# Flow-Based Conformal Predictive Distributions

arXiv: https://arxiv.org/abs/2602.07633 · HTML: https://mlsanity.com/paper/2602.07633/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 49 references that are in the map.
- First posted on arXiv 7 Feb 2026.

## Builds on (most cited first, 30 of 30)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Implicit Quantile Networks for Distributional Reinforcement Learning](https://mlsanity.com/paper/1806.06923/) (2018)
- [The Vendi Score: A Diversity Evaluation Metric for Machine Learning](https://mlsanity.com/paper/2210.02410/) (2022)
- [Distribution-Free, Risk-Controlling Prediction Sets](https://mlsanity.com/paper/2101.02703/) (2021)
- [Conformal Language Modeling](https://mlsanity.com/paper/2306.10193/) (2023)
- [Distributional conformal prediction](https://mlsanity.com/paper/1909.07889/) (2019)
- [Large language model validity via enhanced conformal prediction methods](https://mlsanity.com/paper/2406.09714/) (2024)
- [Conformal Prediction for Natural Language Processing: A Survey](https://mlsanity.com/paper/2405.01976/) (2024)
- [How to Trust Your Diffusion Model: A Convex Optimization Approach to Conformal Risk Control](https://mlsanity.com/paper/2302.03791/) (2023)
- [Probabilistic Conformal Prediction Using Conditional Random Samples](https://mlsanity.com/paper/2206.06584/) (2022)
- [Conformal Prediction Bands for Multivariate Functional Data](https://mlsanity.com/paper/2106.01792/) (2021)
- [Calibrated Uncertainty Quantification for Operator Learning via Conformal Prediction](https://mlsanity.com/paper/2402.01960/) (2024)
- [Valid model-free spatial prediction](https://mlsanity.com/paper/2006.15640/) (2020)
- [Generative Conformal Prediction with Optimized Coverage Allocation](https://mlsanity.com/paper/2410.13735/) (2024)
- [Conformal Prediction Bands for Two-Dimensional Functional Time Series](https://mlsanity.com/paper/2207.13656/) (2022)
- [Elastic depths for detecting shape anomalies in functional data](https://mlsanity.com/paper/1907.06759/) (2019)
- [Conformalized Prediction of Post-Fault Voltage Trajectories Using Pre-trained and Finetuned Attention-Driven Neural Operators](https://mlsanity.com/paper/2410.24162/) (2024)
- [Beyond Uncertainty Sets: Leveraging Optimal Transport to Extend Conformal Predictive Distributions to Multivariate Settings](https://mlsanity.com/paper/2511.15146/) (2025)
- [Quantifying uncertainty in climate projections with conformal ensembles](https://mlsanity.com/paper/2408.06642/) (2024)
- [Conformal Bounds on Full-Reference Image Quality for Imaging Inverse Problems](https://mlsanity.com/paper/2505.09528/) (2025)
- [Locally Adaptive Conformal Inference for Operator Models](https://mlsanity.com/paper/2507.20975/) (2025)
- [Image Super-Resolution with Guarantees via Conformalized Generative Models](https://mlsanity.com/paper/2502.09664/) (2025)

## Built on by (most cited first, 2 of 2)

- [Manifold Constrained Conformal Prediction for Spatial Events](https://mlsanity.com/paper/2607.10008/) (2026)
- [Sample-Efficient Generative Conformal Prediction](https://mlsanity.com/paper/2610.11349/) (2026)

## Abstract (arXiv)

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.