Stochastic Methods for Modeling and Predicting Complex Dynamical Systems: Uncertainty Quantification, State Estimation, and Reduced-Order Models (Synthesis Lectures on Mathematics & Statistics)

Description

This Second Edition is an essential guide to understanding, modeling, and predicting complex dynamical systems using new methods with stochastic tools. Expanding upon the original book, the author covers a unique combination of qualitative and quantitative modeling skills, novel efficient computational methods, rigorous mathematical theory, as well as physical intuitions and thinking. The author presents mathematical tools for understanding, modeling, and predicting complex dynamical systems using various suitable stochastic tools. The book provides practical examples and motivations when introducing these tools, merging mathematics, statistics, information theory, computational science, and data science. The author emphasizes the balance between computational efficiency and modeling accuracy while equipping readers with the skills to choose and apply stochastic tools to a wide range of disciplines. This second edition includes updated discussion of combining stochastic models with machine learning and addresses several additional topics, including importance sampling, regression, and maximum likelihood estimate. The author also introduces a new chapter on optimal control. Preface Acknowledgements Contents 1 Stochastic Toolkits 1.1 Review of Basic Probability Concepts 1.1.1 Random Variable, Probability Measure, and Probability Density Function (PDF) 1.1.2 Gaussian Distribution 1.1.3 Moments and Non-Gaussian Distributions 1.1.4 Law of Large Numbers and Central Limit Theorem 1.2 Stochastic Processes 1.3 Stochastic Differential Equations (SDEs) 1.3.1 Itô Stochastic Integral 1.3.2 Itô's Formula 1.3.3 Fokker-Planck Equation 1.4 Markov Jump Processes 1.4.1 A Motivating Example of Intermittent Time Series 1.4.2 Finite-State Markov Jump Process 1.4.3 Master Equation 1.4.4 Switching Times 1.5 Chaotic Systems 2 Introduction to Information Theory 2.1 Shannon's Entropy and Maximum Entropy Principle 2.1.1 Shannon's Intuition from the Theory of Communication 2.1.2 Definition of Shannon's Entropy 2.1.3 Shannon's Entropy in Gaussian Framework 2.1.4 Maximum Entropy Principle 2.1.5 Coarse Graining and the Loss of Information 2.2 Relative Entropy, Quantifying the Model Error and Additional Lack of Information 2.2.1 Definition of Relative Entropy 2.2.2 Relative Entropy in Gaussian Framework 2.2.3 Maximum Relative Entropy Principle 2.3 Mutual Information 2.3.1 Definition of Mutual Information 2.3.2 Mutual Information in Gaussian Framework 2.4 Relationship Between Information and Path-Wise Measurements 2.4.1 Two Widely Used Path-Wise Measurements: Root-Mean-Square Error (RMSE) and Pattern Correction 2.4.2 Relationship Between Shannon's Entropy of Residual and RMSE 2.4.3 Relationship Between Mutual Information and Pattern Correlation 2.4.4 Why Relative Entropy Is Important? 3 Basic Stochastic Computational Methods 3.1 Monte Carlo Method 3.1.1 The Basic Idea 3.1.2 A Simple Example 3.1.3 Error Estimates for the Monte Carlo Method 3.2 Importance Sampling 3.2.1 The Method 3.2.2 Example: Estimating the Tail Probability 3.2.3 Example: Importance Sampling with Different Proposal Distributions 3.3 Numerical Schemes for Solving SDEs 3.3.1 Euler-Maruyama Scheme 3.3.2 Milstein Scheme 3.4 Ensemble Method for Solving the Statistics of SDEs 3.5 Kernel Density Estimation 4 Simple Gaussian and Non-Gaussian SDEs 4.1 Linear Gaussian SDEs 4.1.1 Reynolds Decomposition and Time Evolution of the Moments 4.1.2 Statistical Equilibrium State and Decorrelation Time 4.1.3 Fokker-Planck Equation 4.2 Non-Gaussian SDE: Linear Model with Multiplicative Noise 4.2.1 Solving the Exact Path-Wise Solution 4.2.2 Equilibrium Distribution 4.2.3 Time Evolution of the Moments 4.3 Non-Gaussian SDE: Scalar Model with Cubic Nonlinearity … 4.3.1 Equilibrium PDF 4.3.2 Time Evolution of the Moments 4.3.3 Quasi-Gaussian Closure for the Moment Equations Associated with the Cubic Model 4.4 Nonlinear SDEs with Exactly Solvable Conditional Moments 4.4.1 The Coupled Nonlinear SDE System 4.4.2 Derivation of the Exact Solvable Conditional Moments 5 Data Assimilation 5.1 Introduction 5.2 Kalman Filter 5.2.1 One-dimensional Kalman Filter: Basic Idea of Data Assimilation 5.2.2 A Simple Example 5.2.3 Multi-dimensional Case 5.2.4 Some Remarks 5.3 Nonlinear Filters 5.3.1 Extended Kalman Filter 5.3.2 Ensemble Kalman Filter 5.3.3 A Numerical Example 5.3.4 Particle Filter 5.4 Continuous-in-Time Version: Kalman-Bucy Filter 5.5 Other Data Assimilation Methods and Applications 6 Optimal Control 6.1 Introduction 6.2 Linear Quadratic Regulator (LQR) 6.2.1 Problem Setup 6.2.2 Dynamic Programming (DP) 