Approximation of Lyapunov functions from noisy data

Giesl, Peter; Hamzi, Boumediene; Rasmussen, Martin; Webster, Kevin

doi:10.3934/jcd.2020003

Cited by 27 publications

(19 citation statements)

References 24 publications

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“…The training and prediction results are shown in the following table with R 1 the RMSE corresponding to 50,000 points with initial conditions (0.5, 1.5, 2.5). i. Convergence results that characterize the error estimates of the difference between a dynamical system and its approximation from data using kernel methods can be found in [7,18].…”

Section: Example 4 (The Lorenz System)mentioning

confidence: 99%

“…On the other hand, RNN-LSTM were observed to be accurate for estimating Lyapunov exponents but not as good as reservoir computing for predictions (see [14] for a survey). Although Reproducing Kernel Hilbert Spaces (RKHS) [16] have provided strong mathematical foundations for analyzing dynamical systems [5,6,8,20,24,7,18,4,22,23,2], the accuracy of these emulators depends on the kernel and the problem of selecting a good kernel has received less attention.…”

Section: Introductionmentioning

confidence: 99%

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Learning dynamical systems from data: A simple cross-validation perspective, part I: Parametric kernel flows

Hamzi

Owhadi²

2021

Physica D: Nonlinear Phenomena

Self Cite

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This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Section: Example 4 (The Lorenz System)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Learning dynamical systems from data: A simple cross-validation perspective, part I: Parametric kernel flows

Hamzi

Owhadi²

2021

Physica D: Nonlinear Phenomena

Self Cite

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show abstract

“…Learning safety certificates A wide body of work addresses learning Lyapunov [9,13,7,28,24,6,27] and barrier [37,29,12] functions, as well as contraction metrics [33,23,32] and contracting vector fields [31,14] from data. While the generality and strength of guarantees provided vary (see the literature review of Boffi et al [4] for a detailed exposition), all of the aforementioned works consider nominally specified systems without uncertainty, whereas our approach explicitly considers perturbations that can capture model uncertainty and process noise.…”

Section: Related Workmentioning

confidence: 99%

Adversarially Robust Stability Certificates can be Sample-Efficient

Zhang¹,

Tu²,

Boffi³

et al. 2021

Preprint

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Motivated by bridging the simulation to reality gap in the context of safety-critical systems, we consider learning adversarially robust stability certificates for unknown nonlinear dynamical systems. In line with approaches from robust control, we consider additive and Lipschitz bounded adversaries that perturb the system dynamics. We show that under suitable assumptions of incremental stability on the underlying system, the statistical cost of learning an adversarial stability certificate is equivalent, up to constant factors, to that of learning a nominal stability certificate. Our results hinge on novel bounds for the Rademacher complexity of the resulting adversarial loss class, which may be of independent interest. To the best of our knowledge, this is the first characterization of sample-complexity bounds when performing adversarial learning over data generated by a dynamical system. We further provide a practical algorithm for approximating the adversarial training algorithm, and validate our findings on a damped pendulum example.

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“…The first two issues will cause trouble for existing Lyapunov-based ROA analysis methods using semidefinite programming (Yin et al, 2021;Hu et al, 2020;Jin and Lavaei, 2020;Aydinoglu et al, 2021) or mixed-integer programs (Chen et al, 2020(Chen et al, , 2021Dai et al, 2021). Due to the last issue, the methods of Lyapunov neural networks (Richards et al, 2018;Chang et al, 2019; or other stability certificate learning methods (Kenanian et al, 2019;Giesl et al, 2020;Ravanbakhsh and Sankaranarayanan, 2019) may also be not applicable since these methods typically require the control action to depend on the current state. Our goal is to develop a ROA analysis method which can address the above three issues simultaneously.…”

Section: Introductionmentioning

confidence: 99%

Revisiting PGD Attacks for Stability Analysis of Large-Scale Nonlinear Systems and Perception-Based Control

Havens¹,

Keivan²,

Seiler³

et al. 2022

Preprint

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Many existing region-of-attraction (ROA) analysis tools find difficulty in addressing feedback systems with large-scale neural network (NN) policies and/or high-dimensional sensing modalities such as cameras. In this paper, we tailor the projected gradient descent (PGD) attack method developed in the adversarial learning community as a general-purpose ROA analysis tool for large-scale nonlinear systems and end-to-end perception-based control. We show that the ROA analysis can be approximated as a constrained maximization problem whose goal is to find the worst-case initial condition which shifts the terminal state the most. Then we present two PGD-based iterative methods which can be used to solve the resultant constrained maximization problem. Our analysis is not based on Lyapunov theory, and hence requires minimum information of the problem structures. In the model-based setting, we show that the PGD updates can be efficiently performed using backpropagation. In the model-free setting (which is more relevant to ROA analysis of perception-based control), we propose a finite-difference PGD estimate which is general and only requires a blackbox simulator for generating the trajectories of the closed-loop system given any initial state. We demonstrate the scalability and generality of our analysis tool on several numerical examples with large-scale NN policies and high-dimensional image observations. We believe that our proposed analysis serves as a meaningful initial step toward further understanding of closed-loop stability of large-scale nonlinear systems and perception-based control.

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Approximation of Lyapunov functions from noisy data

Cited by 27 publications

References 24 publications

Learning dynamical systems from data: A simple cross-validation perspective, part I: Parametric kernel flows

Learning dynamical systems from data: A simple cross-validation perspective, part I: Parametric kernel flows

Adversarially Robust Stability Certificates can be Sample-Efficient

Revisiting PGD Attacks for Stability Analysis of Large-Scale Nonlinear Systems and Perception-Based Control

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