# DrBoumedieneHamzi

Faculty of Natural SciencesDepartment of Mathematics

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### Contact

+44 (0)20 7594 1424b.hamzi

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### Location

654Huxley BuildingSouth Kensington Campus

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## Publications

Publication Type
Year
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46 results found

Yang L, Sun X, Hamzi B, Owhadi H, Xie Net al., 2023, Learning Dynamical Systems from Data: A Simple Cross-Validation Perspective, Part V: Sparse Kernel Flows for 132 Chaotic Dynamical Systems

Regressing the vector field of a dynamical system from a finite number ofobserved states is a natural way to learn surrogate models for such systems. Asimple and interpretable way to learn a dynamical system from data is tointerpolate its vector-field with a data-adapted kernel which can be learned byusing Kernel Flows. The method of Kernel Flows is a trainable machine learningmethod that learns the optimal parameters of a kernel based on the premise thata kernel is good if there is no significant loss in accuracy if half of thedata is used. The objective function could be a short-term prediction or someother objective for other variants of Kernel Flows). However, this method islimited by the choice of the base kernel. In this paper, we introduce themethod of \emph{Sparse Kernel Flows } in order to learn the best'' kernel bystarting from a large dictionary of kernels. It is based on sparsifying akernel that is a linear combination of elemental kernels. We apply thisapproach to a library of 132 chaotic systems.

Journal article

Lee J, De Brouwer E, Hamzi B, Owhadi Het al., 2023, Learning dynamical systems from data: A simple cross-validation perspective, Part III: Irregularly-sampled time series, PHYSICA D-NONLINEAR PHENOMENA, Vol: 443, ISSN: 0167-2789

Journal article

Dingle K, Kamal R, Hamzi B, 2023, A note on a priori forecasting and simplicity bias in time series, Physica A: Statistical Mechanics and its Applications, Vol: 609, Pages: 128339-128339, ISSN: 0378-4371

Journal article

Haasdonk B, Hamzi B, Santin G, Wittwar Det al., 2021, Kernel methods for center manifold approximation and a weak data-based version of the Center Manifold Theorem, PHYSICA D-NONLINEAR PHENOMENA, Vol: 427, ISSN: 0167-2789

Journal article

Hamzi B, Owhadi H, 2021, Learning dynamical systems from data: A simple cross-validation perspective, part I: Parametric kernel flows, PHYSICA D-NONLINEAR PHENOMENA, Vol: 421, ISSN: 0167-2789

Journal article

Hamzi B, Maulik R, Owhadi H, 2021, Data-driven geophysical forecasting: Simple, low-cost, and accurate baselines with kernel methods

Modeling geophysical processes as low-dimensional dynamical systems andregressing their vector field from data is a promising approach for learningemulators of such systems. We show that when the kernel of these emulators isalso learned from data (using kernel flows, a variant of cross-validation),then the resulting data-driven models are not only faster than equation-basedmodels but are easier to train than neural networks such as the long short-termmemory neural network. In addition, they are also more accurate and predictivethan the latter. When trained on geophysical observational data, for example,the weekly averaged global sea-surface temperature, considerable gains are alsoobserved by the proposed technique in comparison to classical partialdifferential equation-based models in terms of forecast computational cost andaccuracy. When trained on publicly available re-analysis data for the dailytemperature of the North-American continent, we see significant improvementsover classical baselines such as climatology and persistence-based forecasttechniques. Although our experiments concern specific examples, the proposedapproach is general, and our results support the viability of kernel methods(with learned kernels) for interpretable and computationally efficientgeophysical forecasting for a large diversity of processes.

Journal article

Bittracher A, Klus S, Hamzi B, Koltai P, Schuette Cet al., 2021, Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds, Publisher: SPRINGER

Working paper

Colonius F, Hamzi B, 2020, Entropy for practical stabilization, Publisher: arXiv

For deterministic continuous-time nonlinear control systems,epsilon-practical stabilization entropy and practical stabilization entropy areintroduced. Here the rate of attraction is specified by a KL-function. Upperand lower bounds for the diverse entropies are proved, with special attentionto exponential KL-functions. Two scalar examples are analyzed in detail.

