SFO：通过谱滤波学习 PDE 解算算子

Jul 16, 2026

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Neural operators learn mappings from input functions to PDE solution fields but struggle to capture long-range, nonlocal interactions efficiently. Spectral Filtering Operator (SFO) parameterizes integral kernels in a fixed Universal Spectral Basis (USB) derived from Hilbert-matrix eigenmodes. For an important PDE class, the discrete Green’s kernel has a spatial Linear Dynamical System (LDS) structure, yielding exponential USB spectral decay and an ε-accuracy truncation bound. Learning only spectral coefficients makes SFO highly efficient, delivering state-of-the-art results on six PDE benchmarks with up to 40% lower error using fewer parameters.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages19

Reading Time48 min

Abstract

1 Introduction

2 Related Work

SFO: Learning PDE Solution Operators via Hilbert Spectral Basis Noam Koren Department of Computer Science Technion-Israel Institute of Technology [noam.koren@campus.technion.ac.il](noam.koren@campus.technion.ac.il) Kira Radinsky Department of Computer Science Technion-Israel Institute of Technology [kiraradinsky@gmail.com](kiraradinsky@gmail.com) Rafael Moschopoulos Department of Computer Science Princeton University [gm3460@princeton.edu](gm3460@princeton.edu) Elad Hazan Department of Computer Science Princeton University [elad.hazan@gmail.com](elad.hazan@gmail.com) arXiv :2601 . 17090v2 [ cs .LG] 13 Jul 2026 Abstract Neural operators have emerged as a powerful framework for learning PDE solution operators directly from data. However, efficiently capturing the long-range, nonlocal interactions that characterize these operators remains challenging. We introduce Spectral Filtering Operator (SFO), a neural operator that parameterizes integral kernels in the Universal Spectral Basis (USB), a fixed global orthonormal basis derived from the eigenmodes of the Hilbert matrix in spectral filtering theory. We show that, for an important class of PDEs, the discrete Green’s kernel function has a spatial Linear Dynamical System (LDS) structure, implying that it admitsa compact USB approximation with an ϵ-accuracy error bound. By learning only the spectral coefficients, SFO yields a highly efficient representation. Across six benchmarks spanning reaction-diffusion, fluid dynamics, and electromagnetics, SFO achieves state-of-the-art accuracy, reducing error by up to 40% relative to strong baselines while using substantially fewer parameters. 1 Introduction Many phenomena in physics, engineering, and natural science are governed by partial differential equations (PDEs) . Classical numerical solvers (e.g., finite differences Morton and Mayers [2005]) are often expensive at high resolution, where very fine discretizations are needed Tang and Lin [2017] . This has motivated neural operators: learned models that approximate the PDE solution operator, i.e., a mapping from an input function to the corresponding solution field. Once trained, a neural operator can amortize simulation cost across many queries, enabling fast inference for new PDE instances. Furthermore, unlike traditional approaches that require explicit knowledge of the governing equations, neural operators are data-driven and can learn the solution map directly from observations. A central challenge in neural operators is efficiently capturing nonlocal interactions (e.g., FNO Liet al. [2021], MPNN Brandstetter et al. [2022]) . Many PDE solution operators exhibit long-range dependencies, where the solution at a point depends on distant regions of the domain (e.g., elliptic PDEs and fractional diffusion Herman) . Capturing such behavior requires propagating information globally while maintaining a compact and efficient representation. In this work, we introduce the SPECTRAL FILTERING OPERATOR (SFO), a neural operator that explicitly learns the underlying kernel, whereas most neural operators (e.g., FNO Li et al. [2021]) parameterize the operator implicitly without explicitly learning the kernel. SFO parameterizes the kernel using the Universal Spectral Basis (USB), a fixed, globally supported orthogonal basis given Preprint. by the eigenvectors of the Hilbert matrix. The kernel is represented as an expansion over the leading USB modes, and only the corresponding spectral coefficients are learned. The USB was originally proposed for modeling Linear Dynamical Systems (LDS) Hazan et al. [2017] . Unlike Fourier bases, which bias toward smooth periodic functions, or learned bases, which may overfit, the USB provides a fixed global representation with rapid spectral decay. This choice aligns with the structure of PDE solution operators. We analyze a representative setting in which the discrete PDE kernel exhibits an LDS structure and show that its USB

FAQs

What is SFO and what problem does it address in neural operators?

SFO (Spectral Filtering Operator) is a neural operator architecture designed to learn PDE solution operators while capturing the long-range, nonlocal interactions that are difficult for many existing neural-operator models.

How does SFO represent the integral kernel in its method?

SFO parameterizes the kernel using a Universal Spectral Basis (USB) formed from eigenvectors of the Hilbert matrix, and it learns only the leading spectral coefficients in the USB expansion.

What theoretical insight is provided for a class of PDEs?

For an important class of PDEs, the discrete Green’s kernel has a spatial Linear Dynamical System (LDS) structure, which implies exponential decay of USB modes and an ε-accuracy bound requiring only logarithmically many leading modes.

Jul 16, 2026

Upload by Finn