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This research demonstrates the first successful extrapolation of Hamiltonian neural networks from regular dynamics into chaotic regimes using random-feature architectures, overcoming limitations where conventional models fail outside their training parameter intervals. By leveraging specific geometric continuities in control parameters rather than just structural constraints, these networks accurately predict long-time dynamical behavior and chaos emergence without direct data from those states.
Extrapolating the emergence of Hamiltonian chaos with random-feature Hamiltonian neural networks
arXiv:2607.28977v1 Announce Type: cross Abstract: Machine learning of Hamiltonian dynamics has driven growing interest in Hamiltonian neural networks (HNNs), which encode Hamilton's equations of motion into the learning architecture. Despite this progress, it remains unknown whether such networks can predict dynamical regimes absent from their training data, in particular the broad chaotic sea that emerges beyond the observed parameter interval. We address this question using a parameter-aware random-feature Hamiltonian neural network (RF-HNN). Trained using data from only a small number of control-parameter values at which invariant tori dominate, the RF-HNN predicts autonomous long-time dynamics at unseen parameter values where mixed phase space develops and chaotic regions expand, with no data from that regime used in training or model selection. The method is demonstrated across four two-degree-of-freedom Hamiltonian families, including the H\'enon-Heiles system. Using Poincar\'e-section geometry and finite-time Lyapunov exponents, we show that the RF-HNN reproduces the breakup of regular structures and the emergence and growth of chaotic regions, whereas conventionally trained HNNs with the same Hamiltonian structure remain too regular. These results show that what decides parameter extrapolation is not Hamiltonian structure alone but how the fitted Hamiltonian continues in the control parameter. To our knowledge, this is the first demonstration that a learned Hamiltonian can qualitatively extrapolate from predominantly regular dynamics into a broad chaotic sea absent from training.
Abstract:Machine learning of Hamiltonian dynamics has driven growing interest in Hamiltonian neural networks (HNNs), which encode Hamilton's equations of motion into the learning architecture. Despite this progress, it remains unknown whether such networks can predict dynamical regimes absent from their training data, in particular the broad chaotic sea that emerges beyond the observed parameter interval. We address this question using a parameter-aware random-feature Hamiltonian neural network (RF-HNN). Trained using data from only a small number of control-parameter values at which invariant tori dominate, the RF-HNN predicts autonomous long-time dynamics at unseen parameter values where mixed phase space develops and chaotic regions expand, with no data from that regime used in training or model selection. The method is demonstrated across four two-degree-of-freedom Hamiltonian families, including the Hénon-Heiles system. Using Poincaré-section geometry and finite-time Lyapunov exponents, we show that the RF-HNN reproduces the breakup of regular structures and the emergence and growth of chaotic regions, whereas conventionally trained HNNs with the same Hamiltonian structure remain too regular. These results show that what decides parameter extrapolation is not Hamiltonian structure alone but how the fitted Hamiltonian continues in the control parameter. To our knowledge, this is the first demonstration that a learned Hamiltonian can qualitatively extrapolate from predominantly regular dynamics into a broad chaotic sea absent from training.
Chaotic Dynamics (nlin.CD); Machine Learning (cs.LG); Computational Physics (physics.comp-ph)
arXiv:2607.28977 [nlin.CD]
(or arXiv:2607.28977v1 [nlin.CD] for this version)
https://doi.org/10.48550/arXiv.2607.28977 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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