Slow manifold reduction as a systematic tool for revealing the geometry of phase space

Burby, J. W.

doi:10.1063/5.0084543

Cited by 3 publications

(4 citation statements)

References 31 publications

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“…Symplectic gyroceptrons provide a promising class of architectures for surrogate modeling of non-dissipative dynamical systems that automatically steps over short timescales without introducing spurious instabilities, and could have potential future applications for the Klein-Gordon equation in the weakly-relativistic regime, for charged particles moving through a strong magnetic field, and for the rotating inviscid Euler equations in quasi-geostrophic scaling 7 . Symplectic gyroceptrons could also be used for structure-preserving simulation of non-canonical Hamiltonian systems on exact symplectic manifolds 11 , which have numerous applications across the physical sciences, for instance in modeling weakly-dissipative plasma systems [36][37][38][39][40][41][42] .…”

Section: Discussionmentioning

confidence: 99%

“…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 . 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 .…”

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: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Valentin

Burby

2023

Sci Rep

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“…In addition, the first-order averaged system satisfies the Hamilton-like equation Proof The conservation law (14) follows immediately from ( 12) and the R 0 -average of ( 13). (Notice that �L X 0 µ 1 � = ω 0 �L R 0 µ 1 � = 0 and �µ * � = µ * .)…”

Section: Lemmamentioning

confidence: 99%

“…From a different perspective, Duruisseaux, Burby, and Tang 11 introduced a neural-network architecture to specifically handle systems possessing an adiabatic invariant, by which the dynamics may be reduced to an approximate Hamiltonian system of lower dimension. The latter is part of a series of recent works aimed at exploiting near-periodicity to design more efficient means of understanding and simulating dynamics relevant to plasma physics [12][13][14][15][16] . The current work can be seen as a continuation of this series, with the purpose of offering computationally efficient, noise-robust, and interpretable discovery of reduced Hamiltonian systems to complement neural network-based approaches 11 .…”

Section: Literature Reviewmentioning

confidence: 99%

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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Coarse-Graining Hamiltonian Systems Using WSINDy

Messenger,

Burby,

Bortz

2023

Preprint

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The Weak-form Sparse Identification of Nonlinear Dynamics algorithm (WSINDy) has been demonstrated to offer coarse-graining capabilities in the context of interacting particle systems (https://doi.org/10.1016/j.physd.2022.133406). In this work we extend this capability to the problem of coarse-graining Hamiltonian dynamics which possess approximate symmetries associated with timescale separation. A smooth ε-dependent Hamiltonian vector field Xε possesses an approximate symmetry if the limiting vector field X0 = limε→0 Xε 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 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 computational 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. 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. We provide physically relevant examples, namely coupled oscillator dynamics, the Hénon-Heiles system for stellar motion within a galaxy, and the dynamics of charged particles.

show abstract

Slow manifold reduction as a systematic tool for revealing the geometry of phase space

Cited by 3 publications

References 31 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

Coarse-Graining Hamiltonian Systems Using WSINDy

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