Variational Grey-Box Dynamics Matching

2026-02-19

AI Summary

This work addresses the challenge of accurately fitting observational data when physical models are incomplete or contain unknown terms. The authors propose a gray-box generative modeling approach that embeds an incomplete physical model into a generative framework, learning dynamics directly from observed trajectories without requiring ground-truth parameters or numerical simulations. The method introduces a structured variational distribution, employing two latent encodings to separately capture missing stochasticity and multimodal velocity structures, while incorporating physical priors to support second-order dynamical extensions. Built upon a flow-matching framework, it integrates variational inference with physics-informed priors, thereby circumventing the stability and scalability limitations of neural ODEs. Experiments demonstrate that the approach matches or exceeds the performance of purely data-driven and existing gray-box methods on representative ODE/PDE tasks, while preserving model interpretability.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Deep Generative Models & Autoencoders
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

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 show 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.

Problem

• grey-box modeling
• physics-informed generative models
• incomplete dynamics
• variational inference
• flow matching

Innovation

• grey-box modeling
• flow matching
• physics-informed generative models
• variational inference
• dynamical systems