Data driven science and engineering

Preface
Acknowledgments
Optimization, Equations, Symbols, and Acronyms
I Dimensionality Reduction and Transforms
Singular Value Decomposition (SVD)
Overview
Matrix Approximation
Mathematical Properties and Manipulations
Pseudo-Inverse, Least-Squares, and Regression
Principal Component Analysis (PCA)
Eigenfaces Example
Truncation and Alignment
Randomized Singular Value Decomposition
Tensor Decompositions and N-Way Data Arrays
Fourier and Wavelet Transforms
Fourier Series and Fourier Transforms
Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT)
Transforming Partial Differential Equations
Gabor Transform and the Spectrogram
Laplace Transform
Wavelets and Multi-Resolution Analysis
Two-Dimensional Transforms and Image Processing
Sparsity and Compressed Sensing
Sparsity and Compression
Compressed Sensing
Compressed Sensing Examples
The Geometry of Compression
Sparse Regression
Sparse Representation
Robust Principal Component Analysis (RPCA)
Sparse Sensor Placement
II Machine Learning and Data Analysis
Regression and Model Selection
Classic Curve Fitting
Nonlinear Regression and Gradient Descent
Regression and Ax=b: Over- and Under-Determined Systems
Optimization as the Cornerstone of Regression
The Pareto Front and Lex Parsimoniae
Model Selection: Cross-Validation
Model Selection: Information Criteria
Clustering and Classification
Feature Selection and Data Mining
Supervised versus Unsupervised Learning
Unsupervised Learning: k-Means Clustering
Unsupervised Hierarchical Clustering: Dendrogram
Mixture Models and the Expectation-Maximization Algorithm
Supervised Learning and Linear Discriminants
Support Vector Machines (SVM)
Classification Trees and Random Forest
Top 10 Algorithms of Data Mining circa 2008 (Before the Deep Learning Revolution)
Neural Networks and Deep Learning
Neural Networks: Single-Layer Networks
Multi-Layer Networks and Activation Functions
The Backpropagation Algorithm
The Stochastic Gradient Descent Algorithm
Deep Convolutional Neural Networks
Neural Networks for Dynamical Systems
Recurrent Neural Networks
Autoencoders
Generative Adversarial Networks (GANs)
The Diversity of Neural Networks
III Dynamics and Control
Data-Driven Dynamical Systems
Overview, Motivations, and Challenges
Dynamic Mode Decomposition (DMD)
Sparse Identification of Nonlinear Dynamics (SINDy)
Koopman Operator Theory
Data-Driven Koopman Analysis
Linear Control Theory
Closed-Loop Feedback Control
Linear Time-Invariant Systems
Controllability and Observability
Optimal Full-State Control: Linear–Quadratic Regulator (LQR)
Optimal Full-State Estimation: the Kalman Filter
Optimal Sensor-Based Control: Linear–Quadratic Gaussian (LQG)
Case Study: Inverted Pendulum on a Cart
Robust Control and Frequency-Domain Techniques
Balanced Models for Control
Model Reduction and System Identification
Balanced Model Reduction
System Identification
IV Advanced Data-Driven Modeling and Control
Data-Driven Control
Model Predictive Control (MPC)
Nonlinear System Identification for Control
Machine Learning Control
Adaptive Extremum-Seeking Control
Reinforcement Learning
Overview and Mathematical Formulation
Model-Based Optimization and Control
Model-Free Reinforcement Learning and Q-Learning
Deep Reinforcement Learning
Applications and Environments
Optimal Nonlinear Control
Reduced-Order Models (ROMs)
Proper Orthogonal Decomposition (POD) for Partial Differential Equations
Optimal Basis Elements: the POD Expansion
POD and Soliton Dynamics
Continuous Formulation of POD
POD with Symmetries: Rotations and Translations
Neural Networks for Time-Stepping with POD
Leveraging DMD and SINDy for POD-Galerkin
Interpolation for Parametric Reduced-Order Models
Gappy POD
Error and Convergence of Gappy POD
Gappy Measurements: Minimize Condition Number
Gappy Measurements: Maximal Variance
POD and the Discrete Empirical Interpolation Method (DEIM)
DEIM Algorithm Implementation
Decoder Networks for Interpolation
Randomization and Compression for ROMs
Machine Learning ROMs
Physics-Informed Machine Learning
Mathematical Foundations
SINDy Autoencoder: Coordinates and Dynamics
Koopman Forecasting
Learning Nonlinear Operators
Physics-Informed Neural Networks (PINNs)
Learning Coarse-Graining for PDEs
Deep Learning and Boundary Value Problems
Glossary
References
Index