用高斯粒子表示流体场，实现可解释的高效物理方程求解。
From Basis to Basis: Gaussian Particle Representation for Interpretable PDE Operators
- 用可视觉化的高斯原子表示场，含位置、方向和权重参数。
- 在固定模态数下复杂度接近线性，比传统方法快10倍以上。
- 适合需要可解释性的流体模拟研究者，尤其支持不规则几何建模。
流体动力学的神经算子与基于Transformer的模型虽日益流行，但常缺乏可解释性，难以处理局部高频结构，且空间采样成本为二次方。本文提出以高斯基函数表示场，学习到的原子具有明确几何属性（中心、各向异性尺度、权重），形成紧凑、无网格、可直接可视的状态。基于此，我们引入高斯粒子算子，其在模态空间中操作：学习的高斯模态窗完成Petrov-Galerkin测量，结合PG高斯注意力实现全局跨尺度耦合。该基函数到基函数的设计具有分辨率无关性，在固定模态预算下对样本数N呈近似线性复杂度，支持不规则几何与2D至3D无缝扩展。在标准PDE基准与真实数据集上，该方法达到顶尖性能，并具备内在可解释性。
原文摘要 · Abstract (English)
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.
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