# Bayesian machine learning for quantum molecular dynamics

## TL;DR

This paper explores Bayesian machine learning approaches to quantum molecular dynamics, enabling uncertainty quantification, sensitivity analysis, and potential extrapolation of Schrödinger equation solutions to improve and accelerate quantum simulations.

## Contribution

It introduces a Bayesian machine learning framework for quantum dynamics that models potential energy surfaces and predicts quantum outcomes with uncertainty estimates.

## Key findings

- Bayesian models provide error bars for quantum predictions.
- Models identify sensitivity to potential energy surface variations.
- Potential for physically extrapolating Schrödinger equation solutions.

## Abstract

This article discusses applications of Bayesian machine learning for quantum molecular dynamics. One particular formulation of quantum dynamics advocated here is in the form of a machine learning simulator of the Schr\"{o}dinger equation. If combined with the Bayesian statistics, such a simulator allows one to obtain not only the quantum predictions but also the error bars of the dynamical results associated with uncertainties of inputs (such as the potential energy surface or non-adiabatic couplings) into the nuclear Schr\"{o}dinger equation. Instead of viewing atoms as undergoing dynamics on a given potential energy surface, Bayesian machine learning allows one to formulate the problem as the Schr\"{o}dinger equation with a non-parametric distribution of potential energy surfaces that becomes conditioned by the desired dynamical properties (such as the experimental measurements). Machine learning models of the Schr\"{o}dinger equation solutions can identify the sensitivity of the dynamical properties to different parts of the potential surface, the collision energy, angular momentum, external field parameters and basis sets used for the calculations. This can be used to inform the design of efficient quantum dynamics calculations. Machine learning models can also be used to correlate rigorous results with approximate calculations, providing accurate interpolation of exact results. Finally, there is evidence that it is possible to build Bayesian machine learning models capable of physically extrapolating the solutions of the Schr\"{o}dinger equation. This is particularly valuable as such models could complement common discovery tools to explore physical properties at Hamiltonian parameters not accessible by rigorous quantum calculations or experiments, and potentially be used to accelerate the numerical integration of the nuclear Schr\"{o}dinger equation.