Title: Joint Velocity-Growth Flow Matching for Single-Cell Dynamics Modeling
Venue: NeurIPS 2025
Accepted: yes
Keywords: flow matching

single-cell

optimal transport

dynamics modeling
Abstract: Learning the underlying dynamics of single cells from snapshot data has gained increasing attention in scientific and machine learning research. The destructive measurement technique and cell proliferation/death result in unpaired and unbalanced data between snapshots, making the learning of the underlying dynamics challenging. In this paper, we propose joint Velocity-Growth Flow Matching (VGFM), a novel paradigm that jointly learns state transition and mass growth of single-cell populations via flow matching. VGFM builds an ideal single-cell dynamics containing velocity of state and growth of mass, driven by a presented two-period dynamic understanding of the static semi-relaxed optimal transport, a mathematical tool that seeks the coupling between unpaired and unbalanced data. To enable practical usage, we approximate the ideal dynamics using neural networks, forming our joint velocity and growth matching framework. A distribution fitting loss is also employed in VGFM to further improve the fitting performance for snapshot data. Extensive experimental results on both synthetic and real datasets demonstrate that VGFM can capture the underlying biological dynamics accounting for mass and state variations over time, outperforming existing approaches for single-cell dynamics modeling.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes VGFM (Velocity-Growth Flow Matching), a novel method for modeling single-cell dynamics from snapshot data that accounts for both state transitions and mass variation (e.g., due to cell proliferation or death). The authors introduce a dynamic reinterpretation of semi-relaxed optimal transport (SROT) as a two-phase process: first, mass growth occurs (modeled via a growth function), and then cells transition in state space (modeled via a velocity field). From this decomposition, the authors derive closed-form supervision targets for learning velocity and growth via neural networks using a flow matching approach. An additional distribution fitting loss is incorporated to improve the match between predicted and observed snapshots. The method outperforms existing state-of-the-art (SOTA) methods on both synthetic and real datasets, with superior accuracy, interpretability, and scalability.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The paper is technically sound and builds on a solid mathematical foundation (SROT and continuity equations).\n- The formulation of joint mass and state dynamics is well-posed, with provable equivalence to SROT objectives (Proposition 1, Theorem 1).\n- Empirical results are strong across diverse settings, including high-dimensional synthetic data and real scRNA-seq datasets.\n- Ablation studies and comparisons to unbalanced transport baselines (e.g., DeepRUOT, TIGON) are well done and validate all components.\n\nWeeknesses:\n- The estimation of the Monge map via barycentric projection introduces approximation error, especially when entropy regularization is non-negligible.\n- The growth function $g(x, t)$ is derived from empirical densities and may absorb technical artifacts (sampling bias, dropout), but this is not fully analyzed.\n- The method assumes aligned timepoint snapshots, limiting its direct application to asynchronous or pseudotime datasets.\n\nQUESTIONS:\n- Could the authors provide correlation analyses between predicted growth rates and known proliferation markers or synthetic ground truth?\n-  Have the authors considered estimating uncertainty in predicted velocity/growth (e.g., via dropout or ensembles)? \n-  Can the method be extended to scenarios where timepoint alignment is unknown or uncertain, possibly by integrating OT-based time alignment?\n- Effect of Sampling Bias: Since the method relies on observed cell counts for growth learning, have the authors tested robustness under non-uniform or subsampled cell distributions across time?\n\nLIMITATIONS:\nYes\nAdditional suggestions:\n- Consider including diagnostics or visualizations of growth error spatially or per-cell type.\n- Investigate how growth modeling could be regularized or debiased using biological priors.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a flow and growth model for modeling interpolation of unbalanced snapshots of an evolving distribution. The evolution is derived from a two-stage process of growth and then flow, with the assumption that $p_0$ and the marginal of the coupling for the flow part have the same support. The resulting growth factor and flow vector field is then mixed to occur simultaneously, and proven to result in $p_1$.