SFO: Learning PDE Operators via Spectral Filtering

About

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.

Noam Koren, Rafael Moschopoulos, Kira Radinsky, Elad Hazan · 2026

Related benchmarks

Task	Dataset	Result
Learning PDE Solution Operators	2D Shallow Water	Mean L2 Relative Error0.38	20
Learning PDE Solution Operators	Allen-Cahn 1D	Mean L2 Relative Error0.05	12
Learning PDE Solution Operators	1D Diffusion-Reaction	Mean L2 Rel Error22	12
Learning PDE Solution Operators	1D Diffusion-Sorption	Mean L2 Relative Error1.08	8
Learning PDE Solution Operators	1D Cahn-Hilliard	Mean L2 Relative Error0.08	8
Learning PDE Solution Operators	3D Maxwell	Mean L2 Relative Error0.4085	8
Learning PDE Solution Operators	1D Cahn-Hilliard	Training Time per Epoch (s)1.85	7
Learning PDE Solution Operators	1D Diffusion-Sorption	Training Time per Epoch (s)1.64	7
Learning PDE Solution Operators	1D Diffusion-Reaction	Training Time per Epoch (s)2.12	7
Learning PDE Solution Operators	1D Allen-Cahn	Training Time (s)2.09	7