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Curly Flow Matching for Learning Non-gradient Field Dynamics
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Introduces Curly Flow Matching, a simulation-free Schrödinger bridge method with a non-zero (curl-carrying) reference drift built from inferred velocities, to model periodic non-gradient dynamics in cell cycles, CFD, and ocean currents.
Problem
Trajectory inference from population snapshots fails to capture non-gradient, periodic dynamics (e.g., cell cycles) because existing flow and bridge matching methods assume zero-drift or gradient-field reference processes.
Approach
Solve a Schrödinger bridge with a non-zero drift reference process constructed from approximate velocities; a two-stage simulation-free procedure first learns a neural path interpolant matching the reference drift, then learns a mixture of conditional bridges with OT-based couplings minimizing kinetic energy of the interpolant.
Key results
- On the ocean currents task, Curly-FM reaches EMD 0.019 at t2 versus 0.148 for OT-CFM and 0.073 for SBIRR, and runs in minutes versus ~4 hours for SBIRR.
- On the CFD 2DTGV task, Curly-FM improves cosine distance to 0.189 (vs 0.248 OT-CFM) and precision@5 to 0.489 (vs 0.303 OT-CFM) with comparable MSE.
- On human fibroblast cell cycle data at d=2, Curly-FM achieves cosine distance 0.295 versus 0.800 for OT-CFM, although its W2 (1.199) is worse than OT-CFM (0.248).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use a non-zero reference drift and inferred velocity information | Curly-FM on human fibroblast cell-cycle data · Human fibroblasts cell cycle d=2 | Cos. Dist.: 0.800 $\pm$ 0.309 -> 0.295 $\pm$ 0.040 | Helped |
| use Curly-FM rather than SBIRR | Ocean currents · Ocean currents t2/t4/t6/t8 | EMD: t2: 0.073 $\pm$ 0.020 -> 0.019 $\pm$ 0.003; t4: 0.072 $\pm$ 0.012 -> 0.045 $\pm$ 0.005; t6: 0.120 $\pm$ 0.029 -> 0.027 $ | Helped |
| use Curly-FM rather than OT-CFM | Curly-FM on mouse erythroid data · Mouse erythroid d=2 | Cos. Dist.: 0.146 $\pm$ 0.001 -> 0.009 $\pm$ 0.000 | Helped |
| use Curly-FM rather than MFM | Curly-FM on mouse erythroid data · Mouse erythroid d=2 | W2: 0.269 $\pm$ 0.004 -> 0.369 $\pm$ 0.090 | Hurt |
| use Curly-FM rather than OT-CFM | Curly-FM on 2DTGV CFD · 2DTGV CFD | Prec.@5: 0.303 $\pm$ 0.002 -> 0.489 $\pm$ 0.010 | Helped |
| use Curly-FM rather than CFM | Curly-FM on 2DTGV CFD · 2DTGV CFD | MSE: 0.085 $\pm$ 0.002 -> 0.095 $\pm$ 0.003 | Hurt |
| use independent coupling instead of the learned coupling (ablation) | Curly-FM on ocean currents · Ocean currents | Cos. Dist.: 0.230 $\pm$ 0.003 vs 0.231 $\pm$ 0.004 | No clear effect |
| use independent coupling instead of the learned coupling (ablation) | Curly-FM on 2DTGV CFD · 2DTGV CFD | Cos. Dist.: 0.189 $\pm$ 0.027 -> 0.214 $\pm$ 0.007 | Hurt |
| increase the coupling-cost time samples from n=1 to n=5 (ablation) | Curly-FM on 2DTGV CFD · 2DTGV CFD | Cos. Dist.: 0.189 $\pm$ 0.027 -> 0.179 $\pm$ 0.020 | Helped |
| increase stochasticity from sigma=0.01 to sigma=1.00 (ablation) | Curly-FM on ocean currents · Ocean currents | Cos. Dist.: 0.061 $\pm$ 0.003 -> 0.145 $\pm$ 0.009 | Hurt |
Surprising findings
- Curly-FM has worse W2 than OT-CFM on the cell cycle task, showing a trade-off between matching the reference drift and matching marginals.
- Adding sigma>0 stochasticity monotonically degraded performance, so sigma is set to zero in experiments.
Limitations
- Performance depends on accurate inference of the reference velocity field, which is hard over long timescales.
- Stochastic sigma>0 hurts performance, so the method is tuned for the small/zero-noise regime.
- Trades off marginal matching (W2) for reference drift alignment in several settings.
Experimental setup
Datasets: human fibroblast cell cycle scRNA-seq (5,367 cells) mouse erythroid gastrulation scRNA-seq (9,815 cells) HYCOM Gulf of Mexico ocean currents LagrangeBench 2D Taylor-Green vortex
Benchmarks: human fibroblast cell cycle mouse erythroid HYCOM ocean currents LagrangeBench 2DTGV
Metrics: W2 cosine distance L2 cost EMD MSE precision@k MMD
Baselines: CFM OT-CFM MFM TrajectoryNet SBIRR Vanilla-SB
Builds on: conditional flow matching optimal transport conditional flow matching diffusion Schrödinger bridge matching TrajectoryNet SBIRR
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