Hybrid ab initio and empirical machine learning models for the potential energy surface

TL;DR

The usefulness of the proposed methodology to generate hybrid machine learning models for the potential energy surface trained simultaneously on data from ab initio electronic structure calculations and on thermodynamic and/or structural observables from experiment is illustrated by the generation of hybrid models for liquid water that reproduce accurately the experimental density maximum, the density isobar at 1 bar, and the radial distribution function in molecular dynamics simulations.

Abstract

We propose a methodology to generate hybrid machine learning models for the potential energy surface trained simultaneously on data from ab initio electronic structure calculations and on thermodynamic and/or structural observables from experiment. The approach is based on the use of a loss function that includes the mean square error of observables with respect to their experimental values, in addition to the usual terms involving the mean square error of the energies and forces with respect to ab initio data. We employ a reweighting procedure that allows for the calculation of ensemble averages of observables during training for arbitrary values of the model parameters and on the fly. The method is general and can be applied to any set of static observables. We illustrate the usefulness of this approach by applying it to the generation of hybrid models for liquid water that reproduce accurately the experimental density maximum, the density isobar at 1 bar, and the radial distribution function in molecular dynamics simulations.

Figures (4)

• Figure 1: Schematic architecture of our method. Dataset A contains atomic coordinates, and ab initio energies and forces. Dataset B is extracted from an MD simulation driven by a model trained with Dataset A, and contains atomic coordinates, energies, and instantaneous values of observables. Each iteration of the training process is divided into two parts: in part 1, energies and forces are learned from dataset A using their mean square error (MSE) as loss function, whereas in part 2, the ensemble average of the observable is computed using on-the-fly reweighting with weights $\omega_i=e^{-\beta \left( E(\mathbf{R}^i;\boldsymbol{\theta}) - E(\mathbf{R}^i;\boldsymbol{\theta}_0) \right)}$ as shown in Eq. \ref{['eq:reweighting']}, and its MSE with respect to the experimental data is optimized. After $M$ trainings steps, we obtain a hybrid model compatible with both the ab initio and experimental data.
• Figure 2: Training process of a hybrid model for liquid water consistent with the experimental density. A) Evolution of each term in the loss function during the training process, represented using the root mean square error (RMSE). B) Non-reweighted mean (reference) and reweighted mean of the density computed over a mini-batch of 100 configurations during training. C) Effective sampling size (ESS), which quantifies the efficacy of the reweighting process (see text for details).
• Figure 3: Density of water from MD simulations driven by purely ab initio and hybrid models. A) Density at the temperature of maximum density $T_\mathrm{MD}$ for models based on SCAN ($T_\mathrm{MD}=320$ K) and MB-pol ($T_\mathrm{MD}=260$ K). The hybrid models are trained using the experimental density at $T_\mathrm{MD}=297$ K, which is shown as a horizontal dashed line nistwebbook. B) Density isobar at 1 bar for purely ab initio and hybrid models based on SCAN. The hybrid model is trained on the three experimental densities shown with black dots. The experimental data is taken from Refs. nistwebbook and hare1987density. The shaded areas around each curve represent the error estimated using block averages.
• Figure 4: Radial distribution functions $g_{OO}(r)$ at temperature $T=T_\mathrm{MD}+18$ K from MD simulations driven by purely ab initio and hybrid models based on A) SCAN and B) MB-pol. The hybrid models were trained on the experimental $g_{OO}(r)$ at $T=295$ K, which is also shown in the plots and was taken from Ref. skinner2013benchmark.