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This research introduces Spectral Filtering Operator (SFO), a neural architecture that leverages spectral filtering theory to efficiently model long-range interactions in partial differential equations. By parameterizing integral kernels via a Universal Spectral Basis, SFO achieves state-of-the-art accuracy across physics benchmarks like fluid dynamics while significantly reducing parameter count compared to standard baselines. The work represents a significant methodological advance in scientific machine learning for solving complex physical systems.
SFO: Learning PDE Operators via Spectral Filtering
arXiv:2601.17090v3 Announce Type: replace 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.
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.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2601.17090 [cs.LG]
(or arXiv:2601.17090v3 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2601.17090 Focus to learn more arXiv-issued DOI via DataCite
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