Variational Grey-Box Dynamics Matching

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

cs.LGFeb 19, 2026

Variational Grey-Box Dynamics Matching

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

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.

Figures & tables

Panel of Figure 1 — Figure 1: Probabilistic graphical model of variational grey-box models. Shaded circles are observed variables, empty dashed circles are latent variables, and dashed arrows indicate learned inference dependencies.

Figure 2: Overview of the Variational Grey-Box Dynamics Matching (VGB-DM) framework.

Model	Pendulum ( $\times 10^{-2}$ )	RLC ( $\times 10^{-1}$ )	Reaction Diffusion ( $\times 10^{-3}$ )	Lorenz ( $\times 10^{-1}$ )
PhysVAE [ 12 ]	$0.417\pm 0.001$	$0.29\pm 0.01$	$7.4\pm 1.3$	$13.12\pm 1.27$
VBB-DM [ 3 ]	$1.058\pm 0.004$	$0.38\pm 0.03$	$8.0\pm 1.1$	$9.91\pm 1.22$
BB-NODE [ 2 ]	$0.452\pm 0.001$	$0.39\pm 0.04$	$7.5\pm 0.5$	$22.15\pm 7.84$
TFM [ 17 ]	$0.348\pm 0.001$	$0.26\pm 0.01$	$8.5\pm 2.8$	$9.85\pm 1.32$
Ours	$\mathbf{0.283}\pm 0.001$	$\textbf{0.23}\pm 0.01$	$\mathbf{7.1}\pm 0.9$	$\mathbf{4.68}\pm 0.16$

Table 1: Forecasting performance comparison (MSE). Results on four dynamical systems benchmarks. Our method achieves the lowest error across all tasks.

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. Our method 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). This highlights the benefit of avoiding backpropagation through the ODE solver.

Figure 4: Sample efficiency analysis. Forecast MSE across varying training sample sizes for Pendulum, RLC, and Reaction-Diffusion tasks. Grey-box methods (Ours and PhysVAE) consistently outperform their black-box counterparts (VBB-DM and BB-NODE) across all regimes, with our method achieving the lowest error and variance, particularly under limited data.

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)

Figure 6: Forecasting performance over $42$ hours. Latitude-weighted RMSE (top, lower is better) and ACC (bottom, higher is better) for ERA5 dataset. Our method performs better or on par than ClimODE.

Figure 7: Long-term forecasting performance over $5$ months. Latitude-weighted RMSE for ERA5 dataset over an extended period. Our method outperforms ClimODE by mitigating compounding errors over long horizons, with the performance gap widening as the forecast horizon extends.

Figure 8: Overview of the Variational Grey-Box Dynamics Matching (VGB-DM) framework.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 9: Parameter identification (inference) vs. forecasting performance across the three dynamical systems (lower RMSE is better).

Experiment	FM	Ph-KL	Z-KL
Pendulum	$3.9\times 10^{-5}\pm 1.8\times 10^{-5}$	$8.8\times 10^{-4}\pm 1.1\times 10^{-4}$	$2.5\pm 0.22$
RLC	$0.52\pm 0.03$	$3.4\pm 1.4$	$0.70\pm 0.001$
RD	$3\times 10^{-3}\pm 1\times 10^{-3}$	$2.7\pm 1.1$	$2\times 10^{-3}\pm 2\times 10^{-4}$
Lorenz	$3.8\pm 1.5$	$2.3\pm 0.8$	$0.003\pm 0.001$

Appendix

Table 2: Evaluation of the N-ELBO terms

Exp	Phys. Params. CV
RLC	0.0461, 0.0391
Pendulum	0.0496
Reaction–Diffusion	0.0072, 0.0031
Lorenz	0.0012, 0.0026

Appendix

Table 3: Median consistency of physics parameters across trajectory segments (test set).

Type	Variable name	Abbrev.	ECMWF ID	Levels
Static	Land-sea mask	lsm	172
Single	2 metre temperature	t2m	167
Single	10 metre U wind component	u10	165
Single	10 metre V wind component	v10	166
Atmospheric	Geopotential	z	129	500
Atmospheric	Temperature	t	130	850

Appendix

Table 4: ECMWF data variables from ERA5 used in our dataset. Static variables are time-independent, Single represents surface-level variables, and Atmospheric represents time-varying atmospheric properties at chosen altitudes.

Figure 10: Diagram of the Velocity model architecture (left) and its lightweight attentions mechanism (right)

Figure 11: Example of Climate forecasting using VGB-DM over $42$ hours.

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