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

arXiv:2609.36942 · cs.LG, cs.AI, cs.SY, eess.SY · Submitted 2026-09-29 · Read on arXiv

cs.LG, cs.AI, cs.SY, eess.SY

Submitted: 2026-09-29

Updated: 2026-09-30

Code: https://github.com/sim1bet/energy-safe-dynamics

License: http://creativecommons.org/licenses/by-sa/4.0/

The gist: Learning neural-network models of dynamical systems with safety guarantees is a fundamental requirement for their deployment in safety-critical settings.

Terminology

Abstract

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.

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