SpectraKAN: Conditioning Spectral Operators

TL;DR

SpectraKAN is a novel neural operator that conditions spectral convolution on input data, enabling adaptive, multi-scale PDE solutions with improved accuracy and efficiency across diverse benchmarks.

Contribution

It introduces input-conditioned spectral operators using cross-attention, allowing dynamic adaptation to system states, which improves over static Fourier kernels.

Findings

Achieves up to 49% RMSE reduction on PDE benchmarks.

Provides theoretical convergence guarantees for the modulation.

Shows significant improvements on complex spatio-temporal tasks.

Abstract

Spectral neural operators, particularly Fourier Neural Operators (FNO), are a powerful framework for learning solution operators of partial differential equations (PDEs) due to their efficient global mixing in the frequency domain. However, existing spectral operators rely on static Fourier kernels applied uniformly across inputs, limiting their ability to capture multi-scale, regime-dependent, and anisotropic dynamics governed by the global state of the system. We introduce SpectraKAN, a neural operator that conditions the spectral operator on the input itself, turning static spectral convolution into an input-conditioned integral operator. This is achieved by extracting a compact global representation from spatio-temporal history and using it to modulate a multi-scale Fourier trunk via single-query cross-attention, enabling the operator to adapt its behaviour while retaining the efficiency of spectral mixing. We provide theoretical justification showing that this modulation converges to a resolution-independent continuous operator under mesh refinement and KAN gives smooth, Lipschitz-controlled global modulation. Across diverse PDE benchmarks, SpectraKAN achieves state-of-the-art performance, reducing RMSE by up to 49% over strong baselines, with particularly large gains on challenging spatio-temporal prediction tasks.