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.

翻译：物理信息神经网络（PINNs）已成为求解偏微分方程（PDEs）所描述物理系统的有力工具。然而，其在动态系统（尤其是涉及具有复杂初始形态的尖锐移动边界的问题）中的精度仍然面临挑战。本研究提出了一种将基于残差的自适应细化（RBAR）与因果信息训练相结合的方法，以提升PINNs求解时空偏微分方程的性能。我们的方法采用三步迭代流程：初始基于因果关系的训练、RBAR引导的区域细化，以及在细化网格上进行的后续因果训练。将该方法应用于相场模拟中广泛使用的Allen-Cahn方程时，相较于传统PINNs，我们的方法在求解精度和计算效率方面均展现出显著提升。值得注意的是，我们在具有复杂形态的动态案例中观察到超调与重定位现象，这体现了该方法的自适应误差修正能力。RBAR与因果训练之间的协同作用使得界面演化能够被精确捕捉，即使在传统PINNs失效的挑战性场景中亦然。我们的框架不仅解决了均匀细化策略的局限性，还为求解广泛的时空偏微分方程提供了可推广的方法论。RBAR-因果联合框架增强的性能，彰显了其在推进具有复杂演化界面特征的物理系统PINN建模方面的巨大潜力。