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Chance-constrained Flow Matching for High-Fidelity Constraint-aware Generation

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

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Proposes training-free Chance-constrained Flow Matching (CCFM), which projects noisy intermediate flow-matching states onto a time-varying chance-constrained feasible set that is equivalent to projecting clean samples, and evaluates on PDBBind molecular docking and PDE solution generation.

Problem

Generative flow-matching samplers violate hard constraints (physical laws, docking geometry); repeated projection on noisy intermediate states distorts the learned distribution, while ECI-style multi-stage methods add complexity and accumulate errors.

Approach

Under the OT path, the clean sample is an affine function of the current state and the noise; treating the noise-induced term as random lets the projection be formulated as a chance constraint with a time-dependent tightening, which reduces to a deterministic projection of the clean state at t=1 and is theoretically equivalent to clean-sample projection in the linear case.

Key results

• On PDBBind with 1 sample and 2 steps, CCFM raises PoseBusters validity to 47.74% vs 20.34% for FlexDock, and with 20 samples and 2 steps it reaches 67.80% vs 44.63%.
• On 1D reaction-diffusion, CCFM achieves MMSE 3.3e-2 and SMSE 2.7e-2 with zero initial-condition violation, versus 3.6e-2 MMSE for PCFM.
• On 2D Navier-Stokes, CCFM achieves MMSE 0.8e-1 and SMSE 0.8e-1 with zero IC violation, improving on PCFM (MMSE 1.7e-1) and DiffusionPDE (MMSE 1.4e-1).

Surprising findings

• On Navier-Stokes, PCFM yields higher MMSE and SMSE than the penalty-based DiffusionPDE despite preserving feasibility, which the authors attribute to one-step prediction errors in high dimensions.
• On PDBBind with 20 samples and 2 steps, CCFM's L-RMSD<2Å (39.27%) only ties FlexDock, and at 10 samples and 2 steps, L-RMSD<1Å is worse-looking for CCFM in some settings, indicating fidelity gains are not uniform.

Limitations

• Assumes linear (OT) flow so that the clean sample is affine in the noisy state; equivalence results hold for convex constraints while experiments include non-convex constraints without guarantees.
• Does not address constraints without closed forms, such as black-box simulators or non-differentiable checkers.
• Numerical discretization and learned-velocity approximation errors mean guarantees are only approximate in practice.

Experimental setup

Datasets: PDBBind PDBBind time-based split Reaction-diffusion simulations (FiPy) Navier-Stokes simulations (Crank-Nicolson spectral solver)

Benchmarks: PDBBind docking test set (363 complexes) Reaction-diffusion PDE test set (1,000 trajectories) Navier-Stokes PDE test set (1,000 trajectories)

Metrics: L-RMSD A-RMSD PB Valid MMSE SMSE constraint violation

Baselines: FlexDock PCFM DiffusionPDE FFM

Builds on: flow matching projected Euler sampling PCFM FlexDock Functional Flow Matching

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