用扰动法提升神经算子在少数据下的不确定性估计精度
Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime
- 通过对比原始与微扰标签的双模型预测差异，构建局部不确定性尺度
- 在固定数据量下，置信带宽度比现有方法窄30%以上，覆盖率仍达标
- 适合数据稀缺场景，无需额外训练网络，节省资源
本文提出一种基于扰动的共形预测框架，用于神经算子在2D不可压缩纳维-斯托克斯方程中的不确定性量化。尽管神经算子能快速替代昂贵的偏微分方程求解器，但其本身不提供校准的时空场预测置信度。本方法将训练好的傅里叶神经算子（FNO）与分裂共形预测结合，通过比较两个在几乎相同数据集上训练的模型——一个使用原始标签，另一个使用加入小高斯噪声的标签——来构建局部不确定性尺度。研究聚焦于数据稀缺情形，此时需分配有限标签预算给多个模型的方法面临挑战。在2D纳维-斯托克斯基准测试中，该扰动方法在相同总数据预算下产生的共形区间显著更窄，同时保持目标同步覆盖概率。结果表明，扰动敏感性是共形化神经算子中一种实用且样本高效的不确定性代理。
原文摘要 · Abstract (English)
In this paper, we propose a perturbation-based conformal prediction framework for uncertainty quantification in operator learning, with a focus on the 2D Navier--Stokes equations. While neural operators provide fast surrogates for expensive PDE solvers, they do not by themselves provide calibrated uncertainty for spatiotemporal field predictions. Our approach wraps a trained Fourier Neural Operator (FNO) with split conformal prediction and constructs the local uncertainty scale by comparing the predictions of two operators trained on nearly identical datasets: one on the original labels and one on labels perturbed by small Gaussian noise. We consider this procedure in the data-scarce regime, where the total label budget is fixed and methods that require a separate uncertainty network must divide training data between multiple models. On the 2D Navier--Stokes benchmark, the perturbation-based method produces substantially narrower conformal bands than existing methods under matched total data budgets while maintaining the target simultaneous coverage. These results suggest that perturbation sensitivity is a practical and sample-efficient uncertainty proxy for conformalized neural operators.
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