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

翻译：神经算子作为科学建模中快速、数据驱动的替代方法，但通常依赖于单次推理的单一流程，难以捕捉高频细节——这一局限性被称为谱偏倚。我们提出迭代精化神经算子（IRNO），该方法通过固定点迭代循环应用一个学习得到的精化模块，对预训练算子进行增强。IRNO将预测过程分解为初始粗预测与后续残差修正的序列操作，与经典数值求解器形式相呼应。在局部假设下，我们证明了诱导算子的压缩性，确保其收敛至唯一固定点。为明确针对高频误差，我们提出渐进式谱损失函数，该函数在训练过程中随精化步骤自适应地增加对高频分量的惩罚。在物理系统的测试中，IRNO持续降低误差，在湍流模拟中误差降幅达56.05%。在活性物质案例中，谱分析表明：相对于基础算子，归一化误差比在低频段降至27.72–36.10%，中频段降至5.07–6.68%，高频段降至1.48–2.04%，且该性能在超训练迭代次数后保持稳定。代码开源于 https://github.com/xiaotianliu-dartmouth/Iterative_Refinement_Neural_Operator