Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

Burby, J. W.; Hirvijoki, Eero; Leok, Melvin

doi:10.1007/s00332-023-09891-4

Cited by 4 publications

(26 citation statements)

References 27 publications

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“…Second is timescale separation, corresponding to the relatively short timescale of oscillations compared with slower secular drifts. Coexistence of these two structural properties implies the existence of an adiabatic invariant 8 – 11 . Adiabatic invariants differ from true constants of motion, in particular energy invariants, which do not change at all over arbitrary time intervals.…”

Section: Introductionmentioning

confidence: 99%

“…There are no learning frameworks available today that exactly preserve the two structural properties whose interplay gives rise to adiabatic invariants. This work addresses this challenge by exploiting a recently-developed theory of nearly-periodic symplectic maps 11 , which can be thought of as discrete-time analogues of highly-oscillatory Hamiltonian systems 9 .…”

Section: Introductionmentioning

confidence: 99%

“…It is well-known that a Hamiltonian system which admits a continuous family of symmetries also admits a corresponding conserved quantity. It is thus not surprising that a nearly-periodic Hamiltonian system, which admits an approximate symmetry, must also have an approximate conservation law 11 , and the approximately conserved quantity is referred to as an adiabatic invariant.…”

Section: Introductionmentioning

confidence: 99%

“…Nearly-periodic maps , first introduced by Burby et al 11 , are natural discrete-time analogues of nearly-periodic systems, and have important applications to numerical integration of nearly-periodic systems. Nearly-periodic maps may also be used as tools for structure-preserving simulation of non-canonical Hamiltonian systems on exact symplectic manifolds 11 , which have numerous applications across the physical sciences. Noncanonical Hamiltonian systems play an especially important role in modeling weakly-dissipative plasma systems 36 – 42 .…”

Section: Introductionmentioning

confidence: 99%

“…Noncanonical Hamiltonian systems play an especially important role in modeling weakly-dissipative plasma systems 36 – 42 . Similarly to the continuous-time case, nearly-periodic maps with a Hamiltonian structure (that is symplecticity) admit an approximate symmetry and as a result also possess an adiabatic invariant 11 . The adiabatic invariants that our networks target only arise in purely Hamiltonian systems.…”

Section: Introductionmentioning

confidence: 99%

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Approximation of nearly-periodic symplectic maps via structure-preserving neural networks

Valentin

Burby

2023

Sci Rep

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A continuous-time dynamical system with parameter $$\varepsilon$$ ε is nearly-periodic if all its trajectories are periodic with nowhere-vanishing angular frequency as $$\varepsilon$$ ε approaches 0. Nearly-periodic maps are discrete-time analogues of nearly-periodic systems, defined as parameter-dependent diffeomorphisms that limit to rotations along a circle action, and they admit formal U(1) symmetries to all orders when the limiting rotation is non-resonant. For Hamiltonian nearly-periodic maps on exact presymplectic manifolds, the formal U(1) symmetry gives rise to a discrete-time adiabatic invariant. In this paper, we construct a novel structure-preserving neural network to approximate nearly-periodic symplectic maps. This neural network architecture, which we call symplectic gyroceptron, ensures that the resulting surrogate map is nearly-periodic and symplectic, and that it gives rise to a discrete-time adiabatic invariant and a long-time stability. This new structure-preserving neural network provides a promising architecture for surrogate modeling of non-dissipative dynamical systems that automatically steps over short timescales without introducing spurious instabilities.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Approximation of nearly-periodic symplectic maps via structure-preserving neural networks

Valentin

Burby

2023

Sci Rep

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Coarse-graining Hamiltonian systems using WSINDy