6.2.3 Solving the LQR Using DP 6.3 Linear Quadratic Stochastic Control with Noisy State Observation 6.3.1 Summary of the Main Results 6.3.2 Using Kalman Filter for the Current State Estimate 6.3.3 Solution via DP 6.4 General Optimal Control Framework 6.4.1 The Theory 6.4.2 Examples 6.5 Statistical Control 7 Prediction 7.1 Ensemble Forecast 7.1.1 Trajectory Forecast Versus Ensemble Forecast 7.1.2 Lead Time and Ensemble Mean Forecast Time Series 7.2 Model Error, Internal Predictability and Prediction Skill 7.2.1 Important Factors for Useful Prediction 7.2.2 Quantifying the Predictability and Model Error via Information Criteria 7.3 Procedure of Designing Suitable Forecast Models 7.4 Predicting Model Response via Fluctuation-Dissipation Theorem 7.4.1 The FDT Framework 7.4.2 Approximate FDT Methods 7.4.3 A Nonlinear Example 7.5 Finding the Most Sensitive Change Directions via Information Theory 7.5.1 The Mathematical Framework Using Fisher Information 7.5.2 Examples 7.5.3 Practical Strategies Utilizing FDT 8 Data-Driven Low-Order Stochastic Models 8.1 Motivations 8.2 Complex Ornstein–Uhlenbeck (OU) Process 8.2.1 Calibration of the OU Process 8.2.2 Compensating the Effect of Complicated Nonlinearity by Simple Stochastic Noise 8.2.3 Application: Reproducing the Statistics of the Two-Layer Quasi-Geostrophic Turbulence 8.3 Combining Stochastic Models with Linear Analysis … 8.4 Linear Stochastic Model with Multiplicative Noise 8.4.1 Exact Formula for Model Calibration 8.4.2 Approximating Highly Nonlinear Time Series 8.4.3 Application: Characterizing an El Niño Index 8.5 Stochastically Parameterized Models 8.5.1 The Necessity of Stochastic Parameterization 8.5.2 The Stochastic Parameterized Extended Kalman Filter (SPEKF) Model 8.5.3 Filtering Intermittent Time Series 8.6 Physics-Constrained Nonlinear Regression Models 8.6.1 Motivations 8.6.2 The General Framework of Physics-Constrained Nonlinear Regression Models 8.6.3 Comparison of the Models with and Without Physics Constraints 8.7 Discussion: Linear and Gaussian Approximations for Nonlinear Systems 9 Conditional Gaussian Nonlinear Systems 9.1 Overview of the Conditional Gaussian Nonlinear System (CGNS) 9.1.1 The Mathematical Framework 9.1.2 Examples of Complex Dynamical Systems Belonging to the CGNS 9.1.3 The Mathematical and Physical Reasonings Behind the CGNS 9.2 Closed Analytic Formulae for Solving the Conditional Statistics 9.2.1 Nonlinear Optimal Filtering 9.2.2 Nonlinear Optimal Smoothing 9.2.3 Nonlinear Optimal Conditional Sampling 9.2.4 Example: Comparison of Filtering, Smoothing and Sampling 9.3 Lagrangian Data Assimilation 9.3.1 The Mathematical Setup 9.3.2 Filtering Compressible and Incompressible Flows 9.3.3 Uncertainty Quantification Using Information Theory 9.4 Solving High-Dimensional Fokker-Planck Equations 9.4.1 The Basic Algorithm 9.4.2 A Simple Example 9.4.3 Block Decomposition 9.4.4 Statistical Symmetry 9.4.5 Application to FDT 9.5 Application: Modeling and Predicting Monsoon Intraseasonal Oscillation (MISO) 9.5.1 MISO Indices from a Nonlinear Data Decomposition Technique 9.5.2 Data-Driven Physics-Constrained Nonlinear Model 9.5.3 Data Assimilation and Prediction 10 Parameter Estimation with Uncertainty Quantification 10.1 Markov Chain Monte Carlo 10.1.1 The Metropolis Algorithm 10.1.2 A Simple Example 10.1.3 Parameter Estimation via MCMC 10.1.4 MCMC with Data Augmentation 10.2 Linear Regression and Maximum Likelihood Estimation 10.2.1 Parameter Estimation Using Linear Regression 10.2.2 Maximum Likelihood Estimation (MLE) 10.3 Expectation-Maximization 10.3.1 The Mathematical Framework of the EM Algorithm 10.3.2 Details of the Quadratic Optimization in the Maximization Step 10.3.3 Incorporating Constraints Into the EM Algorithm 10.3.4 A Numerical Example 10.4 Parameter Estimation via Data Assimilation 10.4.1 Two Parameter Estimation Algorithms 10.4.2 Estimating One Additive Parameter in a Linear Scalar Model 10.4.3 Estimating One Multiplicative Parameter in a Linear Scalar Model 10.4.4 Estimating Parameters in a Cubic Nonlinear Scalar Model 10.4.5 A Numerical Example 10.5 Learning Complex Dynamical Systems with Sparse Identification 10.5.1 Constrained Optimization for Sparse Identification 10.5.2 Using Information Theory for Model Identification with Sparsity 10.5.3 Partial Observations and Stochastic Parameterizations 11 Combining Stochastic Models with Machine Learning 11.1 Overview 11.2 Machine Learning and Data Assimilation 11.3 Machine Learning and Stochastic Modeling 11.3.1 Developing Stochastic Closure and Stochastic Parameterizations Using Machine Learning 11.3.2 Using Stochastic Models to Improve Machine Learning Training Data 11.3.3 Neural Stochastic Differential Equations Appendix A Instruction Manual for the MATLAB Codes References Index