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 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. Boundary samples can be reconformalized to form pointwise prediction sets with controlled risk, and mixing across confidence levels yields conformal predictive distributions whose quantile regions coincide exactly with conformal prediction sets. We evaluate the approach on PDE inverse problems, precipitation downscaling, climate model debiasing, and hurricane trajectory forecasting.

翻译：保形预测提供了一个无需分布假设的框架，通过具有精确有限样本覆盖度的预测集进行不确定性量化。在低维情况下，这些集合易于解释，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，任何可微的非保形度分都会在输出空间上诱导一个确定性流，其轨迹收敛于相应保形预测集的边界。这产生了一种计算高效、无需训练的方法，用于在任意维度中采样保形边界。边界样本可通过重新保形化形成具有可控风险的点态预测集，而跨置信水平的混合则产生保形预测分布，其分位数区域与保形预测集完全重合。我们在偏微分方程反问题、降水降尺度、气候模型偏差校正和飓风轨迹预测等任务上评估了该方法。