DIMENSION DOMAIN CO-DECOMPOSITION: SOLVING PDES WITH INTERPRETABILITY

Abstract: Physics-informed neural networks (PINNs) have demonstrated effectiveness in solving partial differential equations (PDEs), yet they often struggle in high-dimensional regimes and lack interpretable representations and in scenarios involving sharp solution structures. Moreover, existing approaches typically rely on manually specified domain partitions. We propose a unified Dimension–Domain Co-Decomposition (3D) framework that jointly integrates dimension-wise decomposition with mixture-of-experts (MoE)–based domain decomposition. At the dimension level, we introduce an interpretable decomposition mechanism in which coordinate inputs are decoupled within each expert through a shared MLP with indexed inputs, enabling parameter efficiency while preserving expressivity. To quantitatively assess interpretability, we define a Variable Interpretability (VI) metric that measures the alignment between learned latent components and the corresponding solution factors. At the domain level, an MoE-based gating mechanism adaptively partitions the solution space without requiring predefined regions or interface conditions. Extensive experiments on PDE benchmarks demonstrate that the proposed framework achieves improved accuracy and computational efficiency compared to standard PINNs and related baselines, while providing interpretable and scalable representations.

Submission Type: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Yunbo_Wang1

Submission Number: 8033