Error Bounds for Carleman Linearization of General Nonlinear Systems

Amini, Arash; Sun, Qiyu; Motee, Nader

doi:10.1137/1.9781611976847.1

Cited by 11 publications

(13 citation statements)

References 23 publications

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“…Recently, Amini et al [69] showed that the error of the Carleman linearization can be bounded exponentially by the truncation order N , but this statement does not hold for any arbitrary time. As is clearly described in the paper, provided with the assumption that the flow field can be approximated by a Maclaurin expansion, the error for a sufficiently short horizon is exponentially bounded.…”

Section: Discussion and Summarymentioning

confidence: 99%

Koopman von Neumann mechanics and the Koopman representation: A perspective on solving nonlinear dynamical systems with quantum computers

Lowrie¹,

Aslangil²,

Subaşı³

et al. 2022

Preprint

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A number of recent studies have proposed that linear representations are appropriate for solving nonlinear dynamical systems with quantum computers, which fundamentally act linearly on a wave function in a Hilbert space. Linear representations, such as the Koopman representation and Koopman von Neumann mechanics, have regained attention from the dynamical-systems research community. Here, we aim to present a unified theoretical framework, currently missing in the literature, with which one can compare and relate existing methods, their conceptual basis, and their representations. We also aim to show that, despite the fact that quantum simulation of nonlinear classical systems may be possible with such linear representations, a necessary projection into a feasible finite-dimensional space will in practice eventually induce numerical artifacts which can be hard to eliminate or even control. As a result, a practical, reliable and accurate way to use quantum computation for solving general nonlinear dynamical systems is still an open problem.1 Here, by 'solve', we mean to output a quantum state that efficiently encodes the classical solution vector. Thus approaches based on classical encodings such as in Ref.[27] are not covered here. We note that this state will, necessarily, need to be sampled many times to extract the desired information.

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Section: Discussion and Summarymentioning

confidence: 99%

Koopman von Neumann mechanics and the Koopman representation: A perspective on solving nonlinear dynamical systems with quantum computers

Lowrie¹,

Aslangil²,

Subaşı³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…The problem, as formulated above, is difficult to solve for general nonlinear systems. In this paper, inspired by [10], the formulation is restricted to a sub-class of nonlinear systems, defined by the following assumptions. Assumption 1.…”

Section: Problem Statementmentioning

confidence: 99%

“…While this assumption is restrictive, a large subclass of systems fall under this category, since polynomial, trigonometric, and exponential functions have Maclaurin expansions with exponential decay. The idea is to use Carleman lifting [9], [10] to lift the nonlinear system into an infinite-dimensional linear system such that the truncation of the trajectory of the linear system to the first d dimensions approximates the trajectory of the system in (1), in infinity norm. The infinite-dimensional linear system is then truncated to yield a finite-dimensional linear system such that the error between the projected trajectories and the original trajectories is less than the given error bound, ∆.…”

Section: Problem Statementmentioning

confidence: 99%

“…In [10], the authors used Carleman Linearization in conjunction with a known nonlinear model to develop a truncated linear system that approximates the nonlinear system in a region of attraction. Moreover, the trajectory of the truncated linear system is guaranteed to stay within a computable error bound around the trajectory of the nonlinear system.…”

Section: Introductionmentioning

confidence: 99%

“…The error bound can be computed a priori when the decay rate of the Maclaurin expansion of the nonlinear system and the order of the truncation are known. In this paper, the results in [10] are leveraged to develop a Carleman lifting-based approach to data-driven modeling of nonlinear systems. Given a set of trajectories generated by a nonlinear system, a desired terminal time τ * , and an upper bound of the decay rate of the Maclaurin expansion, a linear system of a higher order is identified such that the trajectory of the identified linear system stays within a guaranteed error bound around the trajectory of the nonlinear system, and the error bound can be prescribed a priori.…”

Section: Introductionmentioning

confidence: 99%

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Carleman Lifting for Nonlinear System Identification with Guaranteed Error Bounds

Abudia¹,

Rosenfeld²,

Kamalapurkar³

2022

Preprint

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This paper concerns identification of uncontrolled or closed loop nonlinear systems using a set of trajectories that are generated by the system in a domain of attraction. The objective is to ensure that the trajectories of the identified systems are close to the trajectories of the real system, as quantified by an error bound that is prescribed a priori. A majority of existing methods for nonlinear system identification rely on techniques such as neural networks, autoregressive moving averages, and spectral decomposition that do not provide systematic approaches to meet pre-defined error bounds. The developed method is based on Carleman linearization-based lifting of the nonlinear system to an infinite dimensional linear system. The linear system is then truncated to a suitable order, computed based on the prescribed error bound, and parameters of the truncated linear system are estimated from data. The effectiveness of the technique is demonstrated by identifying an approximation of the Van der Pol oscillator from data within a prescribed error bound.

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Reachability of Koopman Linearized Systems Using Random Fourier Feature Observables and Polynomial Zonotope Refinement

Bak

Bogomolov

Hencey

et al. 2022

Computer Aided Verification

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Koopman operator linearization approximates nonlinear systems of differential equations with higher-dimensional linear systems. For formal verification using reachability analysis, this is an attractive conversion, as highly scalable methods exist to compute reachable sets for linear systems. However, two main challenges are present with this approach, both of which are addressed in this work. First, the approximation must be sufficiently accurate for the result to be meaningful, which is controlled by the choice of observable functions during Koopman operator linearization. By using random Fourier features as observable functions, the process becomes more systematic than earlier work, while providing a higher-accuracy approximation. Second, although the higher-dimensional system is linear, simple convex initial sets in the original space can become complex non-convex initial sets in the linear system. We overcome this using a combination of Taylor model arithmetic and polynomial zonotope refinement. Compared with prior work, the result is more efficient, more systematic and more accurate.

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Error Bounds for Carleman Linearization of General Nonlinear Systems

Cited by 11 publications

References 23 publications

Koopman von Neumann mechanics and the Koopman representation: A perspective on solving nonlinear dynamical systems with quantum computers

Koopman von Neumann mechanics and the Koopman representation: A perspective on solving nonlinear dynamical systems with quantum computers

Carleman Lifting for Nonlinear System Identification with Guaranteed Error Bounds

Reachability of Koopman Linearized Systems Using Random Fourier Feature Observables and Polynomial Zonotope Refinement

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