\n\nImplementation of this framework is done via learning of the vector field and growth weight parameterized as neural networks. Training is done with sample access to the snapshot distributions and solving entropy-regularized semi-relaxed OT between the discrete distributions. An additional $W_1$ loss is also added to the result of the joint growth and flow model to improve alignment to the snapshots. Empirical performance is shown to outperform SOTA methods on synthetic and real datasets in achieving alignment to the snapshots.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. Good theoretical justification of the approach is given.\n2. The method is relatively simple, but nontrivially combines existing tools for entropically-regularized transport and neural ODEs.\n3. Empirical performance and runtime seems to clearly beat existing methods.\n\nWeaknesses:\n1. It still seems unclear to me if the suggested model accurately reflects the potential behavior of genetic populations of cells. In particular, is it not possible for novel populations of cells to grow or appear outside the support of the current (or initial distribution) $p_0$? Or for whole populations to just disappear, and to have the remaining population be transported to the next snapshot? It would nice to comment on this or address this.\n2. As noted by the authors, the method is not fully simulation-free due to the additional $W_1$ loss against snapshots.\n\nQUESTIONS:\n1. See point 1 in Weaknesses.\n2. Is there a problem with non-differentiability of $W_1$ in the pipeline?\n3. For Eq. 5, is there a reason you couldn't incorporate the energy from A.4 in order to get a unique specification for $g_t$? It would nice if it\nmirrored the uniqueness of the velocity field from the Benamou-Brenier formulation.\n4. Will an implementation be released?\n\nLIMITATIONS:\nYes, though Weakness point 1 should be addressed in some way.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes VGFM (Joint Velocity-Growth Flow Matching), a scalable, simulation-free approach to modeling single-cell dynamics from unpaired, unbalanced snapshot data. By framing the dynamics as a two-period semi-relaxed optimal transport problem, the authors decouple and then recombine velocity and growth fields into a joint dynamic system. The method uses neural networks to parameterize these fields and optimizes a flow-matching loss with an additional distribution-fitting term. Experiments demonstrate VGFM’s performance on synthetic Gaussian data, a 5D EB dataset, and CITE-seq data.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths \n- [Originality] The formulation of the dynamics via semi-relaxed OT with separate velocity and growth fields is novel and well-grounded. The manuscript clearly explains the derivation of VGFM and its training objectives.\n- [Significance] The proposed method allows training on very high-dimensional synthetic data (up to 1000D) efficiently, outperforming prior work in terms of scalability.\n\n### Weaknesses \n- [Significance] My biggest concern is the misalignment of motivation and experiments. Although the method is explicitly motivated as being tailored for single-cell dynamics, the experiments largely fail to demonstrate its merits in this domain. The synthetic Gaussian and low-dimensional EB experiments are insufficient to validate biological relevance or single-cell-specific challenges. For this, it is essential to analyse the modeled samples (i.e. gene) space. I would suggest to investigate VGFM in the following contexts: \n     - A comparison of the gene-wise empirical mean-variance trend with that of real data and an analysis of the modeled sparsity in gene space, cf. 5.1 in [1]\n     - The inclusion of the MEF dataset [2] as well as the analysis of the decoded space as done in 5.2 of [3]\n     - An analysis on the Pancreas dataset [4], checking that with VGFM one can identify all terminal states as well as investigating the dynamics of linage drivers. For the linage drivers, in particular, it would be interesting if the growth modelling enables more accurate modeling of the real data, cf. [3].\n- [Quality] I encourage the authors are encouraged to discuss key characteristics of scRNA-seq data, such as sparsity, overdispersion, and its negative binomial noise structure. Since VGFM can readily be combined with alternatives to PCA that better reflect the data’s nature, I encourage the authors to consider and elaborate on this connection (incorporating an NB noise model may also be relevant for the additional experiments suggested above).