The Role of Data in Model Building and Prediction: A Survey Through Examples

TL;DR

This survey explores how data-driven models and predictions are constructed across physics disciplines, highlighting model-free methods, empirical knowledge integration, and dimensional reduction techniques for complex systems.

Contribution

It provides a comparative overview of data-based modeling approaches in physics, emphasizing the importance of empirical knowledge and data-driven dimensionality reduction.

Findings

Analogues method enables model-free predictions in dynamical systems

Empirical knowledge is crucial for realistic biophysical models

Dimensional reduction via Langevin dynamics aids in complex many-body systems

Abstract

The goal of Science is to understand phenomena and systems in order to predict their development and gain control over them. In the scientific process of knowledge elaboration, a crucial role is played by models which, in the language of quantitative sciences, mean abstract mathematical or algorithmical representations. This short review discusses a few key examples from Physics, taken from dynamical systems theory, biophysics, and statistical mechanics, representing three paradigmatic procedures to build models and predictions from available data. In the case of dynamical systems we show how predictions can be obtained in a virtually model-free framework using the methods of analogues, and we briefly discuss other approaches based on machine learning methods. In cases where the complexity of systems is challenging, like in biophysics, we stress the necessity to include part of the empirical knowledge in the models to gain the minimal amount of realism. Finally, we consider many body systems where many (temporal or spatial) scales are at play and show how to derive from data a dimensional reduction in terms of a Langevin dynamics for their slow components.