为神经算子提供可保证的不确定性量化，确保预测区间覆盖真解。
Conformal Uncertainty Quantification Guarantees for Neural Operators
- 基于分层交叉验证框架，用残差场的分位数生成置信带
- 在达西流与纳维-斯托克斯方程上实现更紧致的置信带且保持覆盖率
- 适用于连续域和离散化场景，理论保障覆盖概率不低于1-α
神经算子能快速逼近函数空间间的算子，但其预测常缺乏不确定性量化。本文提出一种分层交叉验证框架，确保在至少1-γ的评估域上，以不低于1-α的概率，校准后的点态置信带包含真实解，其中α,γ∈(0,1)。方法将归一化残差场降至其空间(1-γ)-分位数，并利用独立校准集计算缩放因子。我们证明了在任意概率空间上的可测残差场具有边际覆盖保证，涵盖连续域与固定离散化情形。在数据分布满足弱假设下，条件覆盖概率服从贝塔分布，数值实验验证了在达西流与纳维-斯托克斯方程上，该校准方法所得置信带始终比现有修正方法更紧致，同时保持目标覆盖率。
原文摘要 · Abstract (English)
Neural operators provide fast surrogate models for approximating operators between function spaces, but their predictions often lack uncertainty quantification. We develop a split conformal framework to guarantee that a calibrated pointwise band around the neural operator output contains the true solution on at least a $1-γ$ fraction of the evaluation domain, with probability at least $1-α$ over test and calibration inputs, where $α,γ\in(0,1)$. Our method reduces a normalized residual field to its spatial $(1-γ)$-quantile and computes a scaling factor using a held-out calibration dataset. We prove marginal coverage guarantees for measurable residual fields defined on arbitrary probability spaces, covering both continuum domains and fixed discretizations. Under mild assumptions on the data distribution, we show that the coverage conditional on the calibration set follows a Beta distribution, which we verify with numerical experiments on Darcy flow and Navier--Stokes equations, where our calibration yields bands consistently tighter than existing corrections while retaining the target coverage.
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