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

Learning fluid PDE dynamics has increasingly benefited from neural operators and Transformer-based models, but their latent states often remain opaque, and sample-wise attention can be costly at high resolutions. We propose the Gaussian Particle Operator (GPO), a basis-to-basis neural operator that represents fields with a learned Gaussian basis. Each atom carries explicit geometric parameters, including centers, anisotropic scales, and weights, yielding a compact, mesh-agnostic, and directly visualizable intermediate representation. GPO operates in modal space: learned Gaussian modal windows perform Petrov--Galerkin measurements, and a PG Gaussian Attention module couples the resulting modes globally before scattering them back to the spatial domain. This design is resolution-agnostic, scales near-linearly with the number of samples for a fixed modal budget, and naturally applies to irregular geometries and 3D domains. We separately evaluate Gaussian-basis reconstruction and operator prediction. Across standard PDE benchmarks and real reanalysis datasets, GPO achieves competitive accuracy against neural-operator and Transformer baselines, ranking best on most tasks and close to the best on the remainder. The learned particles and modal couplings further provide representation-level interpretability, positioning GPO as a useful intermediate representation rather than a one-to-one decomposition of physical structures. The code is available at https://github.com/lizhihao2022/GPO.