# Safe-by-Design Learning via Energy-based Neural Networks

arXiv: https://arxiv.org/abs/2609.36942 · HTML: https://mlsanity.com/paper/2609.36942/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 37 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 16 of 16)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Large Associative Memory Problem in Neurobiology and Machine Learning](https://mlsanity.com/paper/2008.06996/) (2020)
- [Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks](https://mlsanity.com/paper/2110.12976/) (2021)
- [Energy Transformer](https://mlsanity.com/paper/2302.07253/) (2023)
- [Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systems](https://mlsanity.com/paper/2107.08024/) (2021)
- [Constructing Neural Network-Based Models for Simulating Dynamical Systems](https://mlsanity.com/paper/2111.01495/) (2021)
- [AI Safety for Everyone](https://mlsanity.com/paper/2502.09288/) (2025)
- [Stable Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2502.02480/) (2025)
- [Nonlinear System Identification Nano-drone Benchmark](https://mlsanity.com/paper/2512.14450/) (2025)
- [Input-to-State Stable Neural Ordinary Differential Equations with Applications to Transient Modeling of Circuits](https://mlsanity.com/paper/2202.06453/) (2022)
- [Hybrid Energy-Based Models for Physical AI: Provably Stable Identification of Port-Hamiltonian Dynamics](https://mlsanity.com/paper/2604.00277/) (2026)

## Abstract (arXiv)

Learning neural-network models of dynamical systems with safety guarantees is a fundamental requirement for their deployment in safety-critical settings. Safety is commonly established by proving the invariance of a desired subset in state-space, ensuring that every trajectory initialized in this subset remains confined to it for all time under admissible inputs. Existing frameworks, however, either rely on computationally expensive post-hoc verification or employ safety-enforcing mechanisms without formal correctness guarantees. In this paper, we introduce a novel neural architecture grounded in energy-based modern Hopfield networks to guarantee safety-by-design while retaining sufficient expressiveness to model complex nonlinear dynamics. Specifically, we integrate modern Hopfield networks with a port-Hamiltonian neural ODE, enabling by design the construction of barrier functions yielding explicit admissible-input sets and quantitative robustness radii. Across several benchmarks, including an 12-dimensional nanodrone model, our framework achieves state-of-the-art performance while producing certified invariant sets that are more robust to external solicitations than comparable existing approaches.