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arXiv Machine Learning

Variational Continuation for Double Pendulum Periodic Orbits

The paper introduces a Hessian‑based, integrator‑free method for numerically continuing periodic orbits in dynamical systems, using a Fourier series representation and automatic differentiation to avoid hand‑derived Jacobians. By exploiting flat directions in the loss landscape, the approach efficiently guides the search for periodic orbits, accurately initializes oscillations around unstable fixed points, and detects orbit family intersections and subharmonic bifurcations. Demonstrated on the double pendulum, the method produces full continuations from fixed points, revealing bifurcations along orbit families and identifying previously undocumented periodic orbits where both pendulum masses are never simultaneously at rest.

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arXiv Machine Learning
Sep 4

Data-efficient Kernel Methods for Learning Hamiltonian Systems

The paper introduces kernel-based methods for learning Hamiltonian systems directly from trajectory data, offering both a 2‑step approach (reconstruct trajectories first, then learn the Hamiltonian) and a 1‑step approach (joint inference). Experiments on mass‑spring dynamics, a nonlinear pendulum, and the Henon‑Heiles system show that the methods achieve accurate, data‑efficient predictions, outperforming 2‑step baselines especially when data are scarce, while preserving the Hamiltonian structure. The authors also provide a priori error estimates and a general numerical framework applicable to arbitrary dynamical systems.

By Yasamin Jalalian, Mostafa Samir, Boumediene Hamzi, Peyman Tavallali, Houman Owhadi
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arXiv Machine Learning
Jun 26

Symplectic Neural Networks for learning Generalized Hamiltonians

arXiv:2606. 27029v1 Announce Type: new Abstract: Hamiltonian Neural Networks (HNNs) integrate physical priors into neural models by learning a system's Hamiltonian, improving generalization and sample efficiency.

By Harsh Choudhary, Vyacheslav Kungurtsev, Chandan Gupta, Melvin Leok, Georgios Korpas
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arXiv Machine Learning
Jul 20

Ptolemy's Equant Equates to a Universal Dynamical Clock via Machine Learning

arXiv:2607. 15472v1 Announce Type: cross Abstract: Oscillatory dynamics arise ubiquitously in nonlinear systems, yet identifying a physically interpretable phase and phase dynamics in nonlinear, high-dimensional oscillations remains a central unresolved problem.

By Jingdong Zhang, Luan Yang, Murilo S. Baptista, Zefeng Zhang, Qunxi Zhu, Wei Lin, Celso Grebogi
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arXiv Machine Learning
Jul 1

Automated Discovery of Operable Dynamics from Videos

arXiv:2410. 11894v3 Announce Type: replace-cross Abstract: Dynamical systems form the foundation of scientific discovery, traditionally modeled with predefined state variables such as the angle and angular velocity, and differential equations such as the equation of motion for a single pendulum.

By Kuang Huang, Dong Heon Cho, Boyuan Chen
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arXiv AI
Oct 9

Bi-FORK: Generative Modeling of High-Dimensional Bifurcating Systems

Bi-FORK is a generative framework designed to learn one-to-many solution maps in high-dimensional systems undergoing symmetry-breaking bifurcations. It generates complete trajectories via latent flow matching, maintains space and time coherence, and employs repulsion-guided sampling to recover distinct solution branches in a single amortized pass. The method is evaluated on buckling beams, mechanical metamaterials, and Allen‑Cahn phase separation, handling discretizations up to 260,000 points and recovering multimodal solution structures while scaling beyond prior approaches.

By Anna Zimmel, Fleur Hendriks, Markus Holzleitner, Florian Sestak, Martin Weichselbaumer, Vlado Menkovski, Johannes Brandstetter
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