Working paper

Klus S, Nueske F, Hamzi B, 2020, Kernel-based approximation of the koopman generator and schrodinger operator, Entropy: international and interdisciplinary journal of entropy and information studies, Vol: 22, Pages: 1-22, ISSN: 1099-4300

Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.

Journal article

Giesl P, Hamzi B, Rasmussen M, Webster KNet al., 2020, Approximation of Lyapunov functions from noisy data, Journal of Computational Dynamics, Vol: 7, Pages: 57-81, ISSN: 2158-2491

Methods have previously been developed for the approximation of Lyapunovfunctions using radial basis functions. However these methods assume that theevolution equations are known. We consider the problem of approximating a givenLyapunov function using radial basis functions where the evolution equationsare not known, but we instead have sampled data which is contaminated withnoise. We propose an algorithm in which we first approximate the underlyingvector field, and use this approximation to then approximate the Lyapunovfunction. Our approach combines elements of machine learning/statisticallearning theory with the existing theory of Lyapunov function approximation.Error estimates are provided for our algorithm.

Journal article

Hamzi B, Abed EH, 2020, Local modal participation analysis of nonlinear systems using Poincare linearization, NONLINEAR DYNAMICS, Vol: 99, Pages: 803-811, ISSN: 0924-090X

Journal article

Haasdonk B, Hamzi B, Santin G, Wittwar Det al., 2020, Greedy Kernel Methods for Center Manifold Approximation, Publisher: Springer International Publishing

<jats:title>Abstract</jats:title><jats:p>For certain dynamical systems it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to a non-hyperbolic equilibrium point, and to obtain meaningful predictions of its behavior by analyzing a reduced dimensional problem. Since the manifold is usually not known, approximation methods are of great interest to obtain qualitative estimates. In this work, we use a data-based greedy kernel method to construct a suitable approximation of the manifold close to the equilibrium. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used to construct a surrogate model of the manifold. The method is tested on different examples which show promising performance and good accuracy.</jats:p>

Working paper

Hamzi B, Colonius F, 2019, Kernel methods for the approximation of discrete-time linear autonomous and control systems, SN Applied Sciences, Vol: 1, ISSN: 2523-3963

Methods from learning theory are used in the state space of linear dynamical and control systems in order to estimate relevant matrices and some relevant quantities such as the topological entropy. An application to stabilization via algebraic Riccati equations is included by viewing a control system as an autonomous system in an extended space of states and control inputs. Kernel methods are the main techniques used in this paper and the approach is illustrated via a series of numerical examples. The advantage of using kernel methods is that they allow to perform function approximation from data and, as illustrated in this paper, allow to approximate linear discrete-time autonomous and control systems from data.

Journal article

Hamzi B, Kuehn C, Mohamed S, 2019, A note on kernel methods for multiscale systems with critical transitions, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Vol: 42, Pages: 907-917, ISSN: 0170-4214

Journal article

Hamzi B, Colonius F, 2019, Kernel Methods for Discrete-Time Linear Equations, 19th Annual International Conference on Computational Science (ICCS), Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 178-191, ISSN: 0302-9743

Conference paper

Hamzi B, Abed E, 2018, A Note on Local Mode-in-State Participation Factors for Nonlinear Systems

The paper studies an extension to nonlinear systems of a recently proposedapproach to the concept of modal participation factors. First, a definition isgiven for local mode-in-state participation factors for smooth nonlinearautonomous systems. The definition is general, and, unlike in the moretraditional approach, the resulting participation measures depend on theassumed uncertainty law governing the system initial condition. The workfollows Hashlamoun, Hassouneh and Abed (2009) in taking a mathematicalexpectation (or set-theoretic average) of a modal contribution measure withrespect to an assumed uncertain initial state. As in the linear case, it isfound that a symmetry assumption on the distribution of the initial stateresults in a tractable calculation and an explicit and simple formula formode-in-state participation factors.

