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

Laboratory of Computational Science and Modeling, Institut des Mat eriaux, Ecole Polytechnique F ed erale de Lausanne, 1015 Lausanne, Switzerland (Dated: August 5, 2025) 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, showing that this method is equivalent to learning the mechanical action of the system of interest. 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. Simulating classical mechanical systems with high accuracy and efficiency is a long-standing challenge in computational physics [31, 32]. Traditional numerical methods typically rely on small time steps to propagate the equations of motion for the dynamical system in order to provide accurate integration.