\n- [Clarity] Key dataset details (e.g., number of cells, timepoints, gene dimensions; see L233–234) are missing. Including this information would take little space but greatly improve the clarity of the experimental setup for the reader.\n- [Minor] The additional distribution loss term (L68–69) appears a bit ad-hoc and lacks justification (beyond empirical performance) compared to the VGFM part.\n\n&nbsp; \n\n_____\n\n- [1] Palma et al. \"Multi-Modal and Multi-Attribute Generation of Single Cells with CFGen\", ICLR 2025\n- [2] Schiebinger et al. \"Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming\", Cell, 176(4):928–943, 2019.\n- [3] Palma et al. \"Enforcing Latent Euclidean Geometry in Single-Cell VAEs for Manifold Interpolation\", ICML 2025.\n- [4] Bastidas-Ponce et al. \"Comprehensive single cell mRNA profiling reveals a detailed roadmap for pancreatic endocrinogenesis\" Development, 146(12):dev173849, 2019.\n\nQUESTIONS:\n- The scalability of VGFM is very interesting. Would training directly on gene-level counts (say 1k-2k highly variable genes) be feasible? Do the authors have an intuition how the sparsity of scRNA-seq would be handled by VGFM? \n\n- Could you elaborate on the motivation behind the distribution-fitting loss? I’m curious whether it was derived from a principled rationale or chosen empirically.\n\n- I would be interested to hear the authors’ perspective on the biological significance of explicitly modeling growth effects. Methodologically, this choice is well motivated and aligns well with the nature of the data. \n\n- Could the authors elaborate on how their model could be made conditional on covariates (e.g., cell type, batch)? Incorporating conditioning is important in single-cell modeling and would enable meaningful applications such as data augmentation and batch correction.\n\n&nbsp; \n_____\nI appreciate the authors’ effort in addressing an important research question and look forward to their clarifications and responses during the rebuttal phase. I am happy to raise my score if the authors adequately address my questions and concerns.\n\nLIMITATIONS:\nWhile the authors acknowledge some limitations of their approach, the discussion remains focused on technical aspects. It would strengthen the manuscript to also address broader challenges in single-cell modeling (noise structure, sparsity, overdispersion, data acquisition). A discussion of where VGFM is likely to succeed and where it may face limitations in the context of real single-cell data would provide valuable perspective (for example, the scalability claim is only demonstrated on synthetic Gaussian data rather than realistic gene-level scRNA-seq datasets).", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a new approach, called Velocity-Growth Flow Matching (VGFM), for learning single-cell population dynamics from snapshot data. Unlike existing methods that either rely on simulation-heavy training or ignore the mass imbalance due to cell proliferation and death, VGFM jointly models cell-state transitions (velocity) and mass variations (growth). The method leverages a theoretical foundation based on a \"two-period\" interpretation of semi-relaxed optimal transport, separating the processes of mass growth and cell-state transitions to derive an ideal dynamics formulation. Neural velocity and mass-variation functions are then trained to approximate these dynamics via flow matching, augmented with a Wasserstein-based distribution fitting loss. Extensive experiments on synthetic datasets (e.g., Gaussian mixtures and Dyngen) and real-world single-cell datasets demonstrate that VGFM reconstructs trajectories and growth dynamics, achieving superior performance compared to multiple strong baselines.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths.**\n\nThis paper has several substantial strengths. First, it addresses an important problem—learning single-cell dynamics from unpaired snapshot data—while explicitly handling the biologically relevant challenge of mass imbalance. The proposed theoretical framing via semi-relaxed optimal transport is both elegant and well-motivated. Experimentally, the authors conduct thorough evaluations using synthetic and real-world datasets, demonstrating notable performance improvements over existing methods, especially in high-dimensional settings (up to 1,000 dimensions), where competing approaches significantly struggle. Additionally, the empirical validation provides compelling visualizations of the learned trajectories and growth rates.\n\n**Weaknesses.