Conformal prediction provides a distribution-free framework for uncertainty quantification via prediction sets with exact finite-sample coverage. In low dimensions these sets are easy to interpret, but in high-dimensional or structured output spaces they are difficult to represent and use, which can limit their ability to integrate with downstream tasks such as sampling and probabilistic forecasting. We show that any sufficiently regular differentiable nonconformity score induces a deterministic flow on the output space whose trajectories converge to the boundary of the corresponding conformal prediction set. This leads to a computationally efficient, training-free method for sampling conformal boundaries in arbitrary dimensions. Mixing across confidence levels yields conformal predictive distributions whose quantile regions coincide with the empirical conformal prediction sets. We provide an approximation bound decomposing CPD predictive error into score-induced distortion, base-measure quality, and gradient flow-induced distortion. We evaluate the approach on PDE inverse problems, precipitation downscaling, climate model debiasing, and hurricane trajectory forecasting.

翻译：共形预测通过具有精确有限样本覆盖率的预测集，为不确定性量化提供了一种无分布框架。在低维空间中，这些预测集易于解释，但在高维或结构化输出空间中，它们难以表示和使用，这可能会限制其与下游任务（如采样和概率预测）的集成能力。我们证明，任何足够正则的可微非一致性得分都会在输出空间上诱导出确定性流，其轨迹收敛到相应共形预测集的边界。这产生了一种计算高效、无需训练的采样方法，可在任意维度下对共形边界进行采样。跨置信水平的混合产生了共形预测分布，其分位数区域与经验共形预测集一致。我们提供了一个近似误差界，将共形预测分布的预测误差分解为得分诱导的畸变、基测度质量和梯度流诱导的畸变。我们在PDE反演问题、降水降尺度、气候模型偏差校正和飓风轨迹预测上评估了该方法。