[Paper Review] Deep learning as closure for irreversible processes: A data-driven generalized Langevin equation

Antonio Russo, Miguel A. Durán-Olivencia+2 more|arXiv (Cornell University)|Mar 22, 2019

Advanced Thermodynamics and Statistical Mechanics4 citations

TL;DR

This paper proposes a deep learning approach to approximate the unknown memory kernels and noise terms in the generalized Langevin equation (GLE), using neural networks to learn closures for irreversible processes. By replacing heuristic fitting functions with data-driven neural networks, the method achieves superior accuracy and robustness across diverse systems, including colloidal dynamics, particle chains, climate models, and financial time series.

ABSTRACT

The ultimate goal of physics is finding a unique equation capable of describing the evolution of any observable quantity in a self-consistent way. Within the field of statistical physics, such an equation is known as the generalized Langevin equation (GLE). Nevertheless, the formal and exact GLE is not particularly useful, since it depends on the complete history of the observable at hand, and on hidden degrees of freedom typically inaccessible from a theoretical point of view. In this work, we propose the use of deep neural networks as a new avenue for learning the intricacies of the unknowns mentioned above. By using machine learning to eliminate the unknowns from GLEs, our methodology outperforms previous approaches (in terms of efficiency and robustness) where general fitting functions were postulated. Finally, our work is tested against several prototypical examples, from a colloidal systems and particle chains immersed in a thermal bath, to climatology and financial models. In all cases, our methodology exhibits an excellent agreement with the actual dynamics of the observables under consideration.

Motivation & Objective

• To address the fundamental challenge of modeling irreversible processes in statistical physics using the generalized Langevin equation (GLE), which is analytically intractable due to its non-Markovian memory and hidden degrees of freedom.
• To overcome the limitations of traditional approaches that rely on ad hoc fitting functions for memory kernels and noise terms in the GLE.
• To develop a data-driven framework that learns the closure terms directly from observable trajectories, enabling accurate prediction of system dynamics without explicit knowledge of underlying degrees of freedom.
• To validate the method across diverse physical and stochastic systems, including colloidal dynamics, particle chains, climate models, and financial time series.

Proposed method

• Employ deep neural networks to learn the memory kernel and noise statistics of the GLE directly from time-series data of the observable of interest.
• Formulate the GLE as a non-Markovian stochastic differential equation with unknown memory and noise terms, which are then parameterized by a deep neural network.
• Train the neural network using observed trajectories to minimize the prediction error of the observable's evolution, effectively learning the closure for the GLE.
• Use a loss function based on the discrepancy between predicted and actual trajectories, enabling end-to-end learning of the GLE's unknown components.
• Integrate the trained neural network closure into the GLE framework to simulate the dynamics of the observable with high fidelity.
• Ensure generalization by testing the method on unseen data and diverse systems, including non-equilibrium and non-Gaussian processes.

Experimental results

Research questions

• RQ1 Can deep neural networks effectively learn the memory kernel and noise statistics of the generalized Langevin equation when the underlying degrees of freedom are unknown?
• RQ2 How does a data-driven neural network closure compare to traditional heuristic fitting functions in terms of accuracy and robustness across different physical and stochastic systems?
• RQ3 To what extent can the proposed method capture non-Markovian and non-Gaussian dynamics in systems such as colloidal particles, particle chains, climate models, and financial time series?
• RQ4 Can the method generalize to unseen data and remain stable under varying initial conditions and noise levels?

Key findings

• The deep learning-based closure for the GLE significantly outperforms traditional methods that rely on postulated fitting functions in terms of prediction accuracy and robustness.
• The method achieves excellent agreement with the true dynamics across all tested systems, including colloidal systems, particle chains, climatological models, and financial time series.
• Neural networks successfully learn the complex, non-Markovian memory effects and non-Gaussian noise structures that are typically intractable with analytical or parametric approaches.
• The framework generalizes well to unseen trajectories and maintains high predictive performance without requiring prior knowledge of the system's underlying dynamics.
• The approach enables accurate long-time prediction of observables in irreversible processes, even when the system exhibits strong memory effects and hidden degrees of freedom.