Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Gradient-Free Generation for Hard-Constrained Systems
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Proposes ECI sampling, a gradient-free, zero-shot method that steers a pre-trained unconditional functional flow matching model to satisfy hard PDE constraints exactly via interleaved extrapolation, correction, and interpolation steps.
Problem
Generating PDE solutions that strictly satisfy hard constraints (initial/boundary conditions, conservation laws) with pre-trained generative models, where gradient-based guidance is expensive and does not guarantee exact satisfaction.
Approach
At each flow-ODE sampling step, extrapolate the final prediction in one step, apply an exact projection/correction to that prediction, and interpolate back to the current time along the OT path; repeat for several mixing iterations. This gives iterative, gradient-free control with exact constraint satisfaction at t=1.
Key results
- On generative tasks, ECI achieves exact constraint satisfaction (CE = 0) across all four generative PDE settings while obtaining the best FPD on Stokes IC/BC and Darcy flow, e.g., Stokes IC FPD 0.076 vs 0.202 for DiffusionPDE.
- ECI-1 is reported as ~440x faster and ~310x more memory-efficient than gradient-based D-Flow on the Stokes problem.
- In zero-shot neural operator learning, ECI achieves Stokes MMSE 0.050e-3 vs 0.033e-3 for a directly trained FNO, and NS MMSE 0.069e-2 vs 0.380e-2 for FNO (ECI better on NS).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply ECI sampling instead of zero-shot baselines | pre-trained FFM · Stokes IC and Stokes BC | FPD: 0.202 -> 0.076; 1.162 -> 0.010 | Helped |
| Apply ECI sampling instead of DiffusionPDE | pre-trained FFM · Heat Equation | FPD: 0.951 -> 1.365 | Hurt |
| Apply ECI sampling instead of DiffusionPDE | pre-trained FFM · Darcy Flow | FPD: 8.027 -> 0.946 | Helped |
| Apply ECI sampling instead of ProbConserv | pre-trained FFM · NS Equation | FPD: 2.171 -> 1.131 | Helped |
| Use ECI rather than the strongest listed baseline | GPME uncertainty quantification · Heat Equation | MSE / 10^-4: 1.37 -> 0.81 | Helped |
| Use ECI rather than ProbConserv | GPME uncertainty quantification · PME | MSE / 10^-4: 0.17 -> 0.19 | Hurt |
| Use ECI rather than ProbConserv | GPME uncertainty quantification · Stefan Problem | MSE / 10^-3: 1.89 -> 1.89 | No clear effect |
| Use ECI instead of directly trained FNO | Stokes Problem regression · Stokes Problem | MMSE / 10^-3: 0.033 -> 0.050 | Hurt |
| Use ECI instead of directly trained FNO | NS Equation regression · NS Equation | MMSE / 10^-2: 0.380 -> 0.069 | Helped |
| Increase mixing iterations (ablation) | ECI sampling · Stokes IC and BC | MSE | Mixed |
| Use no noise re-sampling rather than re-sampling (ablation) | ECI sampling · Stokes IC | MSE | Helped |
| Use no noise re-sampling rather than R=5 or R=2 (ablation) | ECI sampling · Stokes BC | MSE | Hurt |
Surprising findings
- On the NS equation, ECI's zero-shot MSE (0.069e-2) is lower than that of an FNO trained directly on the regression task (0.380e-2).
- Mixing-iteration count does not monotonically improve MSE; task-specific tuning is needed.
Limitations
- Number of mixing iterations and re-sampling interval have task-specific effects and require heuristic selection without theory.
- Gaussian uncertainty quantification results show worse log-likelihood than some baselines despite lower MSE.
- Applicability beyond SciML (e.g., supply chain, time series) is left to future work.
Experimental setup
Datasets: Stokes problem (synthetic) Heat equation (synthetic) Darcy flow (simulated) Navier-Stokes equation (simulated) Porous medium equation (synthetic) Stefan problem (synthetic)
Benchmarks: Stokes problem Heat equation Darcy flow Navier-Stokes equation Generalized Porous Medium Equation (PME, Stefan)
Metrics: MMSE SMSE constraint error (CE) Fréchet Poseidon distance (FPD) log-likelihood (LL) MSE time per sample GPU memory
Baselines: ANP ProbConserv DiffusionPDE D-Flow CondFFM SoftC-ANP HardC-ANP FNO
Builds on: functional flow matching flow matching diffusion posterior sampling ProbConserv DiffusionPDE
Closest papers by meaning
- Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints (NeurIPS 2025 · similarity 0.92)
- Chance-constrained Flow Matching for High-Fidelity Constraint-aware Generation (NeurIPS 2026 · similarity 0.89)
- Optimal Flow Matching: Learning Straight Trajectories in Just One Step (NeurIPS 2024 · similarity 0.88)
- Primal-Dual Flow Matching for Sample-Wise Constrained Generation (NeurIPS 2026 · similarity 0.87)
- Bellman Optimal Stepsize Straightening of Flow-Matching Models (ICLR 2024 · similarity 0.87)
- On the Guidance of Flow Matching (ICML 2025 · similarity 0.86)
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