Cellular-Dynamics-Modeling

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Cellular-Dynamics-Modeling

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(unbalanced) optimal transport, Schrödinger bridge problem, and its application in single-cell development dynamics modeling

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# Cellular-Dynamics-Modeling
(unbalanced) optimal transport,  Schrödinger bridge problem, and its application in single-cell development dynamics modeling
### Fundamental Limits [[Fundamental limits on dynamic inference from single-cell snapshots]](https://www.pnas.org/doi/10.1073/pnas.1714723115)
### Tutorial for Diffusion Models [[The Principles of Diffusion Models]](https://github.com/RobinLufdu/Cellular-Dynamics-Modeling/blob/main/Lai%20et%20al.%20-%202025%20-%20The%20Principles%20of%20Diffusion%20Models.pdf)
### Relevant Toolkits [[Single-Cell Analysis in Python]](https://github.com/scverse/scanpy)

## Week 1
Preliminaries on optimal transport, Schrödinger bridge problem (SBP) and its related dynamics formulations.
### Main Reference
- [x] [[OT&SBP]](https://github.com/RobinLufdu/Cellular-Dynamics-Modeling/blob/main/OT%26SBP/Modeling%20and%20control%20of%20collective%20dynamics%20From%20Schroedinger%20bridges%20to%20Optimal%20Mass%20Transport.pdf) Chapter 3, Chapter 4 4.1~4.3

### Related Reference
- [x] [[Optimal Transport]](https://www.damtp.cam.ac.uk/research/cia/files/teaching/Optimal_Transport_Notes.pdf)
- [x] [[Entropic Optimal Transport (SBP)]](https://www.math.columbia.edu/~mnutz/docs/EOT_lecture_notes.pdf)

## Week 2
Optimal control perspective of OT and SBP; SBP as entropy regularised OT.
### Main Reference
- [x] [[OT&SBP]](https://github.com/RobinLufdu/Cellular-Dynamics-Modeling/blob/main/OT%26SBP/Modeling%20and%20control%20of%20collective%20dynamics%20From%20Schroedinger%20bridges%20to%20Optimal%20Mass%20Transport.pdf) Chapter 5 5.1~5.4
- [x] [[Optimal Transport in Systems and Control]](https://www.annualreviews.org/content/journals/10.1146/annurev-control-070220-100858), alternative reference
### Related Reference
- [x] [[A stochastic control approach to reciprocal diffusion processes]](https://link.springer.com/article/10.1007/BF01442404)
- [x] [[About the analogy between optimal transport and minimal entropy]](https://arxiv.org/abs/1510.08230)
- [x] [[On the Relation Between Optimal Transport and Schrödinger Bridges: A Stochastic Control Viewpoint]](https://arxiv.org/abs/1412.4430)
- [x] [[An optimal control perspective on diffusion-based generative modeling]](https://arxiv.org/abs/2211.01364)

## Week 3
Discuss works for learning stochastic dynamics from samples (refer to as {trjactory inference}).
### Main Reference
- [x] [[TrajectoryNet]](https://arxiv.org/abs/2002.04461)
- [x] [[NLSB]](https://arxiv.org/abs/2204.04853)
- [x] [[PNAS25]](https://www.pnas.org/doi/epdf/10.1073/pnas.2420621122)

### Related Reference
- [x] [[Action Matching]](https://openreview.net/pdf?id=E3BW8pG64Y)

### Mathematical Theories [[Trajectory Inference]](https://arxiv.org/abs/2102.09204)

## Week 4
Learning stochastic dynmaics via score and flow matching (a generalization of the flow matching techinque)
- [x] [[Simulation-Free Schrödinger Bridges via Score and Flow Matching]](https://arxiv.org/abs/2307.03672)
- [x] [[SFM for non-equilibrium diffusions]](https://arxiv.org/abs/2505.16644)
- [x] [[Curly flow matching for non-gradient dynamics]](https://openreview.net/forum?id=7cqKVDgFZQ)

## Week 5
Unbalanced dynamic optimal transport.
### Main Reference
- [x] [[Unbalanced Optimal Transport, from Theory to Numerics]](https://arxiv.org/abs/2211.08775)

### Related Reference
- [x] [[Unbalanced Optimal Transport: Dynamic and Kantorovich Formulations]](https://arxiv.org/abs/1508.05216)
- [x] [[The most likely evolution of diffusing and vanishing particles]](https://arxiv.org/abs/2108.02879)
- [x] [[Entropic regularisation of unbalanced optimal transportation problems]](https://arxiv.org/abs/2305.02410)
- [x] [[Regularized unbalanced optimal transport as entropy minimization with respect to branching Brownian motion]](https://arxiv.org/abs/2111.01666)

