基于统计学习的时空动力系统建模

随机偏微分方程的系统辨识是利用随机分布参数系统的观测数据去重构描述这个系统的未知的随机偏微分方程，它可以看作是对随机偏微分方程的”反向”的研究。偏微分动力学系统的辨识与建模是一个比较前沿，综合性的研究方向。本书对这个方向的研究成果进行了综述，对作者已有的一些重要工作进行了总结与延深，并为相关未来的研究提供有益的启迪。

Preface
Chapter 1 Overview of Statistical Learning Methods
1.1 A brief introduction of statistical learning
1.2 Linear model
1.2.1 Linear regression model
1.2.2 Regularized linear regression
1.2.3 Reproducing kernel model
References
Chapter 2 Online Kernel Learning of Nonlinear Spatiotemporal Systems
2.1 Motivation of this chapter
2.2 Discretization and lattice dynamic systems
2.3 MIMO partially linear model
2.4 The PM-RLS-SVM for MIMO partially linear systems
2.5 Numerical simulations and some discussions
2.6 Summary
References
Chapter 3 Learning of Partially Known Nonlinear Stochastic Spatiotemporal Dynamical Systems
3.1 Motivation of this chapter
3.2 Reproducing kernel methods for partially linear models
3.3 The extended partially linear model for SPDE
3.4 Extended partially ridge regression
3.5 Simulations and comparison
3.6 Summary
References
Chapter 4 Learning of Nonlinear Stochastic Spatiotemporal Dynamical Systems
4.1 Motivation of this chapter
4.2 Stochastic evolution equation and approximation error of FEM
4.3 Learning framework and the kernel learning method
4.4 Learning with irregular observation data
4.5 Simulations and comparison
4.6 Summary
References
Chapter 5 Learning of Nonlinear Spatiotemporal Dynamical Systems with Non-Uniform Observations
5.1 Motivation of this chapter
5.2 Discretization and non-uniform sampling problem
5.3 A multi-step learning method with non-uniform sampling data
5.4 Inverse meshless collocation model and learning algorithm
5.5 Numerical example
5.6 Summary
References
Chapter 6 Online Learning of Nonlinear Stochastic Spatiotemporal System with Multiplicative Noise
6.1 Motivation of this chapter
6.2 Discretization and heterogeneous partially linear model
6.3 Error dynamical system of PLM
6.4 Robust optimal control algorithm for error dynamical system
6.5 Numerical examples
6.6 Summary
References
Chapter 7 Robust Online Learning Method Based on Dynamical Linear Quadratic Regulator
7.1 Motivation of this chapter
7.2 Benchmark online learning methods
7.3 Online learning framework
7.4 Robust online learning method based on LQR
7.5 The online learning in kernel spaces
7.6 Numerical examples
7.7 Summary
References
Chapter 8 Approximate Controllability of Nonlinear Stochastic Partial Di.erential Systems
8.1 Motivation of this chapter
8.2 Basic concepts and preliminaries
8.3 The controllability results
8.4 Illustrative example
8.5 Summary
References