通过迭代修正提升神经算子高频细节精度，有效缓解谱偏差问题
Iterative Refinement Neural Operators are Learned Fixed-Point Solvers: A Principled Approach to Spectral Bias Mitigation
- 将预测分解为粗略初始与逐次残差修正，模仿经典数值求解器
- 在湍流模拟中误差降低56.05%，高频误差占比降至1.48-2.04%
- 适合需要高精度频域响应的物理系统建模，如复杂流体与活性物质
神经算子作为科学建模的快速数据驱动替代方法，通常依赖单一过程推理，难以捕捉高频细节，即存在谱偏差问题。本文提出迭代修正神经算子（IRNO），通过在预训练算子基础上添加可学习的迭代修正模块，以不动点迭代方式逐步优化输出。IRNO将预测分解为粗略初始化与连续残差修正，类比经典数值求解器。在局部假设下，我们证明所诱导算子具有压缩性，确保收敛至唯一不动点。为显式抑制高频误差，提出渐进式频谱损失，在训练中随修正步数动态增强对高频成分的惩罚。在多种物理系统上，IRNO均显著降低误差，湍流场景最高提升56.05%。在活性物质系统中，频谱分析显示：相比基线算子，低、中、高频段归一化误差比分别降至27.72-36.10%、5.07-6.68%、1.48-2.04%，且在超过训练迭代次数后仍保持稳定。代码已开源：https://github.com/xiaotianliu-dartmouth/Iterative_Refinement_Neural_Operator
原文摘要 · Abstract (English)
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
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