# Modeling Stochastic Conditional Dynamics from Sparse Observations via Kernel-Stabilized Flow Matching

arXiv: https://arxiv.org/abs/2411.08314 · HTML: https://mlsanity.com/paper/2411.08314/

## Key facts

- Cited by 3 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 34 of its 77 references that are in the map.
- First posted on arXiv 13 Nov 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 34)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [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)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (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)
- [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)
- [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)
- [Flow Matching in Latent Space](https://mlsanity.com/paper/2307.08698/) (2023)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Guided Flows for Generative Modeling and Decision Making](https://mlsanity.com/paper/2311.13443/) (2023)
- [Learning with minibatch Wasserstein : asymptotic and gradient properties](https://mlsanity.com/paper/1910.04091/) (2019)
- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023)
- [Minibatch optimal transport distances; analysis and applications](https://mlsanity.com/paper/2101.01792/) (2021)
- [The Schr"odinger Bridge between Gaussian Measures has a Closed Form](https://mlsanity.com/paper/2202.05722/) (2022)
- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [Neural ODE control for classification, approximation and transport](https://mlsanity.com/paper/2104.05278/) (2021)
- [Deep Generalized Schr"odinger Bridge](https://mlsanity.com/paper/2209.09893/) (2022)
- [Meta Flow Matching: Integrating Vector Fields on the Wasserstein Manifold](https://mlsanity.com/paper/2408.14608/) (2024)
- [Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matching](https://mlsanity.com/paper/2403.18705/) (2024)
- [Wasserstein Flow Matching: Generative modeling over families of distributions](https://mlsanity.com/paper/2411.00698/) (2024)
- [Dynamic Conditional Optimal Transport through Simulation-Free Flows](https://mlsanity.com/paper/2404.04240/) (2024)
- [Extended Flow Matching: a Method of Conditional Generation with Generalized Continuity Equation](https://mlsanity.com/paper/2402.18839/) (2024)

## Built on by (most cited first, 3 of 3)

- [On Fitting Flow Models with Large Sinkhorn Couplings](https://mlsanity.com/paper/2506.05526/) (2025)
- [Coupled Flow Matching](https://mlsanity.com/paper/2510.23015/) (2025)
- [FlowDet: Unifying Object Detection and Generative Transport Flows](https://mlsanity.com/paper/2512.16771/) (2025)

## Abstract (arXiv)

Learning to transform conditional probability densities over time is a fundamental challenge spanning probabilistic modeling and the natural sciences. This task is paramount when forecasting the evolution of stochastic nonlinear dynamical systems in biological and physical domains. While flow-based models can predict the temporal evolution of probability distributions, existing approaches often assume discrete conditioning with samples that are paired across time, limiting their scientific applicability where frequently only sparse data with unpaired continuous conditioning is available. We propose Conditional Variable Flow Matching (CVFM), a framework for learning flows transforming conditional distributions with amortization across the continuous space of conditional densities. CVFM addresses the high-variance instability of prior methods by jointly sampling flows over state and conditioning variables, utilizing a conditioning mismatch kernel alongside a conditional Wasserstein distance to reweight the conditional optimal transport objective. Collectively, these advances allow for learning dynamics from sparse unpaired measurements of state-condition across time. We evaluate CVFM on conditional mapping benchmarks and a case study modeling the temporal evolution of materials internal structure during manufacturing processes, observing improved performance and convergence characteristics over existing conditional variants. Code is available at https://github.com/agenerale/conditional-variable-flow-matching.