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Flow-Based Conformal Predictive Distributions

NeurIPS 2026 · Learning theory and algorithms › Prediction and decision losses · approach: Conformal prediction sets · 0 citations (OpenAlex)

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Proposes nonconformity flows, a training-free deterministic ODE that transports base samples onto conformal set boundaries, and mixes these across confidence levels to build conformal predictive distributions (CPDs) for structured regression.

Problem

Conformal prediction returns implicit prediction sets that are hard to represent and sample in high-dimensional or structured output spaces, which limits downstream probabilistic forecasting, risk estimation and simulation.

Approach

For any sufficiently regular differentiable nonconformity score, a minimum-norm flow with a linear score controller drives points exponentially fast onto the conformal boundary; mixing boundary samples over confidence levels yields a predictive distribution whose quantile regions are exactly the empirical conformal sets.

Key results

• On the 2D GP regression task, CPD-G achieves energy distance 0.313 and LSD 0.069, the best LSD among compared methods, with MMD 0.023 versus 0.187 for MC Dropout.
• On precipitation downscaling, CPD-G reaches LSD 0.067 and MMD 0.019, versus 0.072 and 0.075 for MC Dropout.
• In risk-controlled band experiments under asymmetric bias, the re-conformalized band achieves 0.900 marginal coverage at alpha=0.1, versus 0.646 for raw samples.

Surprising findings

• Raw envelopes of flow samples do not achieve pointwise coverage (0.646 under asymmetric bias), requiring a reconformalization step.
• The l1 and kNN scores fail to converge to tolerance because they are non-smooth.

Limitations

• Requires a differentiable (sufficiently smooth) nonconformity score; l1 and kNN scores stall.
• Does not guarantee full exploration of the conformal boundary; trajectories cluster near the initialization.
• Exchangeability is violated in the temporal experiments, so coverage is not guaranteed there.

Experimental setup

Datasets: HURDAT2 2D Gaussian process data elliptic PDE data Navier-Stokes data North American precipitation fields global climate model temperature fields

Benchmarks: 2D GP regression elliptic PDE inverse Navier-Stokes precipitation downscaling climate debiasing HURDAT2 hurricane trajectories

Metrics: energy distance log spectral distance local MMD marginal coverage median set width log convergence error spectral entropy ratio

Baselines: MC Dropout Deep Ensembles Mean-Variance Networks Conditional Flow Matching Mixture Density Networks Implicit Quantile Networks RCPS UQNO

Builds on: split conformal prediction conformal predictive systems conditional flow matching fourier neural operator

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