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Inferring stochastic dynamics with growth from cross-sectional data
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Proposes unbalanced probability flow inference (UPFI), a two-step method that infers drift, diffusion-related force and growth rate from cross-sectional snapshots of stochastic dynamics with birth and death, using score matching plus a Lagrangian ODE fit with unbalanced Sinkhorn loss.
Problem
Inferring stochastic dynamics with cell division and death from destructive, cross-sectional single-cell snapshots, where drift, intrinsic noise and growth must be disentangled.
Approach
Extends probability flow inference (PFI) to branching diffusions by writing the Fokker-Planck equation with source as a Lagrangian transport with score-dependent velocity plus a mass ODE for growth, and fitting drift and fitness jointly via unbalanced Sinkhorn divergence on snapshots after a separate denoising score matching step.
Key results
- On the bistable system UPFI achieves the lowest path energy distance at d=2,5,10 (e.g. 0.14 vs 1.41 for PFI at d=2) and best force L2 error at d=2,5,10,25.
- On CLE gene regulatory networks, UPFI (multiplicative) reduces the path energy distance on the bifurcating system to 1.69 versus 4.62 for PFI (mult.) and raises fate correlation to 0.98 vs 0.62.
- On the monocyte-neutrophil lineage-tracing data, UPFI attains the highest fate correlations (Pearson 0.26, Spearman 0.27) and comparable cosine similarity to RNA velocity relative to TIGON++.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| model growth alongside drift | UPFI versus PFI · Bistable system, d=10 | Pearson fate correlation: 0.65 ± 0.02 -> 0.99 ± 0.00 | Helped |
| model growth alongside drift | UPFI versus PFI · Bifurcating CLE regulatory graph | AUPR: 0.33 ± 0.09 -> 0.64 ± 0.03 | Helped |
| model growth alongside drift | UPFI versus PFI · Monocyte-neutrophil development | Kendall's tau fate correlation: 0.07 ± 0.03 -> 0.22 ± 0.03 | Helped |
| use multiplicative rather than additive noise (ablation) | UPFI · Bifurcating CLE | Path energy distance: 1.22 ± 0.38 -> 1.69 ± 0.14 | Hurt |
| use multiplicative rather than additive noise (ablation) | UPFI · Bifurcating CLE | Force error (L²): 12.59 ± 0.39 -> 11.78 ± 0.30 | Helped |
| use multiplicative rather than additive noise (ablation) | UPFI · Bifurcating CLE | Pearson fate correlation: 0.97 ± 0.00 -> 0.98 ± 0.00 | Helped |
| use multiplicative rather than additive noise (ablation) | UPFI · Haematopoietic CLE | Path energy distance: 0.99 ± 0.26 -> 0.98 ± 0.15 | Helped |
| use multiplicative rather than additive noise (ablation) | UPFI · Haematopoietic CLE | Force error (L²): 19.58 ± 0.33 -> 18.13 ± 0.25 | Helped |
| use multiplicative rather than additive noise (ablation) | UPFI · Haematopoietic CLE | Fate prediction error (TV): 0.08 ± 0.00 -> 0.07 ± 0.00 | Helped |
| remove the growth regularisation term (ablation) | UPFI · Bistable system | unspecified regularisation evaluation metric | No clear effect |
Surprising findings
- Growth and drift are non-identifiable: for OU processes with quadratic fitness, the same marginals arise from different autonomous drift and growth combinations, so the regularisation is needed for uniqueness.
- DeepRUOT, which models growth and noise in theory, performs poorly on the bistable benchmark, attributed to its multi-stage training instability.
Limitations
- Identifiability of drift versus growth is fundamentally limited in the non-autonomous case, so consistency is not guaranteed without regularisation.
- Requires learning the score beforehand, and scalability to high dimensions is mildly limited by ODE integration and Sinkhorn cost.
- Assumes the diffusivity is known or of a known functional form; only drift and fitness are inferred.
Experimental setup
Datasets: simulated bistable process Chemical Langevin Equation gene regulatory networks monocyte-neutrophil lineage-tracing scRNA-seq
Benchmarks: simulated bistable systems CLE bifurcating and HSC systems monocyte-neutrophil lineage tracing
Metrics: path energy distance force error (cosine) force error (L2) Pearson fate correlation cosine distance to RNA velocity AUPR AUROC
Baselines: PFI fitness-ODE TIGON++ DeepRUOT OTFM UOTFM
Builds on: probability flow inference (PFI) unbalanced Sinkhorn divergence denoising score matching TIGON DeepRUOT
Closest papers by meaning
- WFR-MFM: One-Step Inference for Dynamic Unbalanced Optimal Transport (ICML 2026 · similarity 0.88)
- Variational Mixtures of ODEs for Inferring Cellular Gene Expression Dynamics (ICML 2022 · similarity 0.88)
- Joint Velocity-Growth Flow Matching for Single-Cell Dynamics Modeling (NeurIPS 2025 · similarity 0.88)
- Learning stochastic dynamics from snapshots through regularized unbalanced optimal transport (ICLR 2025 · similarity 0.88)
- Modeling Cell Dynamics and Interactions with Unbalanced Mean Field Schrödinger Bridge (NeurIPS 2025 · similarity 0.87)
- Curly Flow Matching for Learning Non-gradient Field Dynamics (NeurIPS 2025 · similarity 0.85)
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