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  • Open Access

Data-driven reconstruction of a multivariate Langevin equation to model complex systems

Antonio Malpica-Morales1,*, Miguel A. Durán-Olivencia1,2,†, and Serafim Kalliadasis1,‡

  • 1Department of Chemical Engineering, Imperial College, London SW7 2AZ, United Kingdom
  • 2Research, Vortico Tech, Málaga 29100, Spain

  • *Contact author: a.malpica-morales21@imperial.ac.uk
  • †Contact author: miguel@vortico.tech
  • ‡Contact author: s.kalliadasis@imperial.ac.uk
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Phys. Rev. E 112, 024115 – Published 11 August, 2025

DOI: https://doi.org/10.1103/zncf-n4y3

Abstract

Obtaining an accurate description of complex systems is challenging, particularly when their elements exhibit intricate interactions. We propose a data-driven multivariate Langevin equation (LE) to approximate real-world complex systems' observables. By reconstructing the drift and diffusion terms of the LE through a nonparametric technique following the definition of the Kramers-Moyal coefficients, our approach unravels the main features of a complex system without requiring a priori knowledge about the underlying governing mechanisms. We illustrate our framework's adaptability, reliability, and capability to extract pertinent information through three case studies. First, we benchmark the framework with a simple prototypical example from mechanics, a particle confined by a bistable potential energy well. We then turn to two more involved examples from financial markets, the electricity day-ahead prices and currency-exchange rates, where the nonparametric multivariate LE has not previously been applied. In all cases, our framework accurately identifies the equilibrium values, metastabilty regions, and distinct diffusion behaviors, in a functional agnostic manner, as opposed to price-equation models that require specific domain knowledge.

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Outline

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Phys. Rev. E 112, 024115– Published 11 August, 2025
  • Received 13 September 2024
  • Revised 6 May 2025
  • Accepted 11 July 2025
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