Effective Equations in complex systems: from Langevin to machine learning

TL;DR

This paper reviews the derivation of effective equations in complex systems, extending Langevin equations to Hamiltonian systems, and introduces a numerical data-driven method for inferring and validating these models from both simulated and experimental data.

Contribution

It presents a generalized framework for effective equations in complex systems and introduces a novel numerical protocol for inferring models from data, applicable to experimental scenarios.

Findings

Generalization of Langevin equations to Hamiltonian systems with non-standard kinetic terms

A numerical method for inferring effective equations from data

Discussion of practical challenges in data-driven model building

Abstract

The problem of effective equations is reviewed and discussed. Starting from the classical Langevin equation, we show how it can be generalized to Hamiltonian systems with non-standard kinetic terms. A numerical method for inferring effective equations from data is discussed; this protocol allows to check the validity of our results. In addition we show that, with a suitable treatment of time series, such protocol can be used to infer effective models from experimental data. We briefly discuss the practical and conceptual difficulties of a pure data-driven approach in the building of models.