# GENOT: Entropic (Gromov) Wasserstein Flow Matching with Applications to Single-Cell Genomics

arXiv: https://arxiv.org/abs/2310.09254 · HTML: https://mlsanity.com/paper/2310.09254/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 37 of its 85 references that are in the map.
- First posted on arXiv 13 Oct 2023.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 37)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Denoising Diffusion Bridge Models](https://mlsanity.com/paper/2309.16948/) (2023)
- [Optimal Transport for structured data with application on graphs](https://mlsanity.com/paper/1805.09114/) (2018)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Low-Rank Sinkhorn Factorization](https://mlsanity.com/paper/2103.04737/) (2021)
- [Minibatch optimal transport distances; analysis and applications](https://mlsanity.com/paper/2101.01792/) (2021)
- [Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs](https://mlsanity.com/paper/2106.01128/) (2021)
- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [Score-based Generative Neural Networks for Large-Scale Optimal Transport](https://mlsanity.com/paper/2110.03237/) (2021)
- [The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation](https://mlsanity.com/paper/2009.04266/) (2020)
- [Entropic Neural Optimal Transport via Diffusion Processes](https://mlsanity.com/paper/2211.01156/) (2022)
- [Unbalanced Optimal Transport, from Theory to Numerics](https://mlsanity.com/paper/2211.08775/) (2022)
- [Predicting Cellular Responses to Novel Drug Perturbations at a Single-Cell Resolution](https://mlsanity.com/paper/2204.13545/) (2022)
- [Scalable Unbalanced Optimal Transport using Generative Adversarial Networks](https://mlsanity.com/paper/1810.11447/) (2018)
- [The Monge Gap: A Regularizer to Learn All Transport Maps](https://mlsanity.com/paper/2302.04953/) (2023)
- [Energy-guided Entropic Neural Optimal Transport](https://mlsanity.com/paper/2304.06094/) (2023)

## Built on by (most cited first, 28 of 28)

- [Wasserstein Flow Matching: Generative modeling over families of distributions](https://mlsanity.com/paper/2411.00698/) (2024)
- [Joint Velocity-Growth Flow Matching for Single-Cell Dynamics Modeling](https://mlsanity.com/paper/2505.13413/) (2025)
- [Flow Matching Meets Biology and Life Science: A Survey](https://mlsanity.com/paper/2507.17731/) (2025)
- [Mirror and Preconditioned Gradient Descent in Wasserstein Space](https://mlsanity.com/paper/2406.08938/) (2024)
- [Multi-Modal and Multi-Attribute Generation of Single Cells with CFGen](https://mlsanity.com/paper/2407.11734/) (2024)
- [Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme](https://mlsanity.com/paper/2506.01502/) (2025)
- [Modeling Microenvironment Trajectories on Spatial Transcriptomics with NicheFlow](https://mlsanity.com/paper/2511.00977/) (2025)
- [WFR-FM: Simulation-Free Dynamic Unbalanced Optimal Transport](https://mlsanity.com/paper/2601.06810/) (2026)
- [Branched Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.09007/) (2025)
- [Rethinking Model Selection in VLM Through the Lens of Gromov-Wasserstein Distance](https://mlsanity.com/paper/2605.01325/) (2026)
- [Towards generalizable single-cell perturbation modeling via the Conditional Monge Gap](https://mlsanity.com/paper/2504.08328/) (2025)
- [MapPFN: Learning Causal Perturbation Maps in Context](https://mlsanity.com/paper/2601.21092/) (2026)
- [A Statistical Learning Perspective on Semi-dual Adversarial Neural Optimal Transport Solvers](https://mlsanity.com/paper/2502.01310/) (2025)
- [ContextFlow: Context-Aware Flow Matching For Trajectory Inference From Spatial Omics Data](https://mlsanity.com/paper/2510.02952/) (2025)
- [CT-OT Flow: Estimating Continuous-Time Dynamics from Discrete Temporal Snapshots](https://mlsanity.com/paper/2505.17354/) (2025)
- [Neural Correlates of Language Models Are Specific to Human Language](https://mlsanity.com/paper/2510.03156/) (2025)
- [Transfer Learning on Edge Connecting Probability Estimation under Graphon Model](https://mlsanity.com/paper/2510.05527/) (2025)
- [WFR-MFM: One-Step Inference for Dynamic Unbalanced Optimal Transport](https://mlsanity.com/paper/2601.20606/) (2026)
- [CellBRIDGE: Learning Cellular Trajectories via Interaction-Aware Alignment](https://mlsanity.com/paper/2605.30635/) (2026)
- [TracingFlow: A Simulation-Free Trajectory Inference Framework Based on Second-Order Dynamics](https://mlsanity.com/paper/2608.21070/) (2026)
- [Federated Flow Matching](https://mlsanity.com/paper/2509.21250/) (2025)
- [Multiscale Supervised Unbalanced Optimal Transport Flow Matching](https://mlsanity.com/paper/2605.16529/) (2026)
- [Multimodal Alignment Through Joint Kernel Entropic Gromov--Wasserstein Optimal Transport](https://mlsanity.com/paper/2608.04234/) (2026)
- [Simulation-free Unbalanced Dynamic Optimal Transport with General Growth Penalty](https://mlsanity.com/paper/2609.04710/) (2026)
- [Dynamic Generalized Gromov-Wasserstein Optimal Transport](https://mlsanity.com/paper/2609.20008/) (2026)
- [Averaged Mirror Descent and Dual Gradient Methods: Convergent Algorithms for Entropic Gromov-Wasserstein Problems](https://mlsanity.com/paper/2609.31848/) (2026)
- [DRIFT: Disentangled Responsive-Invariant Flow Transport for Single-Cell Perturbation Prediction](https://mlsanity.com/paper/2609.35106/) (2026)
- [TempoBridge: Source-Conditioned Flow Matching with Optimal Transport Couplings for Single-Cell Population Transitions](https://mlsanity.com/paper/2610.04945/) (2026)

## Abstract (arXiv)

Single-cell genomics has significantly advanced our understanding of cellular behavior, catalyzing innovations in treatments and precision medicine. However, single-cell sequencing technologies are inherently destructive and can only measure a limited array of data modalities simultaneously. This limitation underscores the need for new methods capable of realigning cells. Optimal transport (OT) has emerged as a potent solution, but traditional discrete solvers are hampered by scalability, privacy, and out-of-sample estimation issues. These challenges have spurred the development of neural network-based solvers, known as neural OT solvers, that parameterize OT maps. Yet, these models often lack the flexibility needed for broader life science applications. To address these deficiencies, our approach learns stochastic maps (i.e. transport plans), allows for any cost function, relaxes mass conservation constraints and integrates quadratic solvers to tackle the complex challenges posed by the (Fused) Gromov-Wasserstein problem. Utilizing flow matching as a backbone, our method offers a flexible and effective framework. We demonstrate its versatility and robustness through applications in cell development studies, cellular drug response modeling, and cross-modality cell translation, illustrating significant potential for enhancing therapeutic strategies.