●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 242
Chapter 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
內容簡介
隨機偏微分方程的繫統辨識是利用隨機分布參數繫統的觀測數據去重構描述這個繫統的未知的隨機偏微分方程，它可以看作是對隨機偏微分方程的"反向"的研究。偏微分動力學繫統的辨識與建模是一個比較前沿，綜合性的研究方向。本書對這個方向的研究成果進行了綜述，對作者已有的一些重要工作進行了總結與延深，並為相關未來的研究提供有益的啟迪。
