IRNO cuts spectral bias by 56% with iterative refinement

Researchers (Liu et al.) present Iterative Refinement Neural Operators (IRNO), a principled approach to mitigate spectral bias in neural operators — a common limitation where single-pass models fail to resolve high-frequency details. IRNO augments pre-trained operators with a learned refinement module that is applied iteratively via fixed-point iteration. This mirrors classical numerical solvers: a coarse initial prediction is followed by successive residual corrections. Under local assumptions, the induced operator contracts to a unique fixed point. To explicitly target high-frequency errors, the authors introduce a progressive spectral loss that adaptively increases the penalty on high-frequency components during training.

Experimental results across physical systems show consistent improvements. On turbulent flow, IRNO achieves up to 56.05% lower error compared to base operators. For active matter systems, spectral analysis reveals dramatic reductions in normalized error ratios: from low frequencies (27.72-36.10% down to 5.07-6.68% in mid-, and just 1.48-2.04% in high-frequencies). These gains remain stable even beyond the trained iteration count. The paper has been accepted as a Spotlight at ICML 2026, and code is publicly available. This work bridges deep learning and numerical analysis, offering a theoretically grounded method for high-fidelity scientific simulation.