Stochastic adaptive spectral methods improve predictions for complex equations
Stochastic Adaptive Fourier Decomposition for Operator Learning
Machine Learning
Summary
Predicting how physical systems behave, like weather or fluid flow, often involves solving complex equations called partial differential equations (PDEs). Previous methods used fixed patterns (Fourier modes) to simplify these predictions but struggled with details like sharp changes or local effects. The authors propose a new approach, SAFDNO, that adapts these patterns based on the input, allowing more accurate modeling of tricky features. This method works better on a variety of test problems and can handle higher-resolution data without losing accuracy.
What this means in practice
- •For engineering simulation teams: Improve simulation accuracy for fluid dynamics and heat transfer by using adaptive spectral methods that model complex localized and multiscale phenomena more effectively.
- •For climate modelers: Enhance climate prediction models by incorporating adaptive operator learning methods that better capture fine-scale and non-periodic atmospheric variations.
Authors
Pengqing Shi, Liming Zhang, Tao Qian, Stephen Tierney, Jie Yin, Junbin Gao
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.