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Hybrid Energy-Based Models for Physical AI: Provably Stable Identification of Port-Hamiltonian Dynamics

Simone Betteti, Luca Laurenti

Abstract

Energy-based models (EBMs) implement inference as gradient descent on a learned Lyapunov function, yielding interpretable, structure-preserving alternatives to black-box neural ODEs and aligning naturally with physical AI. Yet their use in system identification remains limited, and existing architectures lack formal stability guarantees that globally preclude unstable modes. We address this gap by introducing an EBM framework for system identification with stable, dissipative, absorbing invariant dynamics. Unlike classical global Lyapunov stability, absorbing invariance expands the class of stability-preserving architectures, enabling more flexible and expressive EBMs. We extend EBM theory to nonsmooth activations by establishing negative energy dissipation via Clarke derivatives and deriving new conditions for radial unboundedness, exposing a stability-expressivity tradeoff in standard EBMs. To overcome this, we introduce a hybrid architecture with a dynamical visible layer and static hidden layers, prove absorbing invariance under mild assumptions, and show that these guarantees extend to port-Hamiltonian EBMs. Experiments on metric-deformed multi-well and ring systems validate the approach, showcasing how our hybrid EBM architecture combines expressivity with sound and provable safety guarantees by design.

Hybrid Energy-Based Models for Physical AI: Provably Stable Identification of Port-Hamiltonian Dynamics

Abstract

Energy-based models (EBMs) implement inference as gradient descent on a learned Lyapunov function, yielding interpretable, structure-preserving alternatives to black-box neural ODEs and aligning naturally with physical AI. Yet their use in system identification remains limited, and existing architectures lack formal stability guarantees that globally preclude unstable modes. We address this gap by introducing an EBM framework for system identification with stable, dissipative, absorbing invariant dynamics. Unlike classical global Lyapunov stability, absorbing invariance expands the class of stability-preserving architectures, enabling more flexible and expressive EBMs. We extend EBM theory to nonsmooth activations by establishing negative energy dissipation via Clarke derivatives and deriving new conditions for radial unboundedness, exposing a stability-expressivity tradeoff in standard EBMs. To overcome this, we introduce a hybrid architecture with a dynamical visible layer and static hidden layers, prove absorbing invariance under mild assumptions, and show that these guarantees extend to port-Hamiltonian EBMs. Experiments on metric-deformed multi-well and ring systems validate the approach, showcasing how our hybrid EBM architecture combines expressivity with sound and provable safety guarantees by design.

Paper Structure

This paper contains 10 sections, 4 theorems, 36 equations, 3 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Lie and Clarke's derivatives
  4. Conjugate (Legendre) transform
  5. PROBLEM STATEMENT
  6. LYAPUNOV STABILITY AND THE STABILITY-EXPRESSIVITY TRADE-OFF
  7. MULTI-LAYER EBMs AND ABSORBING INVARIANCE
  8. MODEL VALIDATION
  9. CONCLUSIONS
  10. ACKNOWLEDGMENTS

Key Result

Proposition 7

Let $\mathcal{F}\in C^1(\mathbb{R}^N)$ and let $f_{E}(x)=-x+W\Psi(x)$ be the vector field associated to the EBM dynamics eq: EBM. Then for all $x\in\mathbb{R}^N$ and $\operatorname{E}(x)$ defined in eq: EBM-ene $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: EBM identification of multi-well potential dynamics. (a) Ground‑truth potential $V_m(x,y)$. (b) EBM energy $\operatorname{E}(x,y)$ accurately recovering the landscape geometry. (c) True vector field. (d) Reconstructed field with true and EBM trajectories, showing consistent flow patterns and convergence to the correct attractors.
  • Figure 2: EBM identification of exotic potential dynamics. (a) Ground‑truth potential $V_e(x,y)$. (b) EBM energy $\operatorname{E}(x,y)$ accurately recovering the landscape ring-geometry. (c) True vector field. (d) Reconstructed field with true and EBM trajectories. EBM dynamics are consistent with the true vector field and reliably recover non-linear, rotational components.
  • Figure 3: Expansiveness radius. Maximal radius of expansiveness given the data $\mathcal{X}$. The radii have similar magnitude for both the (a) multi-well potential $V_{m}(x,y)$ and the (b) exotic potential $V_e(x,y)$.

Theorems & Definitions (18)

  • Definition 1: Lie derivative
  • Definition 2: Generalized gradient
  • Definition 3: Generalized directional derivative
  • Definition 4: Conjugate transform
  • Remark 1: Smooth proper convex functions
  • Definition 5: Energy-based models (EBMs)
  • Definition 6
  • Proposition 7: Negative definiteness of the energy derivative
  • proof
  • Proposition 8: Radial unboudedness of the energy derivative
  • ...and 8 more