**\n\n- As acknowledged by the authors (lines 37–41), incorporating the OT-based distribution fitting loss renders the training procedure simulation-based. Specifically, this requires both simulating the underlying ODE dynamics and backpropagating through these simulations, making the method computationally cumbersome. This design choice diverges from recent advances in generative modeling, which explicitly aim for simulation-free training strategies to improve scalability and efficiency.\n\n- Moreover, the authors do not explicitly describe how they compute the distribution fitting loss. Presumably, they rely on the Sinkhorn algorithm with Euclidean cost computed over mini-batches. If this is indeed the case, it should be clearly stated. This approach presents scalability concerns, as mini-batch OT approximations suffer substantially from the curse of dimensionality. A practical suggestion is to replace the Wasserstein-1 distance with a Sinkhorn divergence [Feydy et al., 2019], which possesses more favorable statistical properties. Nonetheless, even this improvement may not fully resolve scalability issues, particularly when dimensionality exceeds $d > 1,00$.\n\n- The authors do not discuss the critical issue of batch size selection, despite the fact that two of the three loss components (growth function matching and OT distribution fitting) depend explicitly on mini-batch OT computations. Statistically, performance should improve as batch size increases. The current choice of a batch size of 256 for most experiments seems relatively small. Exploring batch sizes of 1,024–2,048 is recommended, especially given that Sinkhorn-type algorithms (including their unbalanced variants) remain computationally feasible at these scales. Furthermore, the authors propose partitioning computations into subsets (Appendix B.6) for larger datasets, which could negatively impact the quality and stability of the learned velocity and growth functions.\n\n- Additionally, concerns arise regarding the stability and scalability of the proposed growth matching loss. While the theoretical justification for its introduction is appealing, the practical implementation—which involves regressing against the right marginal of empirical unbalanced OT couplings computed via mini-batches—is potentially unstable, particularly for smaller batch sizes. Specifically, estimating the growth function effectively involves approximating potentials from the dual unbalanced OT problem [Séjourné et al., 2024], a procedure known to suffer considerably from the curse of dimensionality. Moreover, this instability is exacerbated by the potential inconsistency in how mini-batches handle outliers; some points might have their mass reduced or discarded in one mini-batch, yet retained or increased in another.\n\n- Finally, the notation throughout the paper, particularly in Sections 3.3 and 3.4, is often cumbersome and difficult to follow. Simplifying notation in these sections would significantly improve readability and accessibility.\n\nQUESTIONS:\n- Could the authors provide a discussion regarding the influence of batch size on their method? As both the growth matching loss and the OT fitting loss rely explicitly on mini-batch optimal transport computations, this aspect is crucial. Statistically, we should expect that increasing the batch size would improve performance, as larger batches yield more accurate OT approximations and stable gradient estimates. Providing empirical evidence or a sensitivity analysis on how batch size affects model performance and stability would significantly strengthen the evaluation.\n\n- Could the authors further elaborate on their motivation for choosing a time-independent growth function? Currently, the manuscript briefly states (lines 177–178): \"For simplicity and ease of implementation, we choose a time-independent form...\". However, it would be beneficial to explicitly discuss why this simplification—assuming that mass variations depend solely on spatial variables—is biologically or computationally justified. Clarifying whether this choice could potentially limit model expressiveness or generalization would improve the manuscript’s transparency and reader understanding.\n\n- Could the authors clarify whether they rely on the Sinkhorn algorithm to compute the OT fitting loss (involving the Wasserstein-1 distance)? If so, do you choose a small entropic regularization parameter to closely approximate the true Wasserstein distance? Additionally, have you explored or considered using the Sinkhorn divergence (as introduced by  [Feydy et al., 2019]) instead of the standard entropic Wasserstein distance, given its favorable metric properties?\n\nLIMITATIONS:\nThe authors clearly acknowledge important limitations of their work (section 5, page 9).", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-12"}