Variational Grey-Box Dynamics Matching

TL;DR

This paper introduces a grey-box generative modeling approach that integrates incomplete physics models with deep learning to learn dynamics from observational data without ground-truth physics parameters.

Contribution

It presents a novel structured variational flow matching framework that incorporates physics-informed priors and handles second-order dynamics, improving interpretability and performance.

Findings

Performs on par or better than data-driven and grey-box baselines on ODE/PDE problems.

Learns dynamics without ground-truth physics parameters.

Preserves interpretability of physics models.

Abstract

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.