# Longitudinal Flow Matching for Trajectory Modeling

arXiv: https://arxiv.org/abs/2510.03569 · HTML: https://mlsanity.com/paper/2510.03569/

## Key facts

- Cited by 7 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 76 references that are in the map.
- First posted on arXiv 3 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, 30 of 38)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Align your Latents: High-Resolution Video Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2304.08818/) (2023)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [A large annotated medical image dataset for the development and evaluation of segmentation algorithms](https://mlsanity.com/paper/1902.09063/) (2019)
- [Metrics reloaded: Recommendations for image analysis validation](https://mlsanity.com/paper/2206.01653/) (2022)
- [Fast Symmetric Diffeomorphic Image Registration with Convolutional Neural Networks](https://mlsanity.com/paper/2003.09514/) (2020)
- [Unified Visual Transformer Compression](https://mlsanity.com/paper/2203.08243/) (2022)
- [Efficient and Accurate Gradients for Neural SDEs](https://mlsanity.com/paper/2105.13493/) (2021)
- [A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models](https://mlsanity.com/paper/2408.02320/) (2024)
- [Efficient Video Prediction via Sparsely Conditioned Flow Matching](https://mlsanity.com/paper/2211.14575/) (2022)
- [A Multimodal Transformer: Fusing Clinical Notes with Structured EHR Data for Interpretable In-Hospital Mortality Prediction](https://mlsanity.com/paper/2208.10240/) (2022)
- [Multi-marginal optimal transport: theory and applications](https://mlsanity.com/paper/1406.0026/) (2014)
- [Fourier Neural Operators for Arbitrary Resolution Climate Data Downscaling](https://mlsanity.com/paper/2305.14452/) (2023)
- [Black-box Variational Inference for Stochastic Differential Equations](https://mlsanity.com/paper/1802.03335/) (2018)
- [A Survey of Multimodal Information Fusion for Smart Healthcare: Mapping the Journey from Data to Wisdom](https://mlsanity.com/paper/2306.11963/) (2023)
- [CATS: Enhancing Multivariate Time Series Forecasting by Constructing Auxiliary Time Series as Exogenous Variables](https://mlsanity.com/paper/2403.01673/) (2024)
- [Generative AI for Synthetic Data Across Multiple Medical Modalities: A Systematic Review of Recent Developments and Challenges](https://mlsanity.com/paper/2407.00116/) (2024)
- [Trajectory Flow Matching with Applications to Clinical Time Series Modeling](https://mlsanity.com/paper/2410.21154/) (2024)
- [MedM2G: Unifying Medical Multi-Modal Generation via Cross-Guided Diffusion with Visual Invariant](https://mlsanity.com/paper/2403.04290/) (2024)
- [Learning Spatio-Temporal Model of Disease Progression with NeuralODEs from Longitudinal Volumetric Data](https://mlsanity.com/paper/2211.04234/) (2022)
- [ImageFlowNet: Forecasting Multiscale Image-Level Trajectories of Disease Progression with Irregularly-Sampled Longitudinal Medical Images](https://mlsanity.com/paper/2406.14794/) (2024)
- [Generative modeling for time series via Schr"odinger bridge](https://mlsanity.com/paper/2304.05093/) (2023)
- [Self-Supervised Video Representation Learning via Latent Time Navigation](https://mlsanity.com/paper/2305.06437/) (2023)

## Built on by (most cited first, 7 of 7)

- [Is Flow Matching Just Trajectory Replay for Sequential Data?](https://mlsanity.com/paper/2602.08318/) (2026)
- [ABC: Any-Subset Autoregression via Non-Markovian Diffusion Bridges in Continuous Time and Space](https://mlsanity.com/paper/2604.27443/) (2026)
- [Recursive Flow Matching](https://mlsanity.com/paper/2605.26535/) (2026)
- [MUMINS: Metadata-conditioned Uncertainty-aware Medical Image Next-state Synthesis](https://mlsanity.com/paper/2609.17169/) (2026)
- [FLUX: Geometry-Aware Longitudinal Flow Matching with Mixture of Experts](https://mlsanity.com/paper/2605.08648/) (2026)
- [Physics-Grounded Disentangled Flow Modeling for Brain Disease Progression Trajectory](https://mlsanity.com/paper/2606.28630/) (2026)
- [Scaling Online Optimal Transport Minibatches for Flow Matching](https://mlsanity.com/paper/2610.18838/) (2026)

## Abstract (arXiv)

Generative models for sequential data often struggle with sparsely sampled and high-dimensional trajectories, typically reducing the learning of dynamics to pairwise transitions. We propose Interpolative Multi-Marginal Flow Matching (IMMFM), a framework that learns continuous stochastic dynamics jointly consistent with multiple observed time points. IMMFM employs a piecewise-quadratic interpolation path as a smooth target for flow matching and jointly optimizes drift and a data-driven diffusion coefficient, supported by a theoretical condition for stable learning. This design captures intrinsic stochasticity, handles irregular sparse sampling, and yields subject-specific trajectories. Experiments on synthetic benchmarks and real-world longitudinal neuroimaging datasets show that IMMFM outperforms existing methods in both forecasting accuracy and further downstream tasks.