Physics-informed neural networks (PINNs) are appealing data-driven tools for solving and inferring solutions to nonlinear partial differential equations (PDEs). Unlike traditional neural networks (NNs), which train only on solution data, a PINN incorporates a PDE's residual into its loss function and trains to minimize the said residual at a set of collocation points in the solution domain. This paper explores the use of the Schwarz alternating method as a means to couple PINNs with each other and with conventional numerical models (i.e., full order models, or FOMs, obtained via the finite element, finite difference or finite volume methods) following a decomposition of the physical domain. It is well-known that training a PINN can be difficult when the PDE solution has steep gradients. We investigate herein the use of domain decomposition and the Schwarz alternating method as a means to accelerate the PINN training phase. Within this context, we explore different approaches for imposing Dirichlet boundary conditions within each subdomain PINN: weakly through the loss and/or strongly through a solution transformation. As a numerical example, we consider the one-dimensional steady state advection-diffusion equation in the advection-dominated (high Peclet) regime. Our results suggest that the convergence of the Schwarz method is strongly linked to the choice of boundary condition implementation within the PINNs being coupled. Surprisingly, strong enforcement of the Schwarz boundary conditions does not always lead to a faster convergence of the method. While it is not clear from our preliminary study that the PINN-PINN coupling via the Schwarz alternating method accelerates PINN convergence in the advection-dominated regime, it reveals that PINN training can be improved substantially for Peclet numbers as high as 1e6 by performing a PINN-FOM coupling.

翻译：物理信息神经网络（PINNs）是求解非线性偏微分方程（PDEs）并提供推理解的一种有吸引力的数据驱动工具。与传统仅基于解数据训练的神经网络（NNs）不同，PINN将PDE的残差纳入其损失函数，并通过训练在解域内一组配置点上最小化该残差。本文探讨了采用Schwarz交替法，在物理域分解后，将多个PINN相互耦合以及将PINN与传统数值模型（即通过有限元、有限差分或有限体积法获得的全阶模型，FOMs）耦合的方法。众所周知，当PDE解具有陡峭梯度时，训练PINN可能很困难。本文研究了利用区域分解和Schwarz交替法来加速PINN训练阶段的可能性。在此背景下，我们探索了在每个子域PINN中施加Dirichlet边界条件的不同方法：通过损失函数弱施加和/或通过解变换强施加。作为数值示例，我们考虑了对流主导（高Peclet数）条件下的一维稳态对流-扩散方程。结果表明，Schwarz法的收敛性与所耦合PINN中边界条件实现方式的选择密切相关。令人惊讶的是，强施加Schwarz边界条件并不总是导致该方法收敛更快。尽管我们初步研究并未明确显示通过Schwarz交替法进行的PINN-PINN耦合能在对流主导模式下加速PINN收敛，但揭示出对于高达1e6的Peclet数，通过PINN-FOM耦合可显著提升PINN训练效果。