3D-PINNS: A UNIFIED FRAMEWORK FOR DIMENSION-WISE INTERPRETABILITY AND ADAPTIVE DOMAIN DECOMPOSITION

Shuyuan Shang, Ming Zhong, Ren Wang

Keywords: Partial Differential Equations, Physics-Informed Neural Networks (PINNs), Explainability

Abstract: Physics-informed neural networks (PINNs) provide a flexible framework for solving partial differential equations (PDEs), yet they often face difficulties in high-dimensional settings and in capturing solutions with localized or sharp features. In addition, most existing PINN-based approaches offer limited insight into how each dimensions contribute to the learned solution. We introduce Dimension-Domain Co-Decomposition (3D-PINNs), a structured and unified framework that combines dimension decomposition with mixture-of-experts-based domain decomposition. Within each expert, the solution is modeled through decoupled dimension components, while a router adaptively partitions the domains without requiring predefined subdomains or interface conditions. This formulation encourages structured representations reflecting dimension-wise characteristics of the underlying PDE solution. Empirical results on PDE benchmarks demonstrate that 3D improves solution accuracy and yields interpretable structure.

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Submission Number: 5