Physics-informed neural networks (PINNs) have shown promising potential for solving partial differential equations (PDEs) using deep learning. However, PINNs face training difficulties for evolutionary PDEs, particularly for dynamical systems whose solutions exhibit multi-scale or turbulent behavior over time. The reason is that PINNs may violate the temporal causality property since all the temporal features in the PINNs loss are trained simultaneously. This paper proposes to use implicit time differencing schemes to enforce temporal causality, and use transfer learning to sequentially update the PINNs in space as surrogates for PDE solutions in different time frames. The evolving PINNs are better able to capture the varying complexities of the evolutionary equations, while only requiring minor updates between adjacent time frames. Our method is theoretically proven to be convergent if the time step is small and each PINN in different time frames is well-trained. In addition, we provide state-of-the-art (SOTA) numerical results for a variety of benchmarks for which existing PINNs formulations may fail or be inefficient. We demonstrate that the proposed method improves the accuracy of PINNs approximation for evolutionary PDEs and improves efficiency by a factor of 4-40x.

翻译：物理信息神经网络（PINNs）在利用深度学习求解偏微分方程（PDEs）方面展现出巨大潜力。然而，PINNs在求解演化型PDEs时面临训练困难，尤其对解随时间呈现多尺度或湍流行为的动力系统而言。其根本原因在于PINNs损失函数中所有时间特征被同步训练，可能违反时间因果性。本文提出采用隐式时间差分格式强制满足时间因果性，并利用迁移学习在时域上顺序更新空间中的PINNs，使其作为不同时间框架下PDE解的替代模型。这种渐进式PINNs能更有效地捕捉演化方程的时变复杂性，同时仅需对相邻时间框架的模型进行微调。理论证明当时间步长足够小且各时间框架的PINNs得到充分训练时，该方法具有收敛性。此外，针对现有PINNs公式可能失效或效率低下的多种基准问题，我们提供了最先进的（SOTA）数值结果。实验表明，所提方法将演化型PDEs的PINNs近似精度提升4-40倍的计算效率。