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

2026-02-24

AI Summary

This work proposes an interpretable neural operator framework for learning the dynamics of partial differential equations (PDEs), addressing key limitations of existing neural operators and Transformers—namely poor interpretability, difficulty in capturing local high-frequency structures, and high computational complexity. The approach leverages a Gaussian basis representation field, where Gaussian particles explicitly encode geometric information to yield a compact, mesh-independent, and directly visualizable state representation. By introducing Petrov–Galerkin projection and a novel PG Gaussian attention mechanism in modal space, the method enables efficient cross-scale coupling and computation. It achieves near-linear complexity, naturally supports irregular geometries, and extends seamlessly to both 2D and 3D settings. Experiments on standard PDE benchmarks and real-world datasets demonstrate state-of-the-art accuracy while offering intrinsic interpretability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

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.

Problem

• interpretable PDE operators
• neural operators
• high-frequency structures
• quadratic complexity
• fluid dynamics

Innovation

• Gaussian Particle Representation
• Neural Operators
• Petrov-Galerkin Attention
• Basis-to-Basis Learning
• Interpretable PDE Solvers