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

arXiv: https://arxiv.org/abs/2604.00277 · HTML: https://mlsanity.com/paper/2604.00277/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 34 references that are in the map.
- First posted on arXiv 31 Mar 2026.

## Builds on (most cited first, 15 of 15)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [Large Associative Memory Problem in Neurobiology and Machine Learning](https://mlsanity.com/paper/2008.06996/) (2020)
- [Energy Transformer](https://mlsanity.com/paper/2302.07253/) (2023)
- [Lyapunov-stable neural-network control](https://mlsanity.com/paper/2109.14152/) (2021)
- [In search of dispersed memories: Generative diffusion models are associative memory networks](https://mlsanity.com/paper/2309.17290/) (2023)
- [Energy-Based Cross Attention for Bayesian Context Update in Text-to-Image Diffusion Models](https://mlsanity.com/paper/2306.09869/) (2023)
- [Stable Neural Flows](https://mlsanity.com/paper/2003.08063/) (2020)
- [Stable Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2502.02480/) (2025)
- [Dissipative Deep Neural Dynamical Systems](https://mlsanity.com/paper/2011.13492/) (2020)
- [Neuron-Astrocyte Associative Memory](https://mlsanity.com/paper/2311.08135/) (2023)
- [Dense Associative Memory Through the Lens of Random Features](https://mlsanity.com/paper/2410.24153/) (2024)
- [Learning Dissipative Neural Dynamical Systems](https://mlsanity.com/paper/2309.16032/) (2023)

## Built on by (most cited first, 1 of 1)

- [Safe-by-Design Learning via Energy-based Neural Networks](https://mlsanity.com/paper/2609.36942/) (2026)

## Abstract (arXiv)

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.