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  • Open Access

Scalable learning of macroscopic stochastic dynamics

Mengyi Chen1, Pengru Huang2, Kostya S. Novoselov2,3, and Qianxiao Li1,2,*

  • *Contact author: qianxiao@nus.edu.sg
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Phys. Rev. Materials 10, 033805 – Published 20 March, 2026

DOI: https://doi.org/10.1103/mlh4-htxv

Abstract

Macroscopic dynamical descriptions of complex physical systems are crucial for understanding and controlling material behavior. With the growing availability of data and compute, machine learning has become a promising alternative to first-principles methods to build accurate macroscopic models from microscopic trajectory simulations. However, for spatially extended systems, direct simulations of sufficiently large microscopic systems that inform macroscopic behavior are prohibitive. In this work, we propose a framework that learns the macroscopic dynamics of large stochastic microscopic systems using only small-system simulations. Our framework employs a partial evolution scheme to generate training data pairs by evolving large-system snapshots within local patches. We subsequently derive the closure variables associated with the macroscopic observables and learn the macroscopic dynamics using a custom loss. Furthermore, we introduce a hierarchical upsampling scheme that enables the efficient generation of large-system snapshots from small-system snapshots. We empirically demonstrate the accuracy and robustness of our framework through a variety of stochastic spatially extended systems, including those described by stochastic partial differential equations, idealized lattice spin systems, and a more realistic NbMoTa alloy system.

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This article appears in the following collection:

Machine Learning for Materials Discovery and Understanding

The Editors of Physical Review Materials are pleased to present the Collection on Machine Learning for Materials Discovery and Understanding, highlighting cutting-edge advances in machine learning method development and applications for materials discovery and fundamental understanding of the structure-property-function relationship. The Collection is being guest-edited by Deyu Lu of Brookhaven National Laboratory (USA) and Jinlan Wang of Southeast University (China). Every article published in this collection underwent a rigorous peer review process, adhering to the same high standards applied to all papers. The Physical Review Materials editorial team managed the peer review and made all editorial decisions.

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Phys. Rev. Materials 10, 033805– Published 20 March, 2026
  • Received 29 September 2025
  • Accepted 26 February 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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