# Multi-Marginal Flow Matching with Adversarially Learnt Interpolants

arXiv: https://arxiv.org/abs/2510.01159 · HTML: https://mlsanity.com/paper/2510.01159/

## Key facts

- Cited by 4 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 13 of its 37 references that are in the map.
- First posted on arXiv 1 Oct 2025.

## 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, 13 of 13)

- [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)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [The relativistic discriminator: a key element missing from standard GAN](https://mlsanity.com/paper/1807.00734/) (2018)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [The GAN is dead; long live the GAN! A Modern GAN Baseline](https://mlsanity.com/paper/2501.05441/) (2025)
- [A Computational Framework for Solving Wasserstein Lagrangian Flows](https://mlsanity.com/paper/2310.10649/) (2023)
- [Towards a Better Global Loss Landscape of GANs](https://mlsanity.com/paper/2011.04926/) (2020)
- [Multi-Marginal Stochastic Flow Matching for High-Dimensional Snapshot Data at Irregular Time Points](https://mlsanity.com/paper/2508.04351/) (2025)
- [Robust Registration of Calcium Images by Learned Contrast Synthesis](https://mlsanity.com/paper/1511.01154/) (2015)

## Built on by (most cited first, 4 of 4)

- [ABC: Any-Subset Autoregression via Non-Markovian Diffusion Bridges in Continuous Time and Space](https://mlsanity.com/paper/2604.27443/) (2026)
- [PACE: Geometry-Aware Bridge Transport for Single-Cell Trajectory Inference](https://mlsanity.com/paper/2605.18587/) (2026)
- [Co-Evolving Paths and Flows via Path-Flow Alignment](https://mlsanity.com/paper/2610.08717/) (2026)
- [Neural Wasserstein Splines: Representation-Invariant Second-Order Priors for Trajectory Inference](https://mlsanity.com/paper/2610.19878/) (2026)

## Abstract (arXiv)

Learning the dynamics of a process given sampled observations at several time points is an important but difficult task in many scientific applications. When no ground-truth trajectories are available, but one has only snapshots of data taken at discrete time steps, the problem of modelling the dynamics, and thus inferring the underlying trajectories, can be solved by multi-marginal generalisations of flow matching algorithms. This paper proposes a novel flow matching method that overcomes the limitations of existing multi-marginal trajectory inference algorithms. Our proposed method, ALI-CFM, uses a GAN-inspired adversarial loss to fit neurally parametrised interpolant curves between source and target points such that the marginal distributions at intermediate time points are close to the observed distributions. The resulting interpolants are smooth trajectories that, as we show, are unique under mild assumptions. These interpolants are subsequently marginalised by a flow matching algorithm, yielding a trained vector field for the underlying dynamics. We showcase the versatility and scalability of our method by outperforming the existing baselines on spatial transcriptomics and cell tracking datasets, while performing on par with them on single-cell trajectory prediction. Code: https://github.com/mmacosha/adversarially-learned-interpolants.