## Week 6, 7
Application of unbalanced optimal transport in modeling cellular population dynamics (with explict growth term considered). (contribution mainly form [[Peijie Zhou's group]](https://cliffzhou92.github.io/))

### Reference
- [x] [[TIGON (Nature Machine Intelligence 2023)]](https://www.nature.com/articles/s42256-023-00763-w) [[Project page]](https://github.com/yutongo/TIGON)
- [x] [[DeepRUOT (ICLR25 Oral)]](https://arxiv.org/abs/2410.00844) [[Project Page]](https://github.com/zhenyiizhang/DeepRUOT) [[ICLR2025 oral]](https://iclr.cc/media/iclr-2025/Slides/31800_i2NXVyQ.pdf)
- [x] [[VGFM (Neurips25)]](https://arxiv.org/abs/2505.13413), based on flow matching
- [x] [[Variational RUOT (Neurips25)]](https://arxiv.org/abs/2505.11823) [[Openreview]](https://openreview.net/forum?id=Iguyg0LULD) 
- [x] [[Single Cell Spatiotemporal Dynamical Generative Modeling (review work)]](https://www.mdpi.com/1099-4300/27/5/453)

### Related Codes
[[DeepRUOT]](https://github.com/zhenyiizhang/DeepRUOT); [[DeepRUOTv2]](https://github.com/zhenyiizhang/DeepRUOTv2); [[A Toolkit for Single Cell Spatiotemporal Dynamical Generative Modeling]](https://github.com/zhenyiizhang/CytoBridge), also see the references within.

### Related Works
- [x] [[Unbalanced Diffusion Schrödinger Bridge]](https://arxiv.org/abs/2306.09099)
- [x] [[Modeling Cell Dynamics and Interactions with Unbalanced Mean Field Schrödinger Bridge]](https://arxiv.org/pdf/2505.11197)
#### Learning stochastic dynmaics via score and flow matching (a generalization of the flow matching techinque)
* [[Simulation-Free Schrödinger Bridges via Score and Flow Matching]](https://arxiv.org/abs/2307.03672)
* [[SFM for non-equilibrium diffusions]](https://arxiv.org/abs/2505.16644)

## Week 8
RNA velocity and its generalizations.
- [x] [[RNA velocity of single cells]](https://www.nature.com/articles/s41586-018-0414-6)
- [x] [[Generalizing RNA velocity to transient cell states through dynamical modeling]](https://www.nature.com/articles/s41587-020-0591-3)
- [x] [[Mapping transcriptomic vector fields of single cells]](https://www.cell.com/cell/fulltext/S0092-8674(21)01577-4) (optional)

### Related Works
- [x] [[Mathematical Foundation]](https://www.biorxiv.org/content/10.1101/2020.09.19.304584v1)
- [x] [[Algorithmic Aspects]](https://arxiv.org/abs/2306.05707)

### Tookits [[scVelo]](https://scvelo.readthedocs.io/en/stable/index.html)

## Week 9
Methods for modeling cell fate decision probabilities.
- [x] [[CellRank for directed single-cell fate mapping]](https://www.nature.com/articles/s41592-021-01346-6)
- [x] [[CellRank 2: unified fate mapping in multiview single-cell data]](https://www.nature.com/articles/s41592-024-02303-9)

# New semester

## Week 1
Deepvelo models for learning single cell dynamics in the latent space.
- [x] [[DeepVelo: Single-cell transcriptomic deep velocity field learning with neural ordinary differential equations]](https://www.science.org/doi/10.1126/sciadv.abq3745)
- [x] [[DeepVelo: deep learning extends RNA velocity to multi-lineage systems with cell-specific kinetics]](https://link.springer.com/article/10.1186/s13059-023-03148-9)  [[Project page]](https://github.com/bowang-lab/DeepVelo)
### Related Works
- [x] [[VeloViz: RNA velocity-informed embeddings for visualizing cellular trajectories]](https://academic.oup.com/bioinformatics/article/38/2/391/6368063)

## Week 2
Learning single cell dynamics in the latent space.
- [x] [[MIOFlow]](https://arxiv.org/pdf/2206.14928)
- [x] [[CellStream: Dynamical Optimal Transport Informed Embeddings for Reconstructing Cellular Trajectories from Snapshots Data]](https://arxiv.org/abs/2511.13786)  [[Project page]](https://github.com/PQ-Zhang/CellStream)

## Week 3
Learning single cell responses under perturbation.

**Review Work** [[Interpretation, extrapolation and perturbation of single cells]](https://www.nature.com/articles/s41576-025-00920-4)

- [x] [[Generative modeling of single-cell time series with PRESCIENT enables prediction of cell trajectories with interventions]](https://www.nature.com/articles/s41467-021-23518-w)
- [x] [[Learning single-cell perturbation responses using neural optimal transport]](https://www.nature.com/articles/s41592-023-01969-x)

