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Optimal Prediction of Stiff Oscillatory Mechanics

Anton Kast

TL;DR

This paper tackles the computational bottleneck of simulating classical mechanical systems with fast and slow time scales by applying optimal prediction. It constructs reduced, mean-evolution equations for a selected set of collective variables using conditional expectations with respect to the invariant density $e^{-H}$, producing a $2n$-dimensional system that is easier to solve than the full $2N$-dimensional dynamics. The method is demonstrated first on a Stuart–Warren bath model and then on a nonlinear generalization with fully coupled quartic springs, showing that accurate mean dynamics can be captured with far fewer degrees of freedom and much larger time steps, often with speed-ups up to $10^6$ in force evaluations. The results establish a Hamiltonian, principled framework for reduced modeling of stiff systems and provide perturbative tools to handle non-Gaussian interactions, broadening the applicability to realistic molecular dynamics scenarios.

Abstract

We consider many-body problems in classical mechanics where a wide range of time scales limits what can be computed. We apply the method of optimal prediction to obtain equations which are easier to solve numerically. We demonstrate by examples that optimal prediction can reduce the amount of computation needed to obtain a solution by several orders of magnitude.

Optimal Prediction of Stiff Oscillatory Mechanics

TL;DR

This paper tackles the computational bottleneck of simulating classical mechanical systems with fast and slow time scales by applying optimal prediction. It constructs reduced, mean-evolution equations for a selected set of collective variables using conditional expectations with respect to the invariant density , producing a -dimensional system that is easier to solve than the full -dimensional dynamics. The method is demonstrated first on a Stuart–Warren bath model and then on a nonlinear generalization with fully coupled quartic springs, showing that accurate mean dynamics can be captured with far fewer degrees of freedom and much larger time steps, often with speed-ups up to in force evaluations. The results establish a Hamiltonian, principled framework for reduced modeling of stiff systems and provide perturbative tools to handle non-Gaussian interactions, broadening the applicability to realistic molecular dynamics scenarios.

Abstract

We consider many-body problems in classical mechanics where a wide range of time scales limits what can be computed. We apply the method of optimal prediction to obtain equations which are easier to solve numerically. We demonstrate by examples that optimal prediction can reduce the amount of computation needed to obtain a solution by several orders of magnitude.

Paper Structure

This paper contains 6 sections, 30 equations, 2 figures.

Table of Contents

  1. Stiff oscillatory mechanics
  2. Optimal prediction
  3. Generalizations of the Stuart-Warren experiments
  4. More general models
  5. Conclusions
  6. Acknowledgements

Figures (2)

  • Figure 1: The evolution of $P(t)$ determined in two ways: by solving the equations of motion (\ref{['original']}) with $N=10,000$ particles and random initial data (exact evolution, $\Delta t=10^{-2}/N$); and by solving the reduced equations (\ref{['reduced']}) with $n=100$ particles and a time step $100$ times longer (optimal prediction, $\Delta t=1/N=10^{-2}/n$). For these calculations, $k_Q=k_q=1$.
  • Figure 2: The average evolution of $p_1(t)$ determined in three ways: by solving the equations of motion (\ref{['new']}) for $100$ different initial conditions, with $N=10^3$ particles, $\Delta t=10^{-2}/N$, and then averaging all $100$ solutions (mean evolution); by solving the reduced equations (\ref{['reduced']}) once, with $n=10$ particles, $\Delta t = 1/N = 10^{-2}/n$ (optimal prediction); and by solving the original equations (\ref{['new']}) once with $N=10$, $\Delta t=10^{-2}/N$ (naive prediction, just neglecting interactions with discarded variables).