Research map / Probabilistic and scientific modeling / Variational and generative sampling
Dynamic Conditional Optimal Transport through Simulation-Free Flows
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The paper proves a dynamic formulation of conditional optimal transport and uses it to build COT-FM, a simulation-free conditional flow-matching model applied to Bayesian inverse problems.
Problem
Learning conditional optimal transport maps for conditional generative modeling without adversarial training or simulation during training, extended to infinite-dimensional settings.
Approach
Dynamic formulation of conditional OT generalizing Benamou-Brenier, with flow matching along triangular COT couplings.
Key results
- COT-FM has the lowest W2 and MMD on all four 2D datasets (e.g., Checkerboard W2 4.69e-2 vs 6.27e-2 for PCP-Map).
- On Lotka-Volterra, COT-FM gets W2 4.02e-2 to MCMC samples vs 5.04e-2 (PCP-Map) and 11.41e-2 (FM).
- On Darcy flow, COT-FFM MSE 5.40e-2 vs 6.55e-2 (WaMGAN) and 7.30e-2 (FFM); CRPS on par with FFM.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use COT-FM instead of PCP-Map | 2D synthetic data · Checkerboard | W2 (1e-2): 6.27 ± 0.81 -> 4.69 ± 1.00 | Helped |
| Use COT-FM instead of PCP-Map | 2D synthetic data · Checkerboard | MMD (1e-3): 0.21 ± 0.13 -> 0.17 ± 0.13 | Helped |
| Use COT-FM instead of PCP-Map | 2D synthetic data · Moons | W2 (1e-2): 8.44 ± 1.09 -> 6.50 ± 1.41 | Helped |
| Use COT-FM instead of PCP-Map | 2D synthetic data · Moons | MMD (1e-3): 0.22 ± 0.10 -> 0.13 ± 0.10 | Helped |
| Use COT-FM instead of PCP-Map | 2D synthetic data · Circles | W2 (1e-2): 6.19 ± 0.43 -> 5.56 ± 0.43 | Helped |
| Use COT-FM instead of PCP-Map | 2D synthetic data · Circles | MMD (1e-3): 0.20 ± 0.17 -> 0.20 ± 0.04 | No clear effect |
| Use COT-FM instead of PCP-Map | 2D synthetic data · Swissroll | W2 (1e-2): 5.35 ± 0.93 -> 4.64 ± 1.26 | Helped |
| Use COT-FM instead of PCP-Map | 2D synthetic data · Swissroll | MMD (1e-3): 0.16 ± 0.13 -> 0.15 ± 0.19 | Helped |
| Use COT-FM instead of COT-Flow | Lotka-Volterra posterior generation · Lotka-Volterra | W2 (1e-2): 4.86 ± 1.1 -> 4.02 ± 0.06 | Helped |
| Use COT-FM instead of COT-Flow | Lotka-Volterra posterior generation · Lotka-Volterra | MMD (1e-3): 0.83 ± 0.50 -> 0.95 ± 0.03 | Hurt |
| Use COT-FFM instead of WaMGAN | Darcy flow posterior generation · Darcy flow | MSE (1e-2): 6.55 ± 0.07 -> 5.40 ± 0.08 | Helped |
| Use COT-FFM instead of FFM | Darcy flow posterior generation · Darcy flow | CRPS (1e-2): 15.47 ± 0.06 -> 15.56 ± 0.08 | No clear effect |
Surprising findings
- PCP-Map is competitive and slightly beats COT-FM on Circles MMD.
Limitations
- Full COT plan is expensive for large datasets, requiring minibatch approximations.
- Small additional cost over standard flow matching.
Experimental setup
Datasets: Checkerboard Moons Circles Swissroll Lotka-Volterra simulated data Darcy flow simulated data
Benchmarks: 2D synthetic distributions (Checkerboard, Moons, Circles, Swissroll) Lotka-Volterra inverse problem Darcy flow inverse problem
Metrics: W2 MMD MSE CRPS
Baselines: PCP-Map COT-Flow WaMGAN flow matching (FM) FFM
Builds on: flow matching McCann interpolants Benamou-Brenier theorem PCP-Map stochastic interpolants
Closest papers by meaning
- Flow Matching from Viewpoint of Proximal Operators (NeurIPS 2026 · similarity 0.88)
- Computing Optimal Transport Maps and Wasserstein Barycenters Using Conditional Normalizing Flows (ICML 2025 · similarity 0.88)
- One-Shot Generative Flows: Existence and Obstructions (NeurIPS 2026 · similarity 0.87)
- Generative Modeling by Value-Driven Transport (NeurIPS 2026 · similarity 0.87)
- Beckmann Transport Models: From Autonomous Flows to One-Step Maps (NeurIPS 2026 · similarity 0.87)
- Generalized Wasserstein Flow Matching: Transport Plans, Everywhere, All at Once (NeurIPS 2026 · similarity 0.87)
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