Data-driven parameterization of the generalized Langevin equation

Abstract

We present a data-driven approach to determine the memory kernel and random noise in generalized Langevin equations. To facilitate practical implementations, we parameterize the kernel function in the Laplace domain by a rational function, with coefficients directly linked to the equilibrium statistics of the coarse-grain variables. We show that such an approximation can be constructed to arbitrarily high order and the resulting generalized Langevin dynamics can be embedded in an extended stochastic model without explicit memory. We demonstrate how to introduce the stochastic noise so that the second fluctuationdissipation theorem is exactly satisfied. Results from several numerical tests are presented to demonstrate the effectiveness of the proposed method.generalized Langevin dynamics | data-driven parameterization | coarse-grained molecular models | reaction rate | model reduction G eneralized Langevin equations (GLEs) have recently reemerged in the area of molecular modeling as a promising description of reduced-dimension coarse-grained variables. In principle, GLEs can be derived using the Mori-Zwanzig projection formalism (1, 2). Examples of such derivations can be found for a variety of applications (3-10), for example, climate modeling (11, 12). The GLE approach eliminates a large number of irrelevant degrees of freedom, reducing system dimensions to make direct computation feasible. Because this elimination often projects out high-frequency modes, the GLE can also extend the time scale of simulations. The GLE does this by describing the dynamics for explicit quantities of interest and implicitly describing the remaining degrees of freedom through a memory term and a random noise term. The random noise term is often strongly correlated in time.However, practical implementations of GLEs require specification of the memory function, which can be difficult to obtain, even when the full dynamics of the system is known. For example, the memory functions obtained in past studies (8,9,13,14) have involved functions of high-dimensional matrices. Darve et al. (14) proposed a more efficient algorithm to compute the memory kernel by solving an equation for the orthogonal dynamics derived from the Mori-Zwanzig formalism. However, the orthogonal dynamics equation can be expensive to solve when the original system is large. Furthermore, even when the memory kernel function is available, direct evaluation of the memory term can be costly because it requires the history of the coarse-grained (CG) variables at every time step and the associated numerical integration. Sampling of the random noise is also a challenging component of GLEs: To generate the correct equilibrium statistics for the CG model, the random noise has to obey the second fluctuation-dissipation theorem (FDT) (15). The theory of stationary processes (16) states that the random process is uniquely determined by the correlation function, which is proportional to the memory kernel; however, sampling the random noise is nontrivial in practice. Methods based ...

Generalized Langevin Equations for Systems with Local Interactions

“…Such consistency conditions carry over a certain number of constraints on the process f (t), which allow for its partial identification. As an example, consider the following MZ model recently proposed by Lei et al in [36] to study the dynamics of a tagged particle in a large particle system…”

“…One of the main features of such formulation is that it allows us to systematically derive formally exact generalized Lagevin equations (GLEs) for quantities of interest (macroscopic observables), based on microscopic equations of motion. Such GLEs can be found in a variety of applications, including particle dynamics [52,66,38,39,36], partial differential equations (PDEs) [11,54,8,10,59], fluid dynamics [46,47], and solid-state physics [65,37,42]. As an example, consider the Brownian motion of a colloidal particle subject to collision interactions with a large number of fluid particles.…”

“…By using the MZ formulation it is possible to derive a low-dimensional system of equations characterizing the dynamics (position and momentum) of the colloidal particle alone. This is at the basis of microscopic theories of Brownian motion built upon first principles [27,36].…”

“…Data-driven methods aim at recovering the MZ memory integral/fluctuation term based on data, usually in the form of sample trajectories of the full system. Typical examples are the NARMAX technique developed by Lu et al [40], the rational function approximation recently proposed by Lei et al [36] (see also [15]), and the conditional expectation technique developed by Brennan and Venturi [7]. On the other hand, methods based on first principles aim at approximating the MZ memory integral and fluctuation term based on the structure of the nonlinear system (microscopic equations of motion), without using any simulation data.…”

“…However, they are often quite involved and they do not generalize well to systems with no scale separation [21]. At the same time, data-driven methods can yield accurate results, but they often require a large number of sample trajectories to faithfully capture memory effects [7,15,36,38,39] In this paper, we present a new method to compute the MZ memory integral and the flucuation term from first principles in nonlinear systems with local polynomial interactions. To this end, we build upon the Faber operator series expansion we recently developed in [68], and a new combinatorial algorithm that allows us to compute the MZ memory kernel to any desired accuracy by using only the structure of the microscopic equations of motion.…”