Peijie Zhou

Yuhao Sun, Zhenyi Zhang, Zihan Wang et al.

Recovering the dynamics from a few snapshots of a high-dimensional system is a challenging task in statistical physics and machine learning, with important applications in computational biology. Many algorithms have been developed to tackle this problem, based on frameworks such as optimal transport and the Schrödinger bridge. A notable recent framework is Regularized Unbalanced Optimal Transport (RUOT), which integrates both stochastic dynamics and unnormalized distributions. However, since many existing methods do not explicitly enforce optimality conditions, their solutions often struggle to satisfy the principle of least action and meet challenges to converge in a stable and reliable way. To address these issues, we propose Variational RUOT (Var-RUOT), a new framework to solve the RUOT problem. By incorporating the optimal necessary conditions for the RUOT problem into both the parameterization of the search space and the loss function design, Var-RUOT only needs to learn a scalar field to solve the RUOT problem and can search for solutions with lower action. We also examined the challenge of selecting a growth penalty function in the widely used Wasserstein-Fisher-Rao metric and proposed a solution that better aligns with biological priors in Var-RUOT. We validated the effectiveness of Var-RUOT on both simulated data and real single-cell datasets. Compared with existing algorithms, Var-RUOT can find solutions with lower action while exhibiting faster convergence and improved training stability. Our code is available at https://github.com/ZerooVector/VarRUOT.

Dongyi Wang, Yuanwei Jiang, Zhenyi Zhang et al.

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.

Zhenyi Zhang, Yuhao Sun, Qiangwei Peng et al.

Understanding the dynamic nature of biological systems is fundamental to deciphering cellular behavior, developmental processes, and disease progression. Single-cell RNA sequencing (scRNA-seq) has provided static snapshots of gene expression, offering valuable insights into cellular states at a single time point. Recent advancements in temporally resolved scRNA-seq, spatial transcriptomics (ST), and time-series spatial transcriptomics (temporal-ST) have further revolutionized our ability to study the spatiotemporal dynamics of individual cells. These technologies, when combined with computational frameworks such as Markov chains, stochastic differential equations (SDEs), and generative models like optimal transport and Schrödinger bridges, enable the reconstruction of dynamic cellular trajectories and cell fate decisions. This review discusses how these dynamical system approaches offer new opportunities to model and infer cellular dynamics from a systematic perspective.

Yue Ling, Peiqi Zhang, Zhenyi Zhang et al.

Single-cell RNA sequencing (scRNA-seq), especially temporally resolved datasets, enables genome-wide profiling of gene expression dynamics at single-cell resolution across discrete time points. However, current technologies provide only sparse, static snapshots of cell states and are inherently influenced by technical noise, complicating the inference and representation of continuous transcriptional dynamics. Although embedding methods can reduce dimensionality and mitigate technical noise, the majority of existing approaches typically treat trajectory inference separately from embedding construction, often neglecting temporal structure. To address this challenge, here we introduce CellStream, a novel deep learning framework that jointly learns embedding and cellular dynamics from single-cell snapshot data by integrating an autoencoder with unbalanced dynamical optimal transport. Compared to existing methods, CellStream generates dynamics-informed embeddings that robustly capture temporal developmental processes while maintaining high consistency with the underlying data manifold. We demonstrate CellStream's effectiveness on both simulated datasets and real scRNA-seq data, including spatial transcriptomics. Our experiments indicate significant quantitative improvements over state-of-the-art methods in representing cellular trajectories with enhanced temporal coherence and reduced noise sensitivity. Overall, CellStream provides a new tool for learning and representing continuous streams from the noisy, static snapshots of single-cell gene expression.