Data-Driven Computational Methods: Parameter and Operator Estimations (Chapter 1)

TL;DR

This book explores data-driven computational methods for modeling dynamical systems, focusing on parameter estimation and operator approximation using statistical learning and big data techniques.

Contribution

It introduces two classes of data-driven methods—parameter fitting and operator estimation—for modeling dynamical systems, bridging classical theory with modern data science.

Findings

Data fitting algorithms effectively estimate parameters in physical models.

Operator estimation provides nonparametric approximations of system dynamics.

The methods are supported by MATLAB examples and accessible for self-study.

Abstract

Modern scientific computational methods are undergoing a transformative change; big data and statistical learning methods now have the potential to outperform the classical first-principles modeling paradigm. This book bridges this transition, connecting the theory of probability, stochastic processes, functional analysis, numerical analysis, and differential geometry. It describes two classes of computational methods to leverage data for modeling dynamical systems. The first is concerned with data fitting algorithms to estimate parameters in parametric models that are postulated on the basis of physical or dynamical laws. The second class is on operator estimation, which uses the data to nonparametrically approximate the operator generated by the transition function of the underlying dynamical systems. This self-contained book is suitable for graduate studies in applied mathematics, statistics, and engineering. Carefully chosen elementary examples with supplementary MATLAB codes and appendices covering the relevant prerequisite materials are provided, making it suitable for self-study.