Domain decomposition-based neural networks for complex shaped domains

Abstract

Physics-informed neural networks (PINNs) provide a powerful framework for solving differential equations but often encounter difficulties when addressing high-frequency solutions. Finite basis physics-informed neural networks (FBPINNs) improve PINN performance through uniform overlapping domain decomposition, yet they may still struggle with problems involving non-uniform frequency solutions. In this work, we introduce a novel framework called adaptive domain decomposition-based FBPINNs (Adaptive DD FBPINNs), which incorporates partition of unity networks (POUnets) to learn domain partitions that adaptively decompose the domain in a data-driven manner. This dynamic decomposition significantly enhances the accuracy and efficiency of PDE solvers, particularly for problems with high-frequency components and complex geometries. Furthermore, the framework integrates a residual-based adaptive distribution (RAD) resampling strategy that concentrates training on regions with high residuals, further boosting performance. Experimental results demonstrate that the Adaptive DD FBPINN outperforms standard FBPINN in terms of accuracy, providing a flexible and robust solution for both regular and complex-shaped domains, while efficiently enforcing Dirichlet boundary conditions as hard constraints. Overall, this work provides an exploratory contribution, presenting a promising approach for adaptively learning partitions by combining data-driven POUnets and FBPINNs, which can be further generalized to complex-shaped domains.