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Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints
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Proposes Physics-Constrained Flow Matching (PCFM), a zero-shot inference-time method that projects intermediate flow states onto nonlinear PDE constraint manifolds to exactly enforce hard constraints in pretrained functional flow matching models.
Problem
Generating PDE solutions with pretrained generative models that exactly satisfy physical constraints (linear and nonlinear conservation laws, initial/boundary conditions), where soft penalties and architectural biases fail to guarantee hard constraint satisfaction.
Approach
Interleave forward ODE shooting to t=1, Gauss-Newton tangent-space projection onto the constraint manifold, a reverse step using the OT displacement interpolant, and a relaxed penalty correction at each flow step, plus a final full projection; generalizes ECI to arbitrary nonlinear constraints.
Key results
- PCFM achieves exact constraint satisfaction (CE = 0 at 10^-2 scale) for IC, BC and mass conservation across all four PDEs while baselines (ECI, DiffusionPDE, D-Flow, FFM) leave residuals.
- PCFM reports up to 99.5% improvement in MSE over state-of-the-art baselines, with best FPD on Burgers IC (0.101 vs 1.31 for ECI).
- On Burgers with IC fixed, PCFM with 5 local flux collocation points captures shock structure and gets MMSE 0.052 vs 10.0 (ECI) and 13.7 (FFM).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| apply PCFM constraint-guided sampling instead of unconstrained FFM | Pretrained FFM · Burgers IC | MMSE / $10^{-2}$: 13.7 -> 0.052 | Helped |
| apply PCFM constraint-guided sampling instead of unconstrained FFM | Pretrained FFM · Heat Equation | MMSE / $10^{-2}$: 4.56 -> 0.241 | Helped |
| apply PCFM constraint-guided sampling instead of unconstrained FFM | Pretrained FFM · Burgers BC | MMSE / $10^{-2}$: 4.86 -> 0.335 | Hurt |
| add constrained local flux collocation points alongside IC and mass constraints (ablation) | PCFM on Burgers · Burgers IC | generation quality | Helped |
| add a total variation diminishing (TVD) constraint (ablation) | PCFM on Heat Equation · Heat Equation | fidelity | Helped |
| add relaxed constraint corrections during sampling (ablation) | PCFM · Linear constraints | solution fidelity | Helped |
Surprising findings
- Adding complementary constraints (local flux, IC, global mass) improved generation quality, unlike PINN-style soft penalties where extra constraints compete.
- Enforcing constraints only at the final state, rather than at every timestep as in PDM, avoids over-constraining the trajectory for nonlinear global constraints.
Limitations
- Currently focuses on equality constraints; inequality constraints not handled.
- Higher runtime and memory than vanilla FFM (e.g., 1371 ms vs 105 ms per sample on Burgers), and stacking constraints increases projection cost.
- Several baselines (D-Flow, PDM) omitted from some tables due to numerical instabilities; FPD on Reaction-Diffusion may be affected by interpolation artifacts.
Experimental setup
Datasets: Heat equation simulations (analytical) Navier-Stokes vorticity simulations (Crank-Nicolson spectral solver) Reaction-diffusion numerical solutions Burgers equation Godunov solutions
Benchmarks: Heat equation Navier-Stokes Reaction-Diffusion Burgers' equation
Metrics: MMSE SMSE FPD (Frechet Poseidon Distance) constraint error (IC, BC, CL) runtime peak memory
Baselines: ECI DiffusionPDE D-Flow PDM FFM
Builds on: Functional Flow Matching ECI D-Flow DiffusionPDE PDM
Closest papers by meaning
- Chance-constrained Flow Matching for High-Fidelity Constraint-aware Generation (NeurIPS 2026 · similarity 0.92)
- Gradient-Free Generation for Hard-Constrained Systems (ICLR 2025 · similarity 0.92)
- Generating Physical Dynamics under Priors (ICLR 2025 · similarity 0.88)
- Optimal Flow Matching: Learning Straight Trajectories in Just One Step (NeurIPS 2024 · similarity 0.88)
- Controlled Generation with Equivariant Variational Flow Matching (ICML 2025 · similarity 0.88)
- Physics-Informed Diffusion Models (ICLR 2025 · similarity 0.87)
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