用谱滤波方法高效建模偏微分方程的非局部关系
SFO: Learning PDE Operators via Spectral Filtering
- 基于谱基构建神经算子，通过快速衰减特征值系数实现紧凑表示
- 在六类任务中误差降低最高达40%，参数量显著减少
- 适合需要高精度且计算资源受限的物理系统模拟场景
偏微分方程（PDE）描述复杂系统，但神经算子常难以高效捕捉解映射中的长程非局部相互作用。我们提出谱滤波算子（SFO），通过通用谱基（USB）参数化积分核，该基为希尔伯特矩阵特征模式生成的固定全局正交基。受理论发现启发：平移不变离散化PDE的离散格林函数具有空间线性动态系统（LDS）结构，我们证明这些核可在USB中实现紧凑近似。仅学习快速衰减特征值的谱系数，即实现高效表示。在六项基准测试中，包括反应-扩散、流体动力学和3D电磁学，SFO达到最先进精度，相对强基线误差降低高达40%，同时使用更少参数。
原文摘要 · Abstract (English)
Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps. We introduce Spectral Filtering Operator (SFO), a neural operator that parameterizes integral kernels using the Universal Spectral Basis (USB), a fixed, global orthonormal basis derived from the eigenmodes of the Hilbert matrix in spectral filtering theory. Motivated by our theoretical finding that the discrete Green's functions of shift-invariant PDE discretizations exhibit spatial Linear Dynamical System (LDS) structure, we prove that these kernels admit compact approximations in the USB. By learning only the spectral coefficients of rapidly decaying eigenvalues, SFO achieves a highly efficient representation. Across six benchmarks, including reaction-diffusion, fluid dynamics, and 3D electromagnetics, SFO achieves state-of-the-art accuracy, reducing error by up to 40% relative to strong baselines while using substantially fewer parameters.
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