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This research introduces Generalized Forced Hamiltonian Neural Networks (GFHNNs) that leverage geometric principles to model deterministic and stochastic physical systems with superior long-term stability. By enforcing structure preservation through Lagrange-d'Alembert maps, the architecture achieves higher accuracy with less training data compared to standard residual networks. The work represents a significant advancement in applying machine learning to complex dynamical systems while maintaining fundamental physical invariants.
Learning Deterministic and Stochastic Forced Hamiltonian Systems
arXiv:2608.19688v1 Announce Type: cross Abstract: We develop a geometric framework for learning deterministic and stochastic forced Hamiltonian systems with neural networks. Motivated by the Lagrange-d'Alembert principle and the theory of variational integrators, we introduce the notion of a Lagrange-d'Alembert map and establish a $C^r$ convergence theorem for first-order one-step methods. Building on these results, we propose Generalized Forced Hamiltonian Neural Networks (GFHNNs), a class of structure-preserving neural networks obtained by concatenating Lagrange-d'Alembert-Euler maps, and prove a universal approximation theorem for this architecture. We further extend the framework to parameter-dependent systems, leading to Parametric Generalized Forced Hamiltonian Neural Networks (PGFHNNs). By interpreting the multiple Stratonovich integrals appearing in the Stratonovich-Taylor expansion as parameters, the same framework can be applied to stochastic forced Hamiltonian systems whenever information about the underlying Wiener process is available. Our numerical experiments demonstrate that the proposed geometric architectures provide significantly improved long-time stability and accuracy compared to non-geometric residual neural networks, while requiring substantially less training data to achieve a comparable level of performance.
Abstract:We develop a geometric framework for learning deterministic and stochastic forced Hamiltonian systems with neural networks. Motivated by the Lagrange-d'Alembert principle and the theory of variational integrators, we introduce the notion of a Lagrange-d'Alembert map and establish a $C^r$ convergence theorem for first-order one-step methods. Building on these results, we propose Generalized Forced Hamiltonian Neural Networks (GFHNNs), a class of structure-preserving neural networks obtained by concatenating Lagrange-d'Alembert-Euler maps, and prove a universal approximation theorem for this architecture. We further extend the framework to parameter-dependent systems, leading to Parametric Generalized Forced Hamiltonian Neural Networks (PGFHNNs). By interpreting the multiple Stratonovich integrals appearing in the Stratonovich-Taylor expansion as parameters, the same framework can be applied to stochastic forced Hamiltonian systems whenever information about the underlying Wiener process is available. Our numerical experiments demonstrate that the proposed geometric architectures provide significantly improved long-time stability and accuracy compared to non-geometric residual neural networks, while requiring substantially less training data to achieve a comparable level of performance.
Numerical Analysis (math.NA); Machine Learning (cs.LG); Symplectic Geometry (math.SG)
arXiv:2608.19688 [math.NA]
(or arXiv:2608.19688v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2608.19688 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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