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Learning non-equilibrium diffusions with Schrödinger bridges: from exactly solvable to simulation-free
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Derives exact Gaussian Schrödinger bridge solutions for multivariate Ornstein-Uhlenbeck (non-gradient) reference processes and proposes mvOU-OTFM, a simulation-free score and flow matching solver for non-Gaussian marginals.
Problem
Schrödinger bridge reconstruction of stochastic dynamics from snapshot marginals when the reference dynamics are non-gradient (non-equilibrium), where existing work assumes Brownian or scalar OU references and simulation-based solvers are slow and inaccurate in high dimensions.
Approach
Uses a multivariate OU reference with a generic (possibly asymmetric) drift matrix, for which Q-bridges, scores and flows are available in closed form, yielding exact Gaussian SB solutions and a simulation-free flow+score matching loss for general marginals after a Sinkhorn coupling with a Gaussian-transition cost.
Key results
- On Gaussian marginals, mvOU-OTFM attains BW2^2 marginal error of 0.19 at d=2 and 2.21 at d=50, versus 5.55 for IPML and 1.21 for NLSB at d=2, and lowest error at all tested dimensions.
- On the repressilator, mvOU-OTFM after iterations reaches EMD 1.40 and energy distance 0.89 at iterate 4, better than SBIRR (mvOU) at 2.10 EMD and SBIRR (MLP) at 1.67 EMD.
- mvOU-OTFM trains in roughly 1-2 minutes on CPU for d=50, versus 15+ minutes for NLSB on GPU.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use mvOU-OTFM instead of the strongest listed baseline | Gaussian Schrödinger bridge solver · Gaussian marginals, d=2, 5, 10, 25, 50, 100 | Marginal error (BW₂²): 0.19$\pm$0.17 vs 1.21$\pm$0.18; 0.23$\pm$0.16 vs 1.16$\pm$0.26; 0.59$\pm$0.36 vs 1.36$\pm$0.13; 0.84$\pm$0.22 vs 2.97$\p | Helped |
| Use mvOU-OTFM instead of the strongest listed baseline | Gaussian Schrödinger bridge solver · Gaussian marginals, d=2, 5, 10, 25, 50, 100 | Force error (L²): 3.56$\pm$0.25 vs 7.59$\pm$0.33; 3.66$\pm$0.19 vs 7.72$\pm$0.15; 3.82$\pm$0.13 vs 8.15$\pm$0.25; 4.67$\pm$0.16 vs 11.02$\ | Helped |
| Add another iteration of the learned reference and bridge procedure (ablation) | mvOU-OTFM on repressilator dynamics · Repressilator leave-one-out | EMD: 3.38 $\pm$ 1.52 -> 2.22 $\pm$ 1.12 -> 1.59 $\pm$ 0.66 -> 1.49 $\pm$ 0.64 -> 1.40 $\pm$ 0.57 | Helped |
| Add another iteration of the learned reference and bridge procedure (ablation) | mvOU-OTFM on repressilator dynamics · Repressilator leave-one-out | Energy distance: 1.86 $\pm$ 1.06 -> 1.29 $\pm$ 0.86 -> 1.03 $\pm$ 0.65 -> 0.95 $\pm$ 0.58 -> 0.89 $\pm$ 0.55 | Helped |
| Use a learned mvOU reference rather than the null Brownian reference (ablation) | mvOU-OTFM on repressilator dynamics · Repressilator leave-one-out | EMD: 3.38 $\pm$ 1.52 -> 1.40 $\pm$ 0.57 | Helped |
| Use a learned mvOU reference rather than the null Brownian reference (ablation) | mvOU-OTFM on repressilator dynamics · Repressilator leave-one-out | Energy distance: 1.86 $\pm$ 1.06 -> 0.89 $\pm$ 0.55 | Helped |
| Use a neural reference family instead of an mvOU reference family | SBIRR on repressilator dynamics · Repressilator leave-one-out | EMD: 2.10 $\pm$ 0.74 -> 1.67 $\pm$ 0.95 | Helped |
| Use a neural reference family instead of an mvOU reference family | SBIRR on repressilator dynamics · Repressilator leave-one-out | Energy distance: 1.39 $\pm$ 0.82 -> 1.10 $\pm$ 0.86 | Helped |
| Set the reference drift scale to zero, yielding a Brownian reference, instead of using the fitted nonzero mvOU drift (ablation) | mvOU-OTFM on cell-cycle single-cell data · Cell-cycle snapshots | Cyclic behavior recovery | Hurt |
Surprising findings
- The learned mvOU reference drift for the repressilator resembles the Jacobian of its linearisation, recovering the cyclic activation/inhibition pattern.
- A Brownian reference (gamma=0) fails to recover the expected cyclic behaviour in the single-cell experiment, while the mvOU reference does.
Limitations
- Matrix inversions and matrix powers scale as O(d^3), limiting applicability to high dimensions (d<=100 works in practice).
- Linear reference restricts the model class; non-linear dynamics only handled via linearisation about a fixed point.
- Minibatch OT couplings introduce bias for flow matching, so the exact coupling over all samples is needed.
Experimental setup
Datasets: synthetic Gaussian marginals repressilator simulated snapshots single-cell RNA-seq cell cycle dataset (N=2793)
Benchmarks: Gaussian SB marginal and force error (d=2 to 100) repressilator leave-one-out interpolation single-cell cell cycle interpolation
Metrics: Bures-Wasserstein distance squared L2 force error earth mover's distance energy distance Bures-Wasserstein distance to interpolant
Baselines: IPML NLSB BM-OTFM SBIRR (mvOU) SBIRR (MLP)
Builds on: Schrödinger bridge conditional flow matching Schrödinger bridge matching ([SF]^2M) IPML SBIRR
Closest papers by meaning
- Go With the Flow: Fast Diffusion for Gaussian Mixture Models (NeurIPS 2025 · similarity 0.91)
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- Momentum Multi-Marginal Schrödinger Bridge Matching (NeurIPS 2025 · similarity 0.90)
- Generalized Schrödinger Bridge Matching (ICLR 2024 · similarity 0.89)
- Improved sampling via learned diffusions (ICLR 2024 · similarity 0.89)
- Adjoint Schrödinger Bridge Sampler (NeurIPS 2025 · similarity 0.89)
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