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

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 to represent fields with a \emph{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 \emph{Gaussian Particle Operator} that acts \emph{in modal space}: learned \emph{Gaussian modal windows} perform a Petrov--Galerkin measurement, a \emph{PG Gaussian Attention} effects global cross-scale coupling. This basis-to-basis design is resolution-agnostic and achieves near-linear complexity in $N$ for 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.