Physics-informed neural networks (PINNs) are a 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 work, FBPINNs are extended by adding multiple levels of domain decompositions to their solution ansatz, inspired by classical multilevel Schwarz domain decomposition methods (DDMs). Analogous to typical tests for classical DDMs, we assess how the accuracy of PINNs, FBPINNs and multilevel FBPINNs scale with respect to computational effort and solution complexity by carrying out strong and weak scaling tests. 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 accuracy of FBPINNs when using large numbers of subdomains by aiding global communication between subdomains.

翻译：物理信息神经网络（PINNs）是解决涉及微分方程问题的强大方法，但在处理高频和/或多尺度解时往往面临挑战。有限基物理信息神经网络（FBPINNs）通过结合重叠域分解方法，在此类问题中提升了PINNs的性能。本文借鉴经典多层级施瓦茨域分解方法（DDMs），将多层域分解引入FBPINNs的解假设中。与经典DDMs的典型测试类似，我们通过强扩展性与弱扩展性测试，评估PINNs、FBPINNs及多层FBPINNs的精度随计算开销和解复杂度的变化规律。数值结果表明，所提出的多层FBPINNs在一系列具有高频和多尺度解的问题中始终显著优于PINNs。此外，与经典DDMs预期一致，多层FBPINNs通过增强子域间的全局通信，在使用大量子域时有效提升了FBPINNs的精度。