Kernel Methods for the Approximation of Nonlinear Systems

DOI10.1137/14096815XzbMath1368.93248arXiv1108.2903OpenAlexW2963969307MaRDI QIDQ5348477

Jake Bouvrie, Boumediene Hamzi

Publication date: 18 August 2017

Published in: SIAM Journal on Control and Optimization (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1108.2903

zbMATH Keywords

nonlinear systemsapproximationbalancingkernel methodsmachine learningGramiansobservability energycontrollability energy

Mathematics Subject Classification ID

Learning and adaptive systems in artificial intelligence (68T05) Nonlinear systems in control theory (93C10) Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) (41A65) Time series analysis of dynamical systems (37M10) Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) (46E22) Control/observation systems governed by ordinary differential equations (93C15) Kernel functions in one complex variable and applications (30C40)

Related Items (18)

A long short-term memory embedding for hybrid uplifted reduced order models ⋮ Balanced Truncation Model Reduction for Lifted Nonlinear Systems ⋮ Kernel methods for the approximation of some key quantities of nonlinear systems ⋮ Kernel Methods for the Approximation of Nonlinear Systems ⋮ Model Reduction for Nonlinear Systems by Balanced Truncation of State and Gradient Covariance ⋮ Learning dynamical systems from data: a simple cross-validation perspective. III: Irregularly-sampled time series ⋮ Learning dynamical systems from data: a simple cross-validation perspective. IV: Case with partial observations ⋮ Approximation of Lyapunov functions from noisy data ⋮ A note on microlocal kernel design for some slow-fast stochastic differential equations with critical transitions and application to EEG signals ⋮ Dimensionality reduction of complex metastable systems via kernel embeddings of transition manifolds ⋮ Block Basis Factorization for Scalable Kernel Evaluation ⋮ \texttt{emgr} -- the empirical Gramian framework ⋮ 9 Kernel methods for surrogate modeling ⋮ Kernel methods for center manifold approximation and a weak data-based version of the center manifold theorem ⋮ Learning dynamical systems from data: a simple cross-validation perspective. I: Parametric kernel flows ⋮ New characterizations of reproducing kernel Hilbert spaces and applications to metric geometry ⋮ One-shot learning of stochastic differential equations with data adapted kernels ⋮ Lift \& learn: physics-informed machine learning for large-scale nonlinear dynamical systems

Uses Software

• emgr

Cites Work

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• Balancing for nonlinear systems
• Principal component analysis.
• Regularization networks and support vector machines
• Computation of nonlinear balanced realization and model reduction based on Taylor series expansion
• Shannon sampling. II: Connections to learning theory
• Remarks to Maurice Frechet's article ``Sur la definition axiomatique d'une classe d'espaces vectoriels distancies applicables vectoriellement sur l'espace de Hilbert
• On certain metric spaces arising from euclidean spaces by a change of metric and their imbedding in Hilbert space
• On the mathematical foundations of learning
• A new set of invariants for linear systems--Application to reduced order compensator design
• Balanced Realization and Model Order Reduction for Nonlinear Systems Based on Singular Value Analysis
• Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems
• Kernel techniques: From machine learning to meshless methods
• Support Vector Machines
• Capacity of reproducing kernel spaces in learning theory
• Reduced Order Modeling of Nonlinear Control Systems
• ONLINE LEARNING WITH MARKOV SAMPLING
• Necessary and sufficient conditions for balancing unstable systems
• Balancing transformations for unstable nonminimal linear systems
• Principal component analysis in linear systems: Controllability, observability, and model reduction
• Numerical Solution of the Stable, Non-negative Definite Lyapunov Equation Lyapunov Equation
• ESTIMATING THE APPROXIMATION ERROR IN LEARNING THEORY
• A subspace approach to balanced truncation for model reduction of nonlinear control systems
• Shannon sampling and function reconstruction from point values
• Nonlinear Time Series Analysis
• Kernel Methods for the Approximation of Nonlinear Systems
• Approximation of Large-Scale Dynamical Systems
• Metric Spaces and Positive Definite Functions
• Theory of Reproducing Kernels

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