# Variational Grey-Box Dynamics Matching

arXiv: https://arxiv.org/abs/2602.17477 · HTML: https://mlsanity.com/paper/2602.17477/

## Key facts

- Cited by 2 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 14 of its 28 references that are in the map.
- First posted on arXiv 19 Feb 2026.

## 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, 14 of 14)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [WeatherBench: A benchmark dataset for data-driven weather forecasting](https://mlsanity.com/paper/2002.00469/) (2020)
- [Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting](https://mlsanity.com/paper/2010.04456/) (2020)
- [ClimODE: Climate and Weather Forecasting with Physics-informed Neural ODEs](https://mlsanity.com/paper/2404.10024/) (2024)
- [Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling](https://mlsanity.com/paper/2102.13156/) (2021)
- [Integrating Expert ODEs into Neural ODEs: Pharmacology and Disease Progression](https://mlsanity.com/paper/2106.02875/) (2021)
- [Variational Rectified Flow Matching](https://mlsanity.com/paper/2502.09616/) (2025)
- [Neural Dynamical Systems: Balancing Structure and Flexibility in Physical Prediction](https://mlsanity.com/paper/2006.12682/) (2020)
- [Robust Hybrid Learning With Expert Augmentation](https://mlsanity.com/paper/2202.03881/) (2022)
- [Deep Grey-Box Modeling With Adaptive Data-Driven Models Toward Trustworthy Estimation of Theory-Driven Models](https://mlsanity.com/paper/2210.13103/) (2022)
- [Hybrid Generative Modeling for Incomplete Physics: Deep Grey-Box Meets Optimal Transport](https://mlsanity.com/paper/2506.22204/) (2025)

## Built on by (most cited first, 2 of 2)

- [Bayesian Nonparametric Mixed-Effect ODEs with Gaussian Processes](https://mlsanity.com/paper/2605.13088/) (2026)
- [Climate Physics Dynamic Matching](https://mlsanity.com/paper/2608.26907/) (2026)

## Abstract (arXiv)

Deep generative models such as flow matching and diffusion models have shown great potential in learning complex distributions and dynamical systems, but often act as black-boxes, neglecting underlying physics. In contrast, physics-based simulation models described by ODEs/PDEs remain interpretable, but may have missing or unknown terms, unable to fully describe real-world observations. We bridge this gap with a novel grey-box method that integrates incomplete physics models directly into generative models. Our approach learns dynamics from observational trajectories alone, without ground-truth physics parameters, in a simulation-free manner that avoids scalability and stability issues of Neural ODEs. The core of our method lies in modelling a structured variational distribution within the flow matching framework, by using two latent encodings: one to model the missing stochasticity and multi-modal velocity, and a second to encode physics parameters as a latent variable with a physics-informed prior. Furthermore, we present an adaptation of the framework to handle second-order dynamics. Our experiments on representative ODE/PDE problems and real-world weather forecasting demonstrate that our method performs on par with or superior to fully data-driven approaches and previous grey-box baselines, while preserving the interpretability of the physics model. Our code is available at https://github.com/DMML-Geneva/VGB-DM.