From Basis to Basis: Gaussian Particle Representation for Interpretable PDE Operators

• URL: http://arxiv.org/abs/2602.21551v1
• Date: Wed, 25 Feb 2026 04:16:44 GMT
• Title: From Basis to Basis: Gaussian Particle Representation for Interpretable PDE Operators
• Authors: Zhihao Li, Yu Feng, Zhilu Lai, Wei Wang,
• Abstract summary: Learning PDE dynamics for fluids increasingly relies on neural operators and Transformer-based models, yet these approaches often lack interpretability and struggle with localized, high-frequency structures.<n>We propose representing fields with a Gaussian basis, where learned atoms carry explicit geometry and form a compact, mesh-agnostic, directly visualizable state.<n>On standard PDE benchmarks and real datasets, our method attains state-of-the-art competitive accuracy while providing intrinsic interpretability.
• Score: 10.039418546901775
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Learning PDE dynamics for fluids increasingly relies on neural operators and Transformer-based models, yet these approaches often lack interpretability and struggle with localized, high-frequency structures while incurring quadratic cost in spatial samples. We propose representing fields with a Gaussian basis, where learned atoms carry explicit geometry (centers, anisotropic scales, weights) and form a compact, mesh-agnostic, directly visualizable state. Building on this representation, we introduce a Gaussian Particle Operator that acts in modal space: learned Gaussian modal windows perform a Petrov-Galerkin measurement, and PG Gaussian Attention enables global cross-scale coupling. This basis-to-basis design is resolution-agnostic and achieves near-linear complexity in N for a fixed modal budget, supporting irregular geometries and seamless 2D-to-3D extension. On standard PDE benchmarks and real datasets, our method attains state-of-the-art competitive accuracy while providing intrinsic interpretability.

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