and subsequent causality training on the refined mesh. Applied to the Allen-Cahn equation, a widely-used
model in phase field simulations, our approach demonstrates significant improvements in solution accuracy and computational
efficiency over traditional PINNs. Notably, we observe an ‘overshoot and relocate’ phenomenon in dynamic cases with complex morphologies, showcasing the method’s adaptive error correction capabilities. This synergistic interaction between RBAR and causality training enables accurate capture of interface evolution, even in challenging scenarios where traditional PINNs fail. Our framework not only resolves the limitations of uniform refinement strategies but also provides a generalizable methodology for solving a broad range of spatio-temporal PDEs. The enhanced performance of the RBAR–causality combined framework demonstrates its strong potential for advancing PINN-based modeling of physical systems characterized by complex, evolving interfaces.