Bayesian Identification of Hamiltonian Dynamics from Symplectic Data

Galioto, Nicholas; Gorodetsky, Alex

doi:10.1109/cdc42340.2020.9303852

Cited by 5 publications

(4 citation statements)

References 19 publications

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“…In addition, a discrete-time model that preserves the energy behaviors was constructed in Matsubara, Ishikawa, and Yaguchi (2020). In Galioto and Gorodetsky (2020), HNNs were combined with a Bayesian approach.…”

Section: Related Workmentioning

confidence: 99%

KAM Theory Meets Statistical Learning Theory: Hamiltonian Neural Networks with Non-zero Training Loss

Chen

Matsubara

Yaguchi

2022

AAAI

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Many physical phenomena are described by Hamiltonian mechanics using an energy function (Hamiltonian). Recently, the Hamiltonian neural network, which approximates the Hamiltonian by a neural network, and its extensions have attracted much attention. This is a very powerful method, but theoretical studies are limited. In this study, by combining the statistical learning theory and KAM theory, we provide a theoretical analysis of the behavior of Hamiltonian neural networks when the learning error is not completely zero. A Hamiltonian neural network with non-zero errors can be considered as a perturbation from the true dynamics, and the perturbation theory of the Hamilton equation is widely known as KAM theory. To apply KAM theory, we provide a generalization error bound for Hamiltonian neural networks by deriving an estimate of the covering number of the gradient of the multi-layer perceptron, which is the key ingredient of the model. This error bound gives a sup-norm bound on the Hamiltonian that is required in the application of KAM theory.

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Section: Related Workmentioning

confidence: 99%

KAM Theory Meets Statistical Learning Theory: Hamiltonian Neural Networks with Non-zero Training Loss

Chen

Matsubara

Yaguchi

2022

AAAI

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“…where Σ indicates the (2N) × (2N) symplectic matrix, [216,234], For a system with such a phase-space structure, it is possible to define a Gaussian state determined by its mean vector ⟨x⟩ and covariance matrix C of the position and momentum variables with the (m, n)-element expressed as…”

Section: Bayesian Quantum Feedbackmentioning

confidence: 99%

Quantum Control Modelling, Methods, and Applications

Dehaghani¹,

Pereira²,

Aguiar³

2022

Extsv. Rev.

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This review concerns quantum control results and methods that, over the years, have been used in the various operations involving quantum systems. Most of these methods have been originally developed outside the context of quantum physics, and, then, adapted to take into account the specificities of the various quantum physical platforms. Quantum control consists in designing adequate control signals required to efficiently manipulate systems conforming the laws of quantum mechanics in order to ensure the associated desired behaviours and performances. This work attempts to provide a thorough and self-contained introduction and review of the various quantum control theories and their applications. It encompasses issues spanning quantum control modelling, problem formulation, concepts of controllability, as well as a selection of the main control theories. Given the vastness of the field, we tried our best to be as concise as possible, and, for the details, the reader is pointed out to a profusion of references. The contents of the review are organized in the three major classes of control problems - open-loop control, closed-loop learning control, and feedback control - and, for each one of them, we present the main developments in quantum control theory. Finally, concerning the importance of attaining robustness and reliability due to inherent fragility of quantum systems, methods for quantum robust control are also surveyed.

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“…In another research direction, techniques based on sparse identification of nonlinear dynamics (SINDy) [10] and orthogonal polynomials [11] have also been used for learning Hamiltonian systems from data, but these techniques tend to break down when the data are noisy/sparse since they rely on numerical approximations of the time derivatives of the data. To handle uncertainty and increase robustness, Bayesian inference techniques based on Gaussian process regression [12], [13] and/or consideration of stochastic dynamics [14] have been developed. However, a majority of these approaches assume that the Hamiltonian system is separable, i.e., the system Hamiltonian can be written as H(q, p) = T (p) + U (q) where q is the position, p is the momentum, T (p) is the kinetic energy, and U (q) is the potential energy.…”

Section: Introductionmentioning

confidence: 99%

Bayesian Identification of Nonseparable Hamiltonian Systems Using Stochastic Dynamic Models

Sharma¹,

Galioto²,

Gorodetsky³

et al. 2022

Preprint

Self Cite

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This paper proposes a probabilistic Bayesian formulation for system identification (ID) and estimation of nonseparable Hamiltonian systems using stochastic dynamic models. Nonseparable Hamiltonian systems arise in models from diverse science and engineering applications such as astrophysics, robotics, vortex dynamics, charged particle dynamics, and quantum mechanics. The numerical experiments demonstrate that the proposed method recovers dynamical systems with higher accuracy and reduced predictive uncertainty compared to state-of-the-art approaches. The results further show that accurate predictions far outside the training time interval in the presence of sparse and noisy measurements are possible, which lends robustness and generalizability to the proposed approach. A quantitative benefit is prediction accuracy with less than 10% relative error for more than 12 times longer than a comparable least-squares-based method on a benchmark problem.

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Bayesian Identification of Hamiltonian Dynamics from Symplectic Data

Cited by 5 publications

References 19 publications

KAM Theory Meets Statistical Learning Theory: Hamiltonian Neural Networks with Non-zero Training Loss

KAM Theory Meets Statistical Learning Theory: Hamiltonian Neural Networks with Non-zero Training Loss

Quantum Control Modelling, Methods, and Applications

Bayesian Identification of Nonseparable Hamiltonian Systems Using Stochastic Dynamic Models

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