AI 中文总结
该研究针对神经算子缺乏不确定性量化的问题，提出拆分保形框架以提供覆盖保证，经数值实验验证其校准带更紧且覆盖达标。
AI 中文摘要
神经算子是用于近似函数空间之间算子的快速替代模型，但其预测往往缺乏不确定性量化。我们开发了一种拆分保形框架，以保证神经算子输出周围经过校准的逐点带，在至少1-γ比例的评估域上包含真实解，且在测试和校准输入上的概率至少为1-α，其中α、γ∈(0,1)。我们的方法将归一化残差场简化为其空间(1-γ)分位数，并使用保留的校准数据集计算缩放因子。我们证明了在任意概率空间上定义的可测残差场的边际覆盖保证，涵盖连续域和固定离散化。在数据分布的温和假设下，我们表明以校准集为条件的覆盖服从Beta分布，这一点通过达西流和纳维-斯托克斯方程的数值实验得到验证，在这些实验中，我们的校准产生的带始终比现有修正更紧，同时保持目标覆盖。
英文摘要
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.
Comments19 pages, 6 figures
