Machine learning stochastic differential equations for the evolution of order parameters of classical many-body systems in and out of equilibrium

Carnazza, Francesco; Carollo, Federico; Martius, Georg; Andergassen, Sabine; Klopotek, Miriam; Lesanovsky, Igor

Abstract

We develop a machine learning algorithm to infer the emergent stochastic equation governing the evolution of an order parameter of a many-body system. We train our neural network to independently learn the directed force acting on the order parameter as well as an effective diffusive noise. We illustrate our approach using the classical Ising model endowed with Glauber dynamics, and the contact process as test cases. For both models, which represent paradigmatic equilibrium and nonequilibrium scenarios, the directed force and noise can be efficiently inferred. The directed force term of the Ising model allows us to reconstruct an effective potential for the order parameter which develops the characteristic double-well shape below the critical temperature. Despite its genuine nonequilibrium nature, such an effective potential can also be obtained for the contact process and its shape signals a phase transition into an absorbing state. Also, in contrast to the equilibrium Ising model, the presence of an absorbing state renders the noise term dependent on the value of the order parameter itself.

Citation

Carnazza, F., Carollo, F., Martius, G., Andergassen, S., Klopotek, M., & Lesanovsky, I. (2024). Machine learning stochastic differential equations for the evolution of order parameters of classical many-body systems in and out of equilibrium. Machine Learning: Science and Technology, 5(4), Article 045002. https://doi.org/10.1088/2632-2153/ad7ad7

• Journal Article Type: Article
• Acceptance Date: Sep 13, 2024
• Online Publication Date: Oct 7, 2024
• Publication Date: 2024-12
• Deposit Date: Oct 21, 2024
• Publicly Available Date: Oct 21, 2024
• Journal: Machine Learning: Science and Technology
• Print ISSN: 2632-2153
• Electronic ISSN: 2632-2153
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 5
• Issue: 4
• Article Number: 045002
• DOI: https://doi.org/10.1088/2632-2153/ad7ad7
• Keywords: stochastic processes, dynamical Ising model, contact process, machine learning
• Public URL: https://nottingham-repository.worktribe.com/output/39465386
• Publisher URL: https://iopscience.iop.org/article/10.1088/2632-2153/ad7ad7