Messenger,

Burby,

Bortz

2024

Sci Rep

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Weak form equation learning and surrogate modeling has proven to be computationally efficient and robust to measurement noise in a wide range of applications including ODE, PDE, and SDE discovery, as well as in coarse-graining applications, such as homogenization and mean-field descriptions of interacting particle systems. In this work we extend this coarse-graining capability to the setting of Hamiltonian dynamics which possess approximate symmetries associated with timescale separation. A smooth $$\varepsilon$$ ε -dependent Hamiltonian vector field $$X_\varepsilon$$ X ε possesses an approximate symmetry if the limiting vector field $$X_0=\lim _{\varepsilon \rightarrow 0}X_\varepsilon$$ X 0 = lim ε → 0 X ε possesses an exact symmetry. Such approximate symmetries often lead to the existence of a Hamiltonian system of reduced dimension that may be used to efficiently capture the dynamics of the symmetry-invariant dependent variables. Deriving such reduced systems, or approximating them numerically, is an ongoing challenge. We demonstrate that WSINDy can successfully identify this reduced Hamiltonian system in the presence of large perturbations imparted in the $$\varepsilon >0$$ ε > 0 regime, while remaining robust to extrinsic noise. This is significant in part due to the nontrivial means by which such systems are derived analytically. WSINDy naturally preserves the Hamiltonian structure by restricting to a trial basis of Hamiltonian vector fields. The methodology is computationally efficient, often requiring only a single trajectory to learn the global reduced Hamiltonian, and avoiding forward solves in the learning process. In this way, we argue that weak-form equation learning is particularly well-suited for Hamiltonian coarse-graining. Using nearly-periodic Hamiltonian systems as a prototypical class of systems with approximate symmetries, we show that WSINDy robustly identifies the correct leading-order system, with dimension reduced by at least two, upon observation of the relevant degrees of freedom. While our main contribution is computational, we also provide a contribution to the literature on averaging theory by proving that first-order averaging at the level of vector fields preserves Hamiltonian structure in nearly-periodic Hamiltonian systems. This provides theoretical justification for our approach as WSINDy’s computations occur at the level of Hamiltonian vector fields. We illustrate the efficacy of our proposed method using physically relevant examples, including coupled oscillator dynamics, the Hénon–Heiles system for stellar motion within a galaxy, and the dynamics of charged particles.

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Structure-preserving algorithms for guiding center dynamics based on the slow manifold of classical Pauli particle

ZHANG 张,

WANG 王,

XIAO 肖

et al. 2024

Plasma Sci. Technol.

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Classical Pauli particle (CPP) serves as a slow manifold, substituting the conventional guiding center dynamics. Based on the CPP, we utilize averaged vector field (AVF) method in the computations of drift orbits. Demonstrating significantly higher efficiency, this advanced method is capable of accomplishing the simulation in less than one-third time of directly computing the guiding center motion. In contrast to the CPP-based Boris algorithm, this approach inherits the advantages of the AVF method, yielding stable trajectories even achieved with a tenfold time step and reducing energy error with two orders of magnitude. By comparing these two CPP algorithms with the traditional RK4 method, numerical results indicate the remarkable performance in terms of both computational efficiency and error elimination. Moreover, we verify the properties of slow manifold integrators and successfully observe the bounce on both sides of the limiting slow manifold with initial conditions chosen off. To evaluate the practical value of the methods, we conduct simulations in non-axisymmetric perturbation magnetic fields as part of the experiments, demonstrating our CPP-based AVF method can handle simulations under complex magnetic field configurations with high accuracy, which the CPP-based Boris algorithm lacks. Through numerical experiments, we demonstrate CPP can replace guiding center dynamics in using energy-preserving algorithms for computations, providing a new, efficient, as well as stable approach for applying structure-preserving algorithms in plasma simulations.

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Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

Cited by 4 publications

References 27 publications

Approximation of nearly-periodic symplectic maps via structure-preserving neural networks

Approximation of nearly-periodic symplectic maps via structure-preserving neural networks

Coarse-graining Hamiltonian systems using WSINDy

Structure-preserving algorithms for guiding center dynamics based on the slow manifold of classical Pauli particle

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