Stochastic Methods for Modeling and Predicting Complex Dynamical Systems

1. Front Matter

   Pages i-xvi

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2. Stochastic Toolkits

   • Nan Chen

   Pages 1-21

3. Introduction to Information Theory

   • Nan Chen

   Pages 23-37

4. Basic Stochastic Computational Methods

   • Nan Chen

   Pages 39-48

5. Simple Gaussian and Non-Gaussian SDEs

   • Nan Chen

   Pages 49-65

6. Data Assimilation

   • Nan Chen

   Pages 67-82

7. Prediction

   • Nan Chen

   Pages 83-98

8. Data-Driven Low-Order Stochastic Models

   • Nan Chen

   Pages 99-118

9. Conditional Gaussian Nonlinear Systems

   • Nan Chen

   Pages 119-142

10. Parameter Estimation with Uncertainty Quantification

    • Nan Chen

    Pages 143-169

11. Combining Stochastic Models with Machine Learning

    • Nan Chen

    Pages 171-177

12. Instruction Manual for the MATLAB Codes

    • Nan Chen

    Pages 179-183

13. Back Matter

    Pages 185-199

    PDF

This book enables readers to understand, model, and predict complex dynamical systems using new methods with stochastic tools. The author presents a unique combination of qualitative and quantitative modeling skills, novel efficient computational methods, rigorous mathematical theory, as well as physical intuitions and thinking. An emphasis is placed on the balance between computational efficiency and modeling accuracy, providing readers with ideas to build useful models in practice. Successful modeling of complex systems requires a comprehensive use of qualitative and quantitative modeling approaches, novel efficient computational methods, physical intuitions and thinking, as well as rigorous mathematical theories. As such, mathematical tools for understanding, modeling, and predicting complex dynamical systems using various suitable stochastic tools are presented. Both theoretical and numerical approaches are included, allowing readers to choose suitable methods in different practical situations. The author provides practical examples and motivations when introducing various mathematical and stochastic tools and merges mathematics, statistics, information theory, computational science, and data science. In addition, the author discusses how to choose and apply suitable mathematical tools to several disciplines including pure and applied mathematics, physics, engineering, neural science, material science, climate and atmosphere, ocean science, and many others. Readers will not only learn detailed techniques for stochastic modeling and prediction, but will develop their intuition as well. Important topics in modeling and prediction including extreme events, high-dimensional systems, and multiscale features are discussed.

• Stochastic Methods
• Complex Systems
• Uncertainty Quantification
• Extreme Events
• Non-Gaussian Features
• Prediction
• Data Assimilation
• Textbook

• Department of Mathematics, University of Wisconsin–Madison, Madison, USA

  Nan Chen

Nan Chen, Ph.D., is an Assistant Professor in the Department of Mathematics at the University of Wisconsin-Madison. He is also a Faculty Affiliate of the Institute for Foundations of Data Science. Dr. Chen received his Ph.D. from the Courant Institute of Mathematical Sciences and the Center of Atmosphere and Ocean Science at New York University.  Dr. Chen's research interests include contemporary applied mathematics, stochastic modeling, data assimilation, uncertainty quantification, geophysical fluids, dynamical systems, scientific computing, machine learning, and general data science. He is also active in developing both dynamical and stochastic models and uses these models to predict real-world phenomena related to atmosphere-ocean science, climate, geophysics, and many other complex systems such as the Madden-Julian Oscillation (MJO), the monsoon, the El Nino-Southern Oscillation (ENSO), and the sea ice based on real observational data.  Dr. Chen's research work has beenpublished in top journals in both applied mathematics and many interdisciplinary areas.

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