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Contents Preface Chapter 1 Overview of Statistical Learning Methods 1 1.1 A brief introduction of statistical learning 1 1.2 Linear model 10 1.2.1 Linear regression model 10 1.2.2 Regularized linear regression 12 1.2.3 Reproducing kernel model 18 References 34 Chapter 2 Online Kernel Learning of Nonlinear Spatiotemporal Systems 38 2.1 Motivation of this chapter 38 2.2 Discretization and lattice dynamic systems 40 2.3 MIMO partially linear model 42 2.4 The PM-RLS-SVM for MIMO partially linear systems 44 2.5 Numerical simulations and some discussions 53 2.6 Summary 70 References 71 Chapter 3 Learning of Partially Known Nonlinear Stochastic Spatiotemporal Dynamical Systems 75 3.1 Motivation of this chapter 75 3.2 Reproducing kernel methods for partially linear models 78 3.3 The extended partially linear model for SPDE 80 3.4 Extended partially ridge regression 84 3.5 Simulations and comparison 92 3.6 Summary 100 References 101 Chapter 4 Learning of Nonlinear Stochastic Spatiotemporal Dynamical Systems 105 4.1 Motivation of this chapter 105 4.2 Stochastic evolution equation and approximation error of FEM 107 4.3 Learning framework and the kernel learning method 115 4.4 Learning with irregular observation data 121 4.5 Simulations and comparison 126 4.6 Summary 133 References 134 Chapter 5 Learning of Nonlinear Spatiotemporal Dynamical Systems with Non-Uniform Observations 139 5.1 Motivation of this chapter 139 5.2 Discretization and non-uniform sampling problem 141 5.3 A multi-step learning method with non-uniform sampling data 144 5.4 Inverse meshless collocation model and learning algorithm 156 5.5 Numerical example 164 5.6 Summary 170 References 171 Chapter 6 Online Learning of Nonlinear Stochastic Spatiotemporal System with Multiplicative Noise 176 6.1 Motivation of this chapter 176 6.2 Discretization and heterogeneous partially linear model 178 6.3 Error dynamical system of PLM 184 6.4 Robust optimal control algorithm for error dynamical system 191 6.5 Numerical examples 198 6.6 Summary 205 References 206 Chapter 7 Robust Online Learning Method Based on Dynamical Linear Quadratic Regulator 211 7.1 Motivation of this chapter 211 7.2 Benchmark online learning methods 213 7.3 Online learning framework 217 7.4 Robust online learning method based on LQR 220 7.5 The online learning in kernel spaces 225 7.6 Numerical examples 231 7.7 Summary 241 References 242Chapter 8 Approximate Controllability of Nonlinear Stochastic Partial Di.erential Systems 246 8.1 Motivation of this chapter 246 8.2 Basic concepts and preliminaries 247 8.3 The controllability results 250 8.4 Illustrative example 271 8.5 Summary 273 References 273