# Meta Flow Matching: Integrating Vector Fields on the Wasserstein Manifold

arXiv: https://arxiv.org/abs/2408.14608 · HTML: https://mlsanity.com/paper/2408.14608/

## Key facts

- Cited by 43 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 38 of its 80 references that are in the map.
- First posted on arXiv 26 Aug 2024.

## 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 38)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Palette: Image-to-Image Diffusion Models](https://mlsanity.com/paper/2111.05826/) (2021)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Meta-Learning in Neural Networks: A Survey](https://mlsanity.com/paper/2004.05439/) (2020)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design](https://mlsanity.com/paper/2402.04997/) (2024)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Task2Vec: Task Embedding for Meta-Learning](https://mlsanity.com/paper/1902.03545/) (2019)
- [GenCast: Diffusion-based ensemble forecasting for medium-range weather](https://mlsanity.com/paper/2312.15796/) (2023)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Flow Matching in Latent Space](https://mlsanity.com/paper/2307.08698/) (2023)
- [Guided Flows for Generative Modeling and Decision Making](https://mlsanity.com/paper/2311.13443/) (2023)
- [Tutorial on amortized optimization](https://mlsanity.com/paper/2202.00665/) (2022)
- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023)
- [ClimODE: Climate and Weather Forecasting with Physics-informed Neural ODEs](https://mlsanity.com/paper/2404.10024/) (2024)
- [Manifold Interpolating Optimal-Transport Flows for Trajectory Inference](https://mlsanity.com/paper/2206.14928/) (2022)

## Built on by (most cited first, 30 of 43)

- [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)
- [Curly Flow Matching for Learning Non-gradient Field Dynamics](https://mlsanity.com/paper/2510.26645/) (2025)
- [Modeling Cell Dynamics and Interactions with Unbalanced Mean Field Schr"odinger Bridge](https://mlsanity.com/paper/2505.11197/) (2025)
- [Variational Regularized Unbalanced Optimal Transport: Single Network, Least Action](https://mlsanity.com/paper/2505.11823/) (2025)
- [Designing a Conditional Prior Distribution for Flow-Based Generative Models](https://mlsanity.com/paper/2502.09611/) (2025)
- [On the minimax optimality of Flow Matching through the connection to kernel density estimation](https://mlsanity.com/paper/2504.13336/) (2025)
- [Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme](https://mlsanity.com/paper/2506.01502/) (2025)
- [Better Source, Better Flow: Learning Condition-Dependent Source Distribution for Flow Matching](https://mlsanity.com/paper/2602.05951/) (2026)
- [Modeling Microenvironment Trajectories on Spatial Transcriptomics with NicheFlow](https://mlsanity.com/paper/2511.00977/) (2025)
- [Distribution estimation via Flow Matching with Lipschitz guarantees](https://mlsanity.com/paper/2509.02337/) (2025)
- [WFR-FM: Simulation-Free Dynamic Unbalanced Optimal Transport](https://mlsanity.com/paper/2601.06810/) (2026)
- [Oh SnapMMD! Forecasting Stochastic Dynamics Beyond the Schr"odinger Bridge's End](https://mlsanity.com/paper/2505.16082/) (2025)
- [Branched Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.09007/) (2025)
- [Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport](https://mlsanity.com/paper/2509.22138/) (2025)
- [A Call to Lagrangian Action: Learning Population Mechanics from Temporal Snapshots](https://mlsanity.com/paper/2605.08550/) (2026)
- [Generative Distribution Embeddings: Lifting autoencoders to the space of distributions for multiscale representation learning](https://mlsanity.com/paper/2505.18150/) (2025)
- [Modeling Stochastic Conditional Dynamics from Sparse Observations via Kernel-Stabilized Flow Matching](https://mlsanity.com/paper/2411.08314/) (2024)
- [MapPFN: Learning Causal Perturbation Maps in Context](https://mlsanity.com/paper/2601.21092/) (2026)
- [Generalized Wasserstein Flow Matching: Transport Plans, Everywhere, All at Once](https://mlsanity.com/paper/2605.08424/) (2026)
- [Optimal Transportation by Orthogonal Coupling Dynamics](https://mlsanity.com/paper/2410.08060/) (2024)
- [ContextFlow: Context-Aware Flow Matching For Trajectory Inference From Spatial Omics Data](https://mlsanity.com/paper/2510.02952/) (2025)
- [GRASP: Guided Residual Adapters with Sample-wise Partitioning](https://mlsanity.com/paper/2512.01675/) (2025)
- [Distribution-Conditioned Transport](https://mlsanity.com/paper/2603.04736/) (2026)
- [Beyond Continuity: Simulation-free Reconstruction of Discrete Branching Dynamics from Single-cell Snapshots](https://mlsanity.com/paper/2605.00545/) (2026)
- [Measure-to-measure Regression with Transformers](https://mlsanity.com/paper/2605.28075/) (2026)
- [Pairwise Optimal Transports for Training All-to-All Flow-Based Condition Transfer Model](https://mlsanity.com/paper/2504.03188/) (2025)
- [Entangled Schr"odinger Bridge Matching](https://mlsanity.com/paper/2511.07406/) (2025)
- [Shortest-Path Flow Matching with Mixture-Conditioned Bases for OOD Generalization to Unseen Conditions](https://mlsanity.com/paper/2601.11827/) (2026)
- [WFR-MFM: One-Step Inference for Dynamic Unbalanced Optimal Transport](https://mlsanity.com/paper/2601.20606/) (2026)

## Abstract (arXiv)

Numerous biological and physical processes can be modeled as systems of interacting entities evolving continuously over time, e.g. the dynamics of communicating cells or physical particles. Learning the dynamics of such systems is essential for predicting the temporal evolution of populations across novel samples and unseen environments. Flow-based models allow for learning these dynamics at the population level - they model the evolution of the entire distribution of samples. However, current flow-based models are limited to a single initial population and a set of predefined conditions which describe different dynamics. We argue that multiple processes in natural sciences have to be represented as vector fields on the Wasserstein manifold of probability densities. That is, the change of the population at any moment in time depends on the population itself due to the interactions between samples. In particular, this is crucial for personalized medicine where the development of diseases and their respective treatment response depend on the microenvironment of cells specific to each patient. We propose Meta Flow Matching (MFM), a practical approach to integrate along these vector fields on the Wasserstein manifold by amortizing the flow model over the initial populations. Namely, we embed the population of samples using a Graph Neural Network (GNN) and use these embeddings to train a Flow Matching model. This gives MFM the ability to generalize over the initial distributions, unlike previously proposed methods. We demonstrate the ability of MFM to improve the prediction of individual treatment responses on a large-scale multi-patient single-cell drug screen dataset.