Iterative Refinement Neural Operators are Learned Fixed-Point Solvers: A Principled Approach to Spectral Bias Mitigation

2026-05-21

AI Summary

Neural operators suffer from spectral bias, limiting their ability to accurately capture high-frequency physical details. This work proposes the Iterative Refinement Neural Operator (IRNO), which, for the first time, formulates neural operators as learnable fixed-point solvers. IRNO refines predictions from a pretrained operator through learnable correction modules via fixed-point iterations, decomposing the solution into an initial coarse estimate followed by successive residual corrections. A progressive spectral loss is introduced to adaptively emphasize high-frequency error during training, complemented by theoretical analysis of local contractivity and frequency-domain error evaluation. Experiments demonstrate that IRNO reduces prediction errors by up to 56.05% in turbulent flow tasks. In active matter simulations, it achieves normalized errors of 1.48–2.04% for low frequencies and 27.72–36.10% for high frequencies, with iterative stability extending beyond the training time horizon.

Abstract

Neural operators serve as fast, data-driven surrogates for scientific modeling but typically rely on a monolithic, single-pass inference procedure that struggles to resolve high-frequency details, a limitation known as spectral bias. We introduce the Iterative Refinement Neural Operator (IRNO), which augments pre-trained operators with a learned refinement module iteratively applied via fixed-point iteration. IRNO decomposes the prediction into a coarse initialization followed by successive residual corrections, paralleling classical numerical solvers. Under local assumptions, we establish contraction of the induced operator, ensuring convergence to a unique fixed point. To explicitly target high-frequency errors, we propose a progressive spectral loss that adaptively increases penalty on high-frequency components over refinement steps during training. Across physical systems, IRNO consistently lowers error, with up to 56.05% improvement on turbulent flow. On Active Matter, spectral analysis reveals that, relative to base operator, the normalized error ratios decrease to 27.72-36.10% in low-, 5.07-6.68% in mid-, and 1.48-2.04% in high-frequencies, remaining stable beyond the trained iteration count. Code is available at https://github.com/xiaotianliu-dartmouth/Iterative_Refinement_Neural_Operator

Problem

• neural operators
• spectral bias
• high-frequency details
• scientific modeling
• fixed-point solvers

Innovation

• Iterative Refinement Neural Operator
• spectral bias mitigation
• fixed-point iteration
• progressive spectral loss
• neural operators