Physics-informed neural networks (PINNs) are a popular and powerful approach for solving problems involving differential equations, yet they often struggle to solve problems with high frequency and/or multi-scale solutions. Finite basis physics-informed neural networks (FBPINNs) improve the performance of PINNs in this regime by combining them with an overlapping domain decomposition approach. In this paper, the FBPINN approach is extended by adding multiple levels of domain decompositions to their solution ansatz, inspired by classical multilevel Schwarz domain decomposition methods (DDMs). Furthermore, analogous to typical tests for classical DDMs, strong and weak scaling studies designed for measuring how the accuracy of PINNs and FBPINNs behaves with respect to computational effort and solution complexity are carried out. Our numerical results show that the proposed multilevel FBPINNs consistently and significantly outperform PINNs across a range of problems with high frequency and multi-scale solutions. Furthermore, as expected in classical DDMs, we show that multilevel FBPINNs improve the scalability of FBPINNs to large numbers of subdomains by aiding global communication between subdomains.

翻译：物理信息神经网络（PINNs）是一种求解含微分方程问题的流行且强大的方法，但在处理高频和/或多尺度解的问题时常面临困难。有限基物理信息神经网络（FBPINNs）通过结合重叠域分解方法，改善了PINNs在此类问题中的表现。本文受经典多级施瓦茨域分解方法（DDMs）启发，在FBPINNs的解假设中引入多级域分解，从而扩展了该方法。此外，类比经典DDMs的典型测试，我们设计了强扩展性与弱扩展性研究，用于衡量PINNs及FBPINNs的精度如何随计算代价和解的复杂度变化。数值结果表明，所提出的多级FBPINNs在一系列具有高频与多尺度解的问题上始终显著优于PINNs。进一步地，与经典DDMs的预期一致，我们证明多级FBPINNs通过促进子域间的全局通信，改善了FBPINNs在大量子域情况下的可扩展性。