Title: Dimension Domain Co-decomposition: Solving PDEs with Interpretability
Primary area: interpretability and explainable AI
Keywords: Physics-informed neural networks (PINNs), Mixture-of-Experts, Interpretability, Domain Decomposition, Dimension Decomposition

Abstract: Physics-informed neural networks (PINNs) have shown promise for solving partial differential equations (PDEs), but they face significant challenges in high-dimensional settings and when modeling solutions with sharp features. Existing approaches also lack interpretable per-dimension representations and depend on manually defined domain partitions. To address these challenges, we propose a unified Dimension Domain Co-Decomposition (3D) framework that integrates dimension decomposition with a Mixture-of-Experts (MoE) based domain decomposition. Our approach achieves three key innovations. First, we introduce an interpretable dimension decomposition strategy that decouples individual coordinate inputs within each expert using a single shared MLP with indexed inputs, significantly reducing the model size. Second, we propose a novel metric, Variable Interpretability ($VI$), that quantifies the alignment between the learned latent representations of each input dimension and their corresponding exact solution components. Third, we present an MoE-driven domain decomposition architecture that automatically partitions the solution space without requiring predefined regions or interface conditions. Extensive experiments demonstrate that our approach improves both computational efficiency and solution accuracy across a range of high-dimensional PDE benchmarks, with interpretable and scalable performance.