Stochastic Adaptive Fourier Decomposition for Operator Learning

2026-10-03

AI Summary

This study addresses the limitation of Fourier Neural Operators in modeling non-periodic and local fine-scale structures due to their fixed spectral bases and hard truncation. To overcome this, we propose SAFDNO, which introduces stochastic adaptive Fourier decomposition theory to construct input-adaptive Takenaka-Malmquist orthogonal systems that replace predefined Fourier modes. Furthermore, a neural pole predictor is designed to amortize the stochastic selection process, enabling learned filtering on the analytic branch coefficients within Hardy spaces. Experimental results demonstrate that SAFDNO achieves superior performance across nine PDE benchmarks, significantly improving solution accuracy for equations such as Darcy flow and enhancing zero-shot super-resolution capabilities.

Abstract

Fourier Neural Operators (FNOs) offer an efficient paradigm for solving partial differential equations (PDEs). However, FNOs rely on a fixed Fourier basis and hard frequency truncation, which inherently limit their ability to model non-periodic, localized, and fine-scale solution structures. We propose the Stochastic Adaptive Fourier Decomposition Neural Operator (SAFDNO), a spectral neural operator that replaces predefined Fourier modes with an adaptive Takenaka-Malmquist (TM) orthonormal system derived from the theory of Stochastic Adaptive Fourier Decomposition (SAFD). Instead of performing an expensive greedy pole search in classical SAFD, SAFDNO amortizes stochastic pole selection through a neural pole predictor and constructs the adaptive TM system directly from latent features. The resulting operator performs learned filtering on coefficients of analytic branches in the Hardy space with respect to the input-adaptive TM system, preserving the global receptive field of spectral operators while providing a more flexible representation than using fixed spectral bases. Across nine PDE benchmark problems, SAFDNO achieves the best performance on all six regular-grid problems among strong neural operator baselines, with especially notable gains on Darcy, Burgers, and Navier-Stokes, where fixed Fourier modes are often less effective at modeling localized oscillations, sharp transitions, and multiscale structures. SAFDNO also exhibits stronger zero-shot super-resolution performance and shows less performance degradation when deployed on finer discretizations. These results suggest that the input-adaptive TM system provides a promising alternative to fixed spectral representations for neural operator learning.

Problem

• Fourier Neural Operators
• Operator Learning
• Partial Differential Equations
• Fixed Fourier Basis
• Adaptive Spectral Representation

Innovation

• Stochastic Adaptive Fourier Decomposition
• Fourier Neural Operator
• Takenaka-Malmquist system
• Neural pole predictor
• Operator learning