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.

翻译：本文提出了一种基于扰动的保形预测框架，用于算子学习中的不确定性量化，重点关注二维Navier-Stokes方程组。虽然神经算子为昂贵的偏微分方程求解器提供了快速替代方案，但其本身无法为时空场预测提供校准的不确定性。我们的方法将训练后的傅里叶神经算子与分裂保形预测相结合，通过比较在近乎相同数据集上训练的两个算子的预测结果——其中一份标签保留原始值，另一份添加小幅度高斯噪声扰动——构建局部不确定性尺度。我们在数据稀缺场景下评估该方法，此时总标签预算固定，而需依赖独立不确定性网络的方法必须将训练数据划分给多个模型。在二维Navier-Stokes基准算例中，在匹配总数据预算的前提下，基于扰动的方法相较于现有方法能产生显著更窄的保形带，同时维持目标同步覆盖水平。这表明扰动敏感性是保形化神经算子的一种实用且样本高效的不确定性代理指标。