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.

中文翻译：

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基于多级域分解的物理信息神经网络架构


物理信息神经网络 (PINN) 是解决涉及微分方程问题的强大方法，但它们通常难以解决高频和/或多尺度解决方案的问题。有限基物理信息神经网络 (FBPINN) 通过将其与重叠域分解方法相结合，提高了 PINN 在这种情况下的性能。在这项工作中，受经典多级 Schwarz 域分解方法 (DDM) 的启发，通过在解决方案 ansatz 中添加多级域分解来扩展 FBPINN。与经典 DDM 的典型测试类似，我们通过执行强和弱缩放测试来评估 PINN、FBPINN 和多级 FBPINN 的准确性如何根据计算工作量和解决方案复杂性进行缩放。我们的数值结果表明，所提出的多级 FBPINN 在一系列具有高频和多尺度解决方案的问题上始终显着优于 PINN。此外，正如经典 DDM 中所预期的那样，我们表明，当使用大量子域时，多级 FBPINN 通过帮助子域之间的全局通信来提高 FBPINN 的准确性。