2020/09/11 by Katharina Rath, Christopher G. Albert, Bernd Bischl +1 · 20 citations
Computer Science · Mathematics · Physics and Astronomy · #Chaotic #Euler's formula #Gaussian #Gaussian Processes and Bayesian Inference #Gaussian process #Hamiltonian (control theory) #Hamiltonian mechanics #Hamiltonian system #Model Reduction and Neural Networks #Numerical methods for differential equations #Orthogonal functions #Symplectic geometry #physics.comp-ph #stat.ML
paper · pdf · doi:10.1063/5.0048129
published in Chaos An Interdisciplinary Journal of Nonlinear Science 31(5), 053121 (American Institute of Physics) · 24 pages, 9 figures
arxiv created 2020/09/11 · openalex created_date 2020/09/21 · openalex publication_date 2021/05/01 · arxiv updated 2021/06/02 · openalex updated_date 2026/08/05
We present an approach to construct structure-preserving emulators for Hamiltonian flow maps and Poincaré maps based directly on orbit data. Intended applications are in moderate-dimensional systems, in particular, long-term tracing of fast charged particles in accelerators and magnetic plasma confinement configurations. The method is based on multi-output Gaussian process (GP) regression on scattered training data. To obtain long-term stability, the symplectic property is enforced via the choice of the matrix-valued covariance function. Based on earlier work on spline interpolation, we observe derivatives of the generating function of a canonical transformation. A product kernel produces an accurate implicit method, whereas a sum kernel results in a fast explicit method from this approach. Both are related to symplectic Euler methods in terms of numerical integration but fulfill a complementary purpose. The developed methods are first tested on the pendulum and the Hénon-Heiles system and results compared to spectral regression of the flow map with orthogonal polynomials. Chaotic behavior is studied on the standard map. Finally, the application to magnetic field line tracing in a perturbed tokamak configuration is demonstrated. As an additional feature, in the limit of small mapping times, the Hamiltonian function can be identified with a part of the generating function and thereby learned from observed time-series data of the system's evolution. For implicit GP methods, we demonstrate regression performance comparable to spectral bases and artificial neural networks for symplectic flow maps, applicability to Poincaré maps, and correct representation of chaotic diffusion as well as a substantial increase in performance for learning the Hamiltonian function compared to existing approaches.