## Week 4
Other velocity models.
- [x] [[TFvelo: gene regulation inspired RNA velocity estimation]](https://www.nature.com/articles/s41467-024-45661-w?fromPaywallRec=false)
- [x] [[PhyloVelo enhances transcriptomic velocity field mapping using monotonically expressed genes]](https://www.nature.com/articles/s41587-023-01887-5)
- [x] [[Multivariate stochastic modeling for transcriptional dynamics with cell-specific latent time using SDEvelo]](https://www.nature.com/articles/s41467-024-55146-5.pdf)

### Related Reference
- [x] [[UniTVelo: temporally unified RNA velocity reinforces single-cell trajectory inference]](https://www.nature.com/articles/s41467-022-34188-7)
- [x] [[Dynamo: Mapping Transcriptomic Vector Fields of Single Cells]](https://www.sciencedirect.com/science/article/pii/S0092867421015774) 
- [x] [[A relay velocity model infers cell-dependent RNA velocity]](https://www.nature.com/articles/s41587-023-01728-5.pdf)
- [x] [[Cell2fate infers RNA velocity modules to improve cell fate prediction]](https://www.nature.com/articles/s41592-025-02608-3)
- [x] [[Deep generative modeling of transcriptional dynamics for RNA velocity analysis in single cells]](https://www.nature.com/articles/s41592-023-01994-w)
- [x] [[Dissection and integration of bursty transcriptional dynamics for complex systems]](https://www.pnas.org/doi/10.1073/pnas.2306901121)

## Week 5, 6
Learning single cell responses under perturbation, continuation.

- [x] [[Doloris]](https://openreview.net/forum?id=rvpDHfoTd2)
- [x] [[MixFlow]](https://arxiv.org/html/2601.11827v2)
- [x] [[scDFM: Distributional Flow Matching Model for Robust Single-Cell Perturbation Prediction]](https://openreview.net/forum?id=QSGanMEcUV)
- [x] [[Squidiff: predicting cellular development and responses to perturbations using a  diffusion model]](https://www.nature.com/articles/s41592-025-02877-y)
- [x] [[Latent Causal Diffusions for Single-Cell Perturbation Modeling]](https://arxiv.org/pdf/2601.15341)

## Week 7
Non-Markovian gene expression model.
- [x] [[Neural network aided approximation and parameter inference of non-Markovian models of gene expression]](https://www.nature.com/articles/s41467-021-22919-1)

## Week 8
Leverage PINN to model the continuity equation of cellular population dynamics.
- [x] [[Pseudodynamics+]](https://www.biorxiv.org/content/10.64898/2025.11.30.691399v1) [[Project Page]](https://github.com/Gottgens-lab/pseudodynamics_plus)

## Week 9
Simulation-free learning of single cell dynamics with growth via flow matching (WFR-FM).

- [x] [[WFR-FM: Simulation-Free Dynamic Unbalanced Optimal Transport]](https://openreview.net/forum?id=1nqu7bK1mm)
- [x] [[WFR-MFM: One-Step Inference for Dynamic Unbalanced Optimal Transport]](https://arxiv.org/pdf/2601.20606)

## Week 10
Modeling cell fate differentiation via branching flow models.

- [x] [[Branching Flows: Discrete, Continuous, and Manifold Flow Matching with Splits and Deletions]](https://arxiv.org/pdf/2511.09465)
- [x] [[Branched Schrödinger Bridge Matching]](https://arxiv.org/pdf/2506.09007)

## Week 11
- [x] [[PHLOWER leverages single-cell multimodal data to infer complex, multi-branching cell differentiation trajectories]](https://www.nature.com/articles/s41592-025-02870-5) [[Project Page]](https://github.com/CostaLab/phlower)

## Week 12
- [x] [[Temporal tissue dynamics from a spatial snapshot]](https://www.nature.com/articles/s41586-025-09876-1)

## Other Works
- [x] [[Inferring population dynamics from single-cell RNA-sequencing time series data]](www.nature.com/articles/s41587-019-0088-0)
- [x] [[Dissecting transition cells from single-cell transcriptome data through multiscale stochastic dynamics]](https://www.nature.com/articles/s41467-021-25548-w)
- [x] [[Learning Geometric Models for Developmental Dynamics]](https://journals.aps.org/prx/abstract/10.1103/8vpj-bj7d)
- [x] [[Spatial transition tensor of single cells]](https://www.nature.com/articles/s41592-024-02266-x)
- [x] [[Learning stochastic processes with intrinsic noise from cross-sectional biological data]](https://www.pnas.org/doi/10.1073/pnas.2420621122)
- [x] [[Cellflow]](https://www.biorxiv.org/content/10.1101/2025.04.11.648220v1) [[Project Page]](https://github.com/theislab/CellFlow)

## Prospect (Unfinished)
Introduce memory effects.

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