SFO: Learning PDE Operators via Spectral Filtering

TL;DR

The paper introduces SFO, a spectral filtering neural operator that efficiently learns PDE solution maps by leveraging spectral basis representations, achieving state-of-the-art accuracy across diverse benchmarks.

Contribution

SFO is the first neural operator to parameterize integral kernels using the Universal Spectral Basis, enabling efficient learning of PDE operators with fewer parameters.

Findings

Achieves up to 40% error reduction compared to baselines.

Requires significantly fewer parameters for high accuracy.

Demonstrates effectiveness across multiple PDE benchmarks.

Abstract

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.