Symplectic integration of learned Hamiltonian systems

@article{Offen2022SymplecticIO,
  title={Symplectic integration of learned Hamiltonian systems},
  author={Christian Offen and Sina Ober-Blobaum},
  journal={Chaos},
  year={2022},
  volume={32 1},
  pages={
          013122
        }
}
Hamiltonian systems are differential equations that describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation laws. To predict Hamiltonian dynamics based on discrete trajectory observations, the incorporation of prior knowledge about Hamiltonian structure greatly improves predictions. This is typically done by learning the system's Hamiltonian and then integrating the… 
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Methods Citations
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References

SHOWING 1-10 OF 35 REFERENCES
Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps
TLDR
GFNN directly learns the symp eclectic evolution map in discrete time by representing the symplectic map via a generating function, which is approximate by a neural network, and it will be proved, under reasonable assumptions, that the global prediction error grows at most linearly with long prediction time, which significantly improves an otherwise exponential growth.
Symplectic Gaussian process regression of maps in Hamiltonian systems.
TLDR
This work presents an approach to construct structure-preserving emulators for Hamiltonian flow maps and Poincaré maps based directly on orbit data based on multi-output Gaussian process (GP) regression on scattered training data, and demonstrates regression performance comparable to spectral bases and artificial neural networks for symplectic flow maps.
Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations
Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control
TLDR
This paper introduces Symplectic ODE-Net, a deep learning framework which can infer the dynamics of a physical system, given by an ordinary differential equation (ODE), from observed state trajectories and proposes a parametrization which can enforce this Hamiltonian formalism even when the generalized coordinate data is embedded in a high-dimensional space or the authors can only access velocity data instead of generalized momentum.
Symplectic Learning for Hamiltonian Neural Networks
TLDR
A significantly improved training method for HNNs is explored, exploiting the symplectic structure of Hamiltonian systems with a different loss function, which mathematically guarantee the existence of an exact Hamiltonian function which the HNN can learn.
On learning Hamiltonian systems from data.
TLDR
This work proposes to explore a particular type of underlying structure in the data: Hamiltonian systems, where an "energy" is conserved, and extracts an underlying phase space as well as the generating Hamiltonian from a collection of movies of a pendulum.
Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation
TLDR
It is shown that symp eclectic integrators preserve bifurcations of Hamiltonian boundary value problems and that non-symplectic integrators do not, and the jet-RATTLE method is introduced, a symplectic integrator that easily computes geodesics and their bifURcations.
SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems
Symplectic integrators for spin systems.
TLDR
A symplectic integrator, based on the implicit midpoint method, for classical spin systems where each spin is a unit vector in R{3}, yields a new integrable discretization of the spinning top.
Modified equations for variational integrators
TLDR
A technique to construct a Lagrangians for the modified equation from the discrete Lagrangian of a variational integrator is presented.
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