Unsupervised Machine Learning Neural Gas Algorithm for Accurate Evaluations of the Hessian Matrix in Molecular Dynamics.

Michele Gandolfi,Michele Ceotto

doi:10.1021/acs.jctc.1c00707

Michele Gandolfi, Michele Ceotto

Open Access

https://doi.org/10.1021/acs.jctc.1c00707

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Abstract

The Hessian matrix of the potential energy of molecular systems is employed not only in geometry optimizations or high-order molecular dynamics integrators but also in many other molecular procedures, such as instantaneous normal mode analysis, force field construction, instanton calculations, and semiclassical initial value representation molecular dynamics, to name a few. Here, we present an algorithm for the calculation of the approximated Hessian in molecular dynamics. The algorithm belongs to the family of unsupervised machine learning methods, and it is based on the neural gas idea, where neurons are molecular configurations whose Hessians are adopted for groups of molecular dynamics configurations with similar geometries. The method is tested on several molecular systems of different dimensionalities both in terms of accuracy and computational time versus calculating the Hessian matrix at each time-step, that is, without any approximation, and other Hessian approximation schemes. Finally, the method is applied to the on-the-fly, full-dimensional simulation of a small synthetic peptide (the 46 atom N-acetyl-l-phenylalaninyl-l-methionine amide) at the level of DFT-B3LYP-D/6-31G* theory, from which the semiclassical vibrational power spectrum is calculated.

Highlights

In standard molecular dynamics (MD) simulations, the atomic positions, velocities, and forces are evolved in time according to Hamilton’s equations and calculated at each time-step
We show that the unsupervised machine learning algorithm “neural gas” can optimally compress the information contained in simple molecular geometries along a MD simulation, and we use the compressed information to approximate the Hessian matrix
Given the importance of an accurate method for approximating instead of calculating the Hessian matrix during MD simulations, we have investigated the possibility to employ a slightly customized neural gas (NGas) algorithm that allows us to compute the Hessian matrix of the potential energy along a MD simulation

Summary

Introduction

In standard molecular dynamics (MD) simulations, the atomic positions, velocities, and forces are evolved in time according to Hamilton’s equations and calculated at each time-step. The task may become prohibitive in ab initio MD12 where the potential and its derivatives are evaluated on-the-fly, that is, by solving the electronic structure problem and using the Hellman−Feynman theorem, or by the finite difference formula using the forces or the potential To address this issue, a number of approximate methods have been introduced.[13−16] Usually, these are of the type of updating schemes, where the Hessian is approximated in a step-wise fashion using the latest information available.[17] These updating schemes were originally developed for optimization[17−21] (see references therein) but have much evolved and improved since . The GPR-PES can be differentiated analytically as many times as required, Received: July 14, 2021 Published: October 27, 2021

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