From Basis to Basis: Gaussian Particle Representation for Interpretable PDE Operators
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866912926085939200 |
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| author | Li, Zhihao Feng, Yu Lai, Zhilu Wang, Wei |
| author_facet | Li, Zhihao Feng, Yu Lai, Zhilu Wang, Wei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21551 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From Basis to Basis: Gaussian Particle Representation for Interpretable PDE Operators Li, Zhihao Feng, Yu Lai, Zhilu Wang, Wei Machine Learning Artificial Intelligence 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. |
| title | From Basis to Basis: Gaussian Particle Representation for Interpretable PDE Operators |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2602.21551 |