Learning the action for long-time-step simulations of molecular dynamics

TL;DR

It is shown that an action-derived ML integrator eliminates the pathological behavior of non-structure-preserving ML predictors, and that the method can be applied iteratively, serving as a correction to computationally cheaper direct predictors.

Abstract

The equations of classical mechanics can be used to model the time evolution of countless physical systems, from the astrophysical to the atomic scale. Accurate numerical integration requires small time steps, which limits the computational efficiency -- especially in cases such as molecular dynamics that span wildly different time scales. Using machine-learning (ML) algorithms to predict trajectories allows one to greatly extend the integration time step, at the cost of introducing artifacts such as lack of energy conservation and loss of equipartition between different degrees of freedom of a system. We propose learning data-driven structure-preserving (symplectic and time-reversible) maps to generate long time-step classical dynamics and show that this method is equivalent to learning the mechanical action of the system of interest. These models can be learned based on short reference trajectories, and be transferred across thermodynamic conditions and chemical composition. We show that an action-derived ML integrator eliminates the pathological behavior of non-structure-preserving ML predictors, and that the method can be applied iteratively, serving as a correction to computationally cheaper direct predictors.

Figures (3)

• Figure 1: Energy profiles and trajectories of direct and symplectic methods for the simulation of a symmetric three body problem with large time steps. A velocity Verlet simulation with the same large time step is also shown.
• Figure 2: Simulations of liquid water performed in the NVT ensemble at 300 K, comparing a velocity Verlet baseline, direct predictions, and symplectic and time-reversible predictions using a variable number of fixed-point iterations per time step. Left: oxygen radial distribution function. Center: mean squared displacement of oxygen atoms. Right: profile of the conserved quantity (total energy plus themostat exchange energy), with an inset representing the average atom-type-resolved kinetic temperatures.
• Figure 3: Potential energy relaxation for long-time simulation of deeply-undercooled GeTe ($T=400$ K), using a cubic box containing 432 atoms. The different curves correspond to the reference velocity Verlet simulations (black), direct trajectory prediction (red) and symplectic corrections with different numbers of fixed-point iterations (shades of green). The curves are smoothed with a moving average with a Gaussian window of 2 ps, and averaged over 4 independent runs. The gray band indicates a range of two standard errors around the mean for the VV reference; error bars for the other curves are hidden, for clarity, but are of a similar magnitude. The inset shows the mean temperature (black) as well as the temperature resolved between Ge (blue) and Te (cyan), for direct predictions ($N_{\text{iter}}=0$) and for different levels of symplectic iterations.