Adaptive partition of unity networks (APUNet): a localized deep learning method for solving PDEs.

In this study, we address the challenge of accurately solving physical problems involving complex partial differential equations using deep learning methods. Traditional approaches typically employ uniformly sampled points from the entire domain as training data. In practical PDE problems, achieving uniform precision across the entire domain is often unnecessary. Instead, it is more advantageous to employ random sampling during neural network training, provided that areas demanding precision can be identified and appropriately refined, akin to the concept of mesh refinement in classical numerical methods. To overcome this limitation, we propose a novel method based on the partition of unity approach. Our method introduces an approximation of the integral in the loss function, enabling localized calculation of the loss function within each subdomain. This innovative approach allows us to focus on areas of the domain that require higher precision while maintaining a fully automated process for selecting random samples. To validate the effectiveness of our proposed method, we conduct extensive testing on various physical problems, including Poisson, heat, and wave equations representing elliptic, parabolic, and hyperbolic problems, respectively. Additionally, we consider Burgers' equation as a nonlinear example. Through benchmarking against the standard Deep Ritz Method (DRM) and Deep Galerkin Method (DGM), we demonstrate that our method consistently outperforms these existing approaches, producing superior results. This approach offers researchers and practitioners the opportunity to significantly improve the accuracy of deep learning models when tackling complex physical problems. [ABSTRACT FROM AUTHOR]