# Extrapolating the emergence of Hamiltonian chaos with random-feature Hamiltonian neural networks

arXiv: https://arxiv.org/abs/2607.28977 · HTML: https://mlsanity.com/paper/2607.28977/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 29 references that are in the map.
- First posted on arXiv 31 Jul 2026.

## Builds on (most cited first, 13 of 13)

- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Machine learning prediction of critical transition and system collapse](https://mlsanity.com/paper/2012.01545/) (2020)
- [Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps](https://mlsanity.com/paper/2103.05632/) (2021)
- [Out-of-Domain Generalization in Dynamical Systems Reconstruction](https://mlsanity.com/paper/2402.18377/) (2024)
- [Adaptable Hamiltonian neural networks](https://mlsanity.com/paper/2102.13235/) (2021)
- [Sampling weights of deep neural networks](https://mlsanity.com/paper/2306.16830/) (2023)
- [Extrapolating tipping points and simulating non-stationary dynamics of complex systems using efficient machine learning](https://mlsanity.com/paper/2312.06283/) (2023)
- [Reconstruction of observed mechanical motions with Artificial Intelligence tools](https://mlsanity.com/paper/2202.11447/) (2022)
- [Neural Ordinary Differential Equations for Learning and Extrapolating System Dynamics Across Bifurcations](https://mlsanity.com/paper/2507.19036/) (2025)
- [Deep Learning of the Evolution Operator Enables Forecasting of Out-of-Training Dynamics in Chaotic Systems](https://mlsanity.com/paper/2502.20603/) (2025)
- [Training Hamiltonian neural networks without backpropagation](https://mlsanity.com/paper/2411.17511/) (2024)
- [Rapid training of Hamiltonian graph networks using random features](https://mlsanity.com/paper/2506.06558/) (2025)

## Abstract (arXiv)

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.