Efficient Discrete Physics-informed Neural Networks for Addressing Evolutionary Partial Differential Equations
CoRR(2023)
摘要
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.
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