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Primal-Dual Flow Matching for Sample-Wise Constrained Generation

NeurIPS 2026 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Flow matching and flow sampling

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Proposes Primal-Dual Flow Matching (PDFM), a training-time primal-dual method that enforces a user-specified sample-wise constraint satisfaction rate using only membership-oracle feedback, with a learned differentiable surrogate for constraint satisfaction and TV-based guarantees.

Problem

Generating samples from a flow-matching model that satisfy a sample-wise constraint set with a user-specified probability, when the constraint is accessible only through a binary membership oracle (no differentiable distance, projection, or convexity).

Approach

Formulate constrained NLL minimization at the population level, derive a primal-dual flow-matching algorithm with TV guarantees, and make it practical by training a neural predictor f_phi of terminal constraint satisfaction from intermediate Euler states (binary cross-entropy on oracle labels) and using its gradient, as in deterministic policy gradients, to update the velocity field; the dual vari

Key results

• On KS Case 1 (Q(C)=1, target 98%), PDFM reaches 97.74% constraint satisfaction versus 14.19% for vanilla FM and 43.17% for FM-RE.
• On 8-D and 20-D l2-ball tasks, PDFM attains P(C) of 99.906% and 99.755% with lower SWD than FM-RE (0.0172 vs 0.0194; 0.0101 vs 0.0132).
• On QM9, PDFM raises molecule stability from 32.77% to 48.75% and lowers NLL from -19.74 to -20.70 relative to FM, below the 60% target.

Surprising findings

• Training FM on the truncated feasible distribution Q' still fails to reach the target constraint rate (e.g., 33.24% vs 90% target in KS Case 2), and in the 2-D example it reaches only 98.74% against a 99.5% target.
• Vanilla FM trained on a fully feasible training set (Q(C)=1) satisfies the constraint only 14.19% of the time on KS, showing local consistency does not transfer to the learned model.

Limitations

• Requires sufficient positive constraint-satisfaction feedback during training to learn the predictor f_phi reliably.
• Theoretical guarantees rely on strong assumptions (bounded velocity error, exact dual gradients, expressivity), and the practical algorithm's error terms are not fully characterized.
• On QM9 the stability rate stays below the 60% target.

Experimental setup

Datasets: synthetic 2-D Gaussian-corrupted two-region data Gaussian mixture 8-D and 20-D with l2-ball constraint Kuramoto-Sivashinsky space-time patches SOCOFing fingerprints QM9

Benchmarks: KS equation patches (case 1 and case 2) 8-D and 20-D l2-ball constrained generation SOCOFing fingerprint ridge connectivity QM9 molecule stability

Metrics: constraint satisfaction rate P(C) sliced Wasserstein distance histogram error molecule stability rate negative log-likelihood

Baselines: FM MDM NAMM PDM PCFM FM-RE

Builds on: flow matching deterministic policy gradient FM-RE mirror diffusion model

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