Physics-informed neural networks (PINNs) [4, 10] are an approach for solving boundary value problems based on differential equations (PDEs). The key idea of PINNs is to use a neural network to approximate the solution to the PDE and to incorporate the residual of the PDE as well as boundary conditions into its loss function when training it. This provides a simple and mesh-free approach for solving problems relating to PDEs. However, a key limitation of PINNs is their lack of accuracy and efficiency when solving problems with larger domains and more complex, multi-scale solutions. In a more recent approach, finite basis physics-informed neural networks (FBPINNs) [8] use ideas from domain decomposition to accelerate the learning process of PINNs and improve their accuracy. In this work, we show how Schwarz-like additive, multiplicative, and hybrid iteration methods for training FBPINNs can be developed. We present numerical experiments on the influence of these different training strategies on convergence and accuracy. Furthermore, we propose and evaluate a preliminary implementation of coarse space correction for FBPINNs.

翻译：物理信息神经网络（PINNs）[4,10]是一种基于微分方程（PDE）求解边值问题的方法。PINNs的核心思想是利用神经网络近似偏微分方程的解，并在训练过程中将PDE残差及边界条件纳入损失函数。这种方法为求解偏微分方程相关问题提供了一种简单无网格的途径。然而，PINNs在求解大区域、复杂多尺度解的问题时存在精度与效率不足的关键局限性。在最近的研究中，有限基物理信息神经网络（FBPINNs）[8]借鉴区域分解思想加速PINNs的学习过程并提升其精度。本研究展示了如何开发训练FBPINNs的Schwarz型加性、乘性及混合迭代方法。我们通过数值实验探讨了这些不同训练策略对收敛性与精度的影响，并进一步提出并评估了针对FBPINNs的粗空间校正预处理实现方案。