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

Zhihao Li, Yu Feng, Zhilu Lai, Wei Wang

TL;DR

This work proposes representing fields with a Gaussian basis, where learned atoms carry explicit geometry and form a compact, mesh-agnostic, directly visualizable state, and introduces a Gaussian Particle Operator that acts in modal space.

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.

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

TL;DR

This work proposes representing fields with a Gaussian basis, where learned atoms carry explicit geometry and form a compact, mesh-agnostic, directly visualizable state, and introduces a Gaussian Particle Operator that acts in modal space.

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.
Paper Structure (68 sections, 4 theorems, 32 equations, 7 figures, 4 tables)

This paper contains 68 sections, 4 theorems, 32 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Gaussian Basis Representation
  4. From physics field to Gaussian field and basis.
  5. Encoder.
  6. Gaussian basis evaluation.
  7. Decoder.
  8. Gaussian particle regularization.
  9. Overview pipeline.
  10. Approximation capacity of the Gaussian basis.
  11. Petrov--Galerkin Gaussian Attention
  12. From Petrov--Galerkin projection to a Gaussian-basis operator
  13. Trial functions (Gaussian basis).
  14. Test functions (Gaussian modal windows).
  15. Modal coupling and scatter.
  16. ...and 53 more sections

Key Result

Lemma 2.1

Let $\Omega\!\subset\!\mathbb{R}^{d}$ be compact. Finite linear combinations of (possibly anisotropic) Gaussian kernels are dense in $C(\Omega)$ and in $L^{r}(\Omega)$ for $1\le r<\infty$. Consequently, for any continuous scalar field $v$ and $\varepsilon>0$, there exist $G$ and parameters $\{(\mu_i Vector-valued fields admit componentwise approximation. (Proof in Appx.sec:exp_gf)

Figures (7)

  • Figure 1: Overview of the Gaussian Basis Field framework. Given the physics field $a(x)$, an encoder $E_\theta$ produces $G$ Gaussian components per spatial location (mean $\mu$, scale $\sigma$, and mixture weight $w$). These define a Gaussian field that is evaluated at queries to form the basis $z$, which is decoded by a fixed decoder $f_\phi^{\mathrm{dec}}$ to reconstruct the output field.
  • Figure 2: Architecture of the Gaussian Particle Operator (GPO). The pipeline encodes $a(\mathbf{x})$ into a Gaussian field $(\mu,\sigma,w)$, evaluates a basis $Z$, applies the modal operator $\mathcal{O}_{\psi}$, and decodes to $\hat{u}(\mathbf{x})$; black arrows denote forward computation and red arrows denote gradients.
  • Figure 3: Interpretable visualization on an in-distribution sample (above) and an out-of-distribution sample (below). Left to right: ground truth, reconstruction from the Gaussian basis, absolute error, and learned Gaussian particles overlaid (ellipses: center $\mu$, axes $\propto\sigma$, color/size $\propto w$).
  • Figure 4: Layer-wise evolution of the Gaussian particle field. Particle activations after successive PG Gaussian Attention layers.
  • Figure 5: Rollout stability schematic: error versus rollout horizon.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Lemma 2.1: Density of Gaussian mixtures
  • Theorem 2.2: Universal approximation in modal form
  • Lemma 2.1: Density of Gaussian mixtures
  • proof
  • Theorem 2.2: Universal approximation in modal form
  • proof