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Variational Grey-Box Dynamics Matching

Gurjeet Sangra Singh, Frantzeska Lavda, Giangiacomo Mercatali, Alexandros Kalousis

TL;DR

The paper tackles the challenge of learning dynamical systems when physical models are incomplete by blending physics priors with data-driven dynamics in a simulation-free framework. It introduces Variational Grey-Box Dynamics Matching (VGB-DM), which uses a structured latent space with latent variables $z$ (stochasticity) and $ heta$ (physics parameters) and a physics-informed prior to infer underlying physics while fitting trajectory data. By extending flow-matching techniques to account for incomplete physics and second-order dynamics, VGB-DM achieves state-of-the-art forecasting performance, faster convergence, and robust parameter identification across diverse ODE/PDE benchmarks and real-world ERA5 weather forecasting, all while preserving interpretability of the physics terms. The approach reduces reliance on numerical solvers during training and offers scalable, transferable modelling capabilities for complex, real-world systems. The work provides open-source code and lays groundwork for broader adoption of simulation-free grey-box dynamics in scientific and engineering domains.

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.

Variational Grey-Box Dynamics Matching

TL;DR

The paper tackles the challenge of learning dynamical systems when physical models are incomplete by blending physics priors with data-driven dynamics in a simulation-free framework. It introduces Variational Grey-Box Dynamics Matching (VGB-DM), which uses a structured latent space with latent variables (stochasticity) and (physics parameters) and a physics-informed prior to infer underlying physics while fitting trajectory data. By extending flow-matching techniques to account for incomplete physics and second-order dynamics, VGB-DM achieves state-of-the-art forecasting performance, faster convergence, and robust parameter identification across diverse ODE/PDE benchmarks and real-world ERA5 weather forecasting, all while preserving interpretability of the physics terms. The approach reduces reliance on numerical solvers during training and offers scalable, transferable modelling capabilities for complex, real-world systems. The work provides open-source code and lays groundwork for broader adoption of simulation-free grey-box dynamics in scientific and engineering domains.

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.
Paper Structure (49 sections, 1 theorem, 39 equations, 10 figures, 4 tables)

This paper contains 49 sections, 1 theorem, 39 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Problem setting
  4. Grey-Box Modelling: Beyond Forecasting
  5. Flow Matching
  6. Method
  7. Variational Grey-Box Dynamics Matching
  8. Integrating Incomplete Physics.
  9. Structured Variational Inference.
  10. Second-Order Dynamics.
  11. Related works
  12. Experiments
  13. Differential-Equation Datasets
  14. Weather modelling
  15. Conclusion
  16. ...and 34 more sections

Key Result

Proposition 1

Under the regularity conditions of Lemma A.1 zhang2024tjfm, if Assumption (A4) holds and $x, \pi(\boldsymbol{x}), p_t(x_t|\boldsymbol{x}),$ and $u_t(x_t|\boldsymbol{x})$, then training with $\mathcal{L}_{\text{VGB-DM}}(\phi, \psi^*)$ preserves the optimal coupling:

Figures (10)

  • Figure 1: Probabilistic graphical model of variational grey-box models. Shaded circles are observable variables. Empty dashed circles represent latent variables, and dashed arrows denote (learned) inference dependencies.
  • Figure 2: Overview of the Variational Grey-Box Dynamics Matching (VGB-DM) framework.
  • Figure 3: Optimization convergence analysis. Forecast $\log$MSE vs. training time (minutes) for the Pendulum (left), RLC (center), and Reaction-Diffusion (right) tasks using the largest training dataset size. VGB-DM demonstrates significantly faster convergence and achieves lower final error compared to simulation-based methods (PhysVAE, BB-NODE), while also exhibiting better stability than the black-box dynamics matching baseline (VBB-DM)
  • Figure 4: Sample efficiency analysis. Forecast MSE across varying training sample sized for for Pendulum, RLC, and Reaction-Diffusion tasks. Grey-box methods (Ours and PhysVAE) consistently outperform black-box approaches, especially in low-data regimes. Our method demonstrates more robust performance compared to all baselines.
  • Figure 5: Residual maps for weather forecasting Visualization of absolute error maps between ground truth and predicted fields for five meteorological variables (Ours vs ClimODE)
  • ...and 5 more figures

Theorems & Definitions (1)

  • Proposition 1: Coupling Preservation with learnt Latents by Marginal Preservation