Journal article

Hamzi B, AlOtaiby TN, AlShebeili S, AlAnqary Aet al., 2018, Kernel Methods and the Maximum Mean Discrepancy for Seizure Detection, 1st International Conference on Computer Applications and Information Security (ICCAIS), Publisher: IEEE

Conference paper

Bouvrie J, Hamzi B, 2017, KERNEL METHODS FOR THE APPROXIMATION OF NONLINEAR SYSTEMS, SIAM JOURNAL ON CONTROL AND OPTIMIZATION, Vol: 55, Pages: 2460-2492, ISSN: 0363-0129

Journal article

Hamzi B, Lamb JSW, Lewis D, 2015, A Characterization of Normal Forms for Control Systems, JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS, Vol: 21, Pages: 273-284, ISSN: 1079-2724

Journal article

Hamzi B, Abed EH, 2014, Local Mode-in-State Participation Factors for Nonlinear Systems, 53rd IEEE Annual Conference on Decision and Control (CDC), Publisher: IEEE, Pages: 43-48, ISSN: 0743-1546

Conference paper

Bouvrie J, Hamzi B, 2012, Empirical Estimators for Stochastically Forced Nonlinear Systems: Observability, Controllability and the Invariant Measure, Proc. American Control Conference (ACC)

We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems - with a reasonable expectation of success - once the nonlinear system has been mapped into a high or infinite dimensional feature space. In particular, we develop computable, non-parametric estimators approximating controllability and observability energy functions for nonlinear systems, and study the ellipsoids they induce. In all cases the relevant quantities are estimated from simulated or observed data. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system.

Conference paper

Bouvrie J, Hamzi B, 2011, Model Reduction for Nonlinear Control Systems using Kernel Subspace Methods

We introduce a data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves linearly when lifted into a high (or infinite) dimensional feature space where balanced truncation may be carried out implicitly. This leads to a nonlinear reduction map which can be combined with a representation of the system belonging to a reproducing kernel Hilbert space to give a closed, reduced order dynamical system which captures the essential input-output characteristics of the original model.Empirical simulations illustrating the approach are also provided.

Journal article

Hamzi B, Krener AJ, 2007, The Controlled Center Systems, IEEE TRANSACTIONS ON AUTOMATIC CONTROL, Vol: 52, Pages: 2188-2192, ISSN: 0018-9286

Journal article

Hamzi B, Krener AJ, Kang W, 2006, The controlled center dynamics of discrete time control bifurcations, SYSTEMS & CONTROL LETTERS, Vol: 55, Pages: 585-596, ISSN: 0167-6911

Journal article

Hamzi B, Krener AJ, 2006, The controlled center systems, 45th IEEE Conference on Decision and Control, Publisher: IEEE, Pages: 6414-6418, ISSN: 0743-1546

Conference paper

Kang W, Hamzi B, Krener AJ, 2006, On the Convergence and Behavior of Three Dimensional Normal Forms, 1st IFAC Conference on the Analysis and Control of Chaotic Systems

Conference paper

Krener AJ, Kang W, Hamzi B, Tall Iet al., 2005, Low codimension control singularities for single input nonlinear systems, Conference in Honor of Clyde Martins 60th Birthday on New Directions and Applications in Control Theory, Publisher: SPRINGER-VERLAG BERLIN, Pages: 181-192, ISSN: 0170-8643

Conference paper

Hamzi B, Kang W, Krener AJ, 2005, The controlled center dynamics, MULTISCALE MODELING & SIMULATION, Vol: 3, Pages: 838-852, ISSN: 1540-3459

Journal article

Krener AJ, Kang W, Hamzi B, Tall IAet al., 2005, Low codimension control singularities for single input nonlinear systems, Publisher: Springer Verlag, ISBN: 9783540239536

This volume contains a collection of papers in control theory and applications presented at a conference in honor of Clyde Martin on the occasion of his 60th ...

Book

Hamzi B, Kang W, Barbot JP, 2004, Analysis and control of Hopf bifurcations, SIAM JOURNAL ON CONTROL AND OPTIMIZATION, Vol: 42, Pages: 2200-2220, ISSN: 0363-0129

Journal article

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