实例介绍
【实例简介】数据驱动的科学与工程-机器学习、动态系统与控制
【实例截图】
【核心代码】
Contents
Preface vi
Common Optimization Techniques, Equations, Symbols, and Acronyms x
I Dimensionality Reduction and Transforms 1
1 Singular Value Decomposition 3
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2 Matrix approximation . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3 Mathematical properties and manipulations . . . . . . . . . . . . 12
1.4 Pseudo-inverse, least-squares, and regression . . . . . . . . . . . . 17
1.5 Principal component analysis (PCA) . . . . . . . . . . . . . . . . . 24
1.6 Eigenfaces example . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.7 Truncation and alignment . . . . . . . . . . . . . . . . . . . . . . . 34
1.8 Randomized singular value decomposition . . . . . . . . . . . . . 42
1.9 Tensor decompositions and N-way data arrays . . . . . . . . . . . 47
2 Fourier and Wavelet Transforms 54
2.1 Fourier series and Fourier transforms . . . . . . . . . . . . . . . . 55
2.2 Discrete Fourier transform (DFT) and fast Fourier transform (FFT) 65
2.3 Transforming partial differential equations . . . . . . . . . . . . . 73
2.4 Gabor transform and the spectrogram . . . . . . . . . . . . . . . . 80
2.5 Wavelets and multi-resolution analysis . . . . . . . . . . . . . . . . 85
2.6 2D transforms and image processing . . . . . . . . . . . . . . . . . 88
3 Sparsity and Compressed Sensing 96
3.1 Sparsity and compression . . . . . . . . . . . . . . . . . . . . . . . 97
3.2 Compressed sensing . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.3 Compressed sensing examples . . . . . . . . . . . . . . . . . . . . 106
3.4 The geometry of compression . . . . . . . . . . . . . . . . . . . . . 110
3.5 Sparse regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.6 Sparse representation . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.7 Robust principal component analysis (RPCA) . . . . . . . . . . . . 123
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CONTENTS
3.8 Sparse sensor placement . . . . . . . . . . . . . . . . . . . . . . . . 125
II Machine Learning and Data Analysis 132
4 Regression and Model Selection 134
4.1 Classic curve fifitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.2 Nonlinear regression and gradient descent . . . . . . . . . . . . . 142
4.3 Regression and Ax = b: Over- and under-determined systems . . 149
4.4 Optimization as the cornerstone of regression . . . . . . . . . . . . 155
4.5 The Pareto front and Lex Parsimoniae . . . . . . . . . . . . . . . . . 162
4.6 Model selection: Cross validation . . . . . . . . . . . . . . . . . . . 166
4.7 Model selection: Information criteria . . . . . . . . . . . . . . . . . 172
5 Clustering and Classifification 178
5.1 Feature selection and data mining . . . . . . . . . . . . . . . . . . 179
5.2 Supervised versus unsupervised learning . . . . . . . . . . . . . . 185
5.3 Unsupervised learning: k-means clustering . . . . . . . . . . . . . 189
5.4 Unsupervised hierarchical clustering: Dendrogram . . . . . . . . 194
5.5 Mixture models and the expectation-maximization algorithm . . 198
5.6 Supervised learning and linear discriminants . . . . . . . . . . . . 203
5.7 Support vector machines (SVM) . . . . . . . . . . . . . . . . . . . . 209
5.8 Classifification trees and random forest . . . . . . . . . . . . . . . . 214
5.9 Top 10 algorithms in data mining 2008 . . . . . . . . . . . . . . . . 220
6 Neural Networks and Deep Learning 226
6.1 Neural networks: 1-Layer networks . . . . . . . . . . . . . . . . . 227
6.2 Multi-layer networks and activation functions . . . . . . . . . . . 232
6.3 The backpropagation algorithm . . . . . . . . . . . . . . . . . . . . 237
6.4 The stochastic gradient descent algorithm . . . . . . . . . . . . . . 242
6.5 Deep convolutional neural networks . . . . . . . . . . . . . . . . . 245
6.6 Neural networks for dynamical systems . . . . . . . . . . . . . . . 250
6.7 The diversity of neural networks . . . . . . . . . . . . . . . . . . . 255
III Dynamics and Control 264
7 Data-Driven Dynamical Systems 266
7.1 Overview, motivations, and challenges . . . . . . . . . . . . . . . . 267
7.2 Dynamic mode decomposition (DMD) . . . . . . . . . . . . . . . . 274
7.3 Sparse identifification of nonlinear dynamics (SINDy) . . . . . . . . 288
7.4 Koopman operator theory . . . . . . . . . . . . . . . . . . . . . . . 299
7.5 Data-driven Koopman analysis . . . . . . . . . . . . . . . . . . . . 312
Copyright © 2017 Brunton & Kutz. All Rights Reserved.CONTENTS
v
8 Linear Control Theory 323
8.1 Closed-loop feedback control . . . . . . . . . . . . . . . . . . . . . 325
8.2 Linear time-invariant systems . . . . . . . . . . . . . . . . . . . . . 330
8.3 Controllability and observability . . . . . . . . . . . . . . . . . . . 336
8.4 Optimal full-state control: linear quadratic regulator (LQR) . . . . 343
8.5 Optimal full-state estimation: The Kalman fifilter . . . . . . . . . . 347
8.6 Optimal sensor-based control:
Linear quadratic Gaussian (LQG) . . . . . . . . . . . . . . . . . . . 350
8.7 Case study: Inverted pendulum on a cart . . . . . . . . . . . . . . 352
8.8 Robust control and frequency domain techniques . . . . . . . . . 362
9 Balanced Models for Control 376
9.1 Model reduction and system identifification . . . . . . . . . . . . . 376
9.2 Balanced model reduction . . . . . . . . . . . . . . . . . . . . . . . 377
9.3 System identifification . . . . . . . . . . . . . . . . . . . . . . . . . . 393
10 Data-Driven Control 405
10.1 Nonlinear system identifification for control . . . . . . . . . . . . . 406
10.2 Machine learning control . . . . . . . . . . . . . . . . . . . . . . . . 414
10.3 Adaptive extremum-seeking control . . . . . . . . . . . . . . . . . 427
IV Reduced Order Models 438
11 Reduced Order Models (ROMs) 440
11.1 POD for partial differential equations . . . . . . . . . . . . . . . . 440
11.2 Optimal basis elements: The POD expansion . . . . . . . . . . . . 447
11.3 POD and soliton dynamics . . . . . . . . . . . . . . . . . . . . . . . 453
11.4 Continuous formulation of POD . . . . . . . . . . . . . . . . . . . 459
11.5 POD with symmetries: Rotations and translations . . . . . . . . . 464
12 Interpolation for Parametric ROMs 473
12.1 Gappy POD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
12.2 Error and convergence of gappy POD . . . . . . . . . . . . . . . . 480
12.3 Gappy measurements: Minimize condition number . . . . . . . . 485
12.4 Gappy measurements: Maximal variance . . . . . . . . . . . . . . 492
12.5 POD and the discrete empirical interpolation method (DEIM) . . 497
12.6 DEIM algorithm implementation . . . . . . . . . . . . . . . . . . . 501
12.7 Machine learning ROMs . . . . . . . . . . . . . . . . . . . . . . . . 504
Glossary 512
References 521
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