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

Zhihao Li; Yu Feng; Zhilu Lai; Wei Wang

cs.LGFeb 25, 2026

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

Authors: Zhihao Li, Yu Feng, Zhilu Lai, Wei Wang

Organizations: The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China · The Hong Kong University of Science and Technology, Hong Kong SAR, China

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.

Figures & tables

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.

Model	NS2D	NS3D	ERA5-temp	ERA5-wind u	Carra	Airfoil	Turbulent	PlanetSWE
(Geo-)FNO	3.24E-02	5.07E-01	7.09E-03	1.02E-01	3.50E-01	5.12E-03	4.82E-01	9.23E-02
M2NO	2.76E-02	4.65E-01	3.35E-03	7.15E-02	3.62E-01	1.98E-03	4.53E-01	8.96E-02
CNO	9.27E-02	8.12E-01	1.24E-02	1.86E-01	6.89E-01	4.65E-03	7.91E-01	1.57E-01
LSM	3.11E-02	3.80E-01	5.86E-03	8.23E-02	4.05E-01	5.37E-03	4.16E-01	9.58E-02
AMG	3.12E-02	4.32E-01	6.11E-03	8.79E-02	3.21E-01	3.09E-03	4.47E-01	8.14E-02
GT	8.81E-02	5.39E-01	5.44E-03	1.55E-01	3.73E-01	8.26E-03	5.41E-01	1.23E-01

Table 1: Performance comparison with baselines on benchmarks.

L_{2}

loss is recorded.

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.

Panel of Figure 5 — Figure 5: Rollout stability schematic: error versus rollout horizon.

Panel of Figure 6 — Figure 6: NS2D energy spectrum (log–log).

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Benchmarks	Model Configurations
Benchmarks	hidden_dim	num_layers	num_heads	num_gaussians
NS2D	128	8	8	32
NS3D	64	8	4	16
ERA5-temp	64	4	4	16
ERA5-wind u	64	4	4	16
Carra	64	4	4	16

Appendix

Table 2: Model configurations of GPO.

Model Configuration	$L_{2}$ Error
w/o PG Operator	8.57E-02
w/o Gaussian Field	7.44E-02
num_gaussian = 1	6.28E-02
num_gaussian = 16	4.21E-02
num_gaussian = 64	3.84E-02
GPO (Baseline)	3.90E-02

Appendix

Table 3: Ablation results comparing the

L_{2}

error of different configurations.

Model	Param Count	Param (MB)	GPU Mem (MiB)	Train (s/epoch)	Inference (s/epoch)
FNO	640,305	4.84	949	28.27	0.5
LSM	19,187,457	73.23	2,875	48.42	1.73
Galerkin Transformer	1,096,321	4.18	4,301	65.29	2.96
GNOT	2,485,901	9.48	8,643	139.42	6.09
Transolver	3,069,889	11.71	4,917	97.03	4.10
ONO	1,596,673	6.09	6,163	94.80	4.27

Appendix

Table 4: Computational efficiency comparison across models (measured with input size 64

\times

\times

3, batch size 16).

Figure 7: Interpretable visualization of ERA5 on an in-distribution sample (above) and an out-of-distribution sample (below) . Left to right: ground truth, reconstruction from the Gaussian basis and learned Gaussian particles overlaid (ellipses: center $\mu$ , axes $\propto\sigma$ , color/size $\propto w$ ).

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Physics-Integrated Operator Learning via Gaussian Splatting Representations

Jihao Zhang, Junyi Guo, Jian-Xun Wang

Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, USA

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Open paper AskPDF arXiv

Neural operators provide efficient surrogates for spatiotemporal PDE systems, but purely data-driven formulations often accumulate substantial errors during long-horizon autoregressive prediction and may fail to exploit available governing-equation structure. Existing approaches incorporate physics primarily through residual-based training objectives or PDE-specific architectural constraints, which can introduce optimization difficulties or limit architectural generality. In this work, we introduce a representation-level approach to physics integration in which a feed-forward Gaussian splatting (FFGS) representation serves as a continuous interface between discretized solution fields and governing operators. The FFGS representation reconstructs the state as a continuous Gaussian field with closed-form spatial derivatives, allowing available physical PDE operators to be integrated directly within the learned evolution map without introducing a physics-residual loss. We evaluate the framework across two- and three-dimensional PDE systems, including advection, diffusion, nonlinear self-advection, and reaction dynamics. Over long-horizon autoregressive rollouts, the proposed framework reduces relative

\ell_2

error by

1.5\times

2.2\times

compared with the strongest purely data-driven baseline across the benchmark suite, while consistently improving spectral fidelity. The framework also remains effective when the governing equations are partially known, demonstrating robustness to incomplete physics. These results demonstrate that continuous field representations can provide a practical interface for incorporating known physical structure into generic neural-operator surrogates.

Abstract

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