SFO: Learning PDE Operators via Spectral Filtering
This addresses the problem of efficiently solving PDEs for complex systems like fluid dynamics, though it appears incremental as an improvement to neural operators.
The paper tackles the challenge of neural operators struggling with long-range interactions in PDEs by introducing SFO, which uses a spectral basis for efficient kernel representation. It achieves state-of-the-art accuracy, reducing error by up to 40% across six benchmarks while using fewer parameters.
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.