标题： 基于域分解的通过Schwarz交替方法耦合物理信息神经网络 显示英文标题

标题： Domain decomposition-based coupling of physics-informed neural networks via the Schwarz alternating method

摘要： 物理信息神经网络（PINNs）是用于求解和推断非线性偏微分方程（PDEs）的有吸引力的数据驱动工具。与仅在解数据上训练的传统神经网络（NNs）不同，PINN在其损失函数中包含了PDE的残差，并在解域内的配点集上训练以最小化该残差。本文探讨了使用Schwarz交替方法作为耦合PINNs彼此以及与传统数值模型（即通过有限元、有限差分或有限体积方法获得的全阶模型，或FOMs）的方法，该方法是在物理域分解之后进行的。众所周知，当PDE解具有陡峭梯度时，训练PINN可能会很困难。本文研究了域分解和Schwarz交替方法作为加速PINN训练阶段的手段。在此背景下，我们探索了在每个子域PINN中施加狄利克雷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。作为一个数值例子，我们考虑了对流主导（高佩克莱特数）情况下的一维稳态对流扩散方程。我们的结果表明，Schwarz方法的收敛性与被耦合的PINNs中边界条件实现的选择密切相关。令人惊讶的是，Schwarz边界条件的强施加并不总能导致方法更快的收敛。虽然从我们的初步研究中尚不清楚通过Schwarz交替方法进行的PINN-PINN耦合是否能在对流主导的情况下加速PINN收敛，但它揭示了通过进行PINN-FOM耦合，可以显著改善佩克莱特数高达1e6时的PINN训练。 显示英文摘要

摘要： 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.