Finite-data error bounds for Koopman-based prediction and control

Nüske, Feliks; Peitz, Sebastian; Philipp, Friedrich; Schaller, Manuel; Worthmann, Karl

doi:10.48550/arxiv.2108.07102

Cited by 5 publications

(11 citation statements)

References 26 publications

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“…A wealth of numerical methods for the solution of the control problem (10) exist, for instance using dynamic programming [30,31] or Monte Carlo sampling [32]. More recently, reformulations as deterministic control problems via the Koopman generator, which allow for a significant reduction of the complexity, were proposed [13,33,34]. For an application to quantum control, see [35].…”

Section: Solution Of the Control Problemmentioning

confidence: 99%

Koopman analysis of quantum systems

Klus,

Nüske,

Peitz

2022

Preprint

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Koopman operator theory has been successfully applied to problems from various research areas such as fluid dynamics, molecular dynamics, climate science, engineering, and biology. Applications include detecting metastable or coherent sets, coarse-graining, system identification, and control. There is an intricate connection between dynamical systems driven by stochastic differential equations and quantum mechanics. In this paper, we compare the ground-state transformation and Nelson's stochastic mechanics and demonstrate how data-driven methods developed for the approximation of the Koopman operator can be used to analyze quantum physics problems. Moreover, we exploit the relationship between Schrödinger operators and stochastic control problems to show that modern data-driven methods for stochastic control can be used to solve the stationary or imaginary-time Schrödinger equation. Our findings open up a new avenue towards solving Schrödinger's equation using recently developed tools from data science. * These authors contributed equally to this work 1 A certain relationship between the basic wave mechanical equation and the Fokker equation, as well as the related statistical concepts, has probably struck anyone who is sufficiently familiar with both ideas.

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Section: Solution Of the Control Problemmentioning

confidence: 99%

Koopman analysis of quantum systems

Klus,

Nüske,

Peitz

2022

Preprint

Self Cite

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“…We briefly sketch the main steps of the bilinear surrogate modeling approach presented in [33], for which a finite-data error estimate was given in [32]. Considering a control u ∈ L ∞ ([0, T ], R nc ), it turns out that by control affinity of the system, also the Koopman generators are control affine, cf.…”

Section: Bilinear Edmd-based Approximation Of Control Systemsmentioning

confidence: 99%

“…In our work [32], we provided first quantitative error bounds for approximating control systems by means of bilinear eDMD-based surrogate modeling of control systems as described in Subsection 2.2. Our results were formulated for the broad class of Stochastic Differential Equations (SDEs) from which, the ODE-dynamics (1) can be obtained as a particular case.…”

Section: Finite-data Error Bounds Koopman-based Controlmentioning

confidence: 99%

“…Constrained data-based control with guarantees. In this paper, we propose an approach based on recently-derived error estimates [32] for control-affine systems governed by ordinary (and/or stochastic) differential equations to successfully accomplish this task. To this end, we proceed as follows.…”

Section: Introduction and Outlinementioning

confidence: 99%

“…data points using [39]. 1 The respective error estimates are generalized to the control setting in [32] using ideas from [33] such that a bilinear surrogate model is obtained. This error decays with an increasing number of data points.…”

Section: Introduction and Outlinementioning

confidence: 99%

See 2 more Smart Citations

Towards efficient and reliable prediction-based control using eDMD

Schaller¹,

Worthmann²,

Philipp³

et al. 2022

Preprint

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We present an approach for guaranteed constraint satisfaction by means of data-based optimal control, where the model is unknown and has to be obtained from measurement data. To this end, we utilize the Koopman framework and an eDMD-based bilinear surrogate modeling approach for control systems to show an error bound on predicted observables, i.e., functions of the state. This result is then applied to the constraints of the optimal control problem to show that satisfaction of tightened constraints in the purely data-based surrogate model implies constraint satisfaction for the original system. Keywords. Approximation error, control of constrained systems, data-based control, eDMD, finite-data error quantization, Koopman operator, MPC, nonlinear predictive control Notation: Let N 0 and R denote the natural (including zero) and the real numbers, respectively. Further, let C k ([0, T ], R), k ∈ N 0 , and L ∞ ([0, T ], R) be the spaces of k-times continuously differentiable and of Lebesgue-measurable, essentially bounded functions, respectively.

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Koopman analysis of quantum systems*

Klus¹,

Nüske²,

Peitz³

2022

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Koopman operator theory has been successfully applied to problems from various research areas such as fluid dynamics, molecular dynamics, climate science, engineering, and biology. Applications include detecting metastable or coherent sets, coarse-graining, system identification, and control. There is an intricate connection between dynamical systems driven by stochastic differential equations and quantum mechanics. In this paper, we compare the ground-state transformation and Nelson’s stochastic mechanics and demonstrate how data-driven methods developed for the approximation of the Koopman operator can be used to analyze quantum physics problems. Moreover, we exploit the relationship between Schrödinger operators and stochastic control problems to show that modern data-driven methods for stochastic control can be used to solve the stationary or imaginary-time Schrödinger equation. Our findings open up a new avenue toward solving Schrödinger’s equation using recently developed tools from data science.

show abstract

Finite-data error bounds for Koopman-based prediction and control

Cited by 5 publications

References 26 publications

Koopman analysis of quantum systems

Koopman analysis of quantum systems

Towards efficient and reliable prediction-based control using eDMD

Koopman analysis of quantum systems*

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