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Machine learning force field: Theory

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Here we present the theory for on-the-fly machine learning force fields, which will be only available from VASP6.0 or higher. The theory will be presented in a very condensed manner and for a more detailed description of the methods we refer the readers to Refs. and [1].


Fig. 1: On-the-fly machine learning force field generation scheme.

Before presenting the algorithm we want to define the following terms:

  • Structure datasets: A single structure dataset consist of the Bravais lattice, the atomic positions, the total energy, the forces and the stress tensor for one specific structure calculated by first principles.
  • Local configurations: Local configuration around each atom mapped onto descriptors. Several structure datasets and local configurations are selected and the machine-learning force field is fitted to those.

The on-the-fly force field generation scheme is given by the following steps (a flowchart of the algorithm is shown in Fig. 1):

  1. The machine predicts the energy, forces and stress tensor and their uncertainties on a given structure using existing force field.
  2. The machine decides whether to do first principles calculations or not. If machine decides not to do then algorithm skips steps 3 and 4.
  3. The first principles calculation is carried out and the data is stored as a candidate for new reference dataset.
  4. If the number of newly collected structures reaches a certain threshold, or if the uncertainty in the prediction becomes too large, the machine updates the set of reference structure datasets and local reference configurations and creates a new force field.
  5. Atomic positions are velocities are updated before returning to step 1 or ending the calculation if the total number of ionic steps is reached. The update is done in the following way:
    1. If machine judges that the force field is not accurate enough, the first principles energy, forces and stress tensor are used.
    2. Otherwise the force field is used.


The potential energy of a structure with atoms is approximated as

The local energies are functionals of the probability density to find another atom at the position around the atom within a cut-off radius defined as

Here is a cut-off function acting outside of . In the Smooth Overlap of Atomic Positions[2] (SOAP) method the delta function is approximated as

Fig. 2: Radial and angular descriptors.

Unfortunately is not rotationally invariant. To deal with this problem intermediate functions or descriptors depending on possessing rotational invariance are introduced:

Radial descriptor

This is the simplest descriptor which relies on the radial distribution function

where denotes the unit vector the vector between atoms and [see Fig. 2 (a)]. The Radial descriptor can also be regarded as a two-body descriptor.

Angular descriptor

In most cases the radial descriptor is not enough to distinguish different probability densities , since two different can yield the same . To improve on this angular information between two radial descriptors is also incorporated within an angular descriptor

where denotes the angle between two vectors and [see Fig. 2 (b)]. The function is also commonly referred to as angular distribution function and it is equivalent to the power spectrum used in the Gaussian Approximation Potential[3] (GAP) method.

Basis set expansion

The atomic probability density can be also expanded in terms of basis functions

where , and denote expansion coefficients, radial basis functions and spherical harmonics, respectively. The indices , and denote main, angular and magnetic quantum numbers, respectively.

By using the above equation the radial descriptor and angular descriptor (power spectrum) can be written as


where represent Legendre polynomials of order .

Potential energy fitting

It is convenient to express the local potential energy of atom in structure in terms of linear coefficients and a kernel as follows

where is the basis set size. The kernel measures the similarity between a local configuration from the training set and basis set . Using the radial and angular descriptors it is written as

Here the vectors and contain all coefficients and , respectively. The notation indicates that it is a normalized vector of . The parameters and control the weighting of the radial and angular terms, respectively. The parameter controls the sharpness of the kernel and the order of the many-body interactions.

Matrix vector form of linear equations

Similarly to the energy the forces and the stress tensor are also described as linear functions of the coefficients . All three are fitted simultaneously which leads to following matrix-vector form

where is a super vector consisting of the sub vectors . Here each contains the first principle energies per atom, forces and stress tensors for each structure . denotes the total number of structures. The size of is .

The matrix is also called as design matrix[4]. The rows of this matrix are blocked for each structure , where the first line of each block consists of the kernel used to calculate the energy. The subsequent lines consist of the derivatives of the kernel with respect to the atomic coordinates used to calculate the forces. The final 6 lines within each structure consist of the derivatives of the kernel with respect to the unit cell coordinates used to calculate the stress tensor components. The overall size of is looking like as follows

Error estimation

We use two different estimators for the errors in the force field:

  • Bayesian error estimation.
  • Spilling factor[5].

Bayesian error estimation

Ultimately for error prediction we want to get the maximized probability of observing a new structure on basis of the training set , which is denoted as . For this we need to get from the error of the linear fitting coefficients in the reproduction of the training data to which is explained in the following.

First we obtain from the Bayesian theorem


where we assume multivariate Gaussian distributions for the likelihood function

and the (conjugate) prior

The parameters and need to be optimized to balance the accuracy and robustness of the force field (see below).The ormalization is obtained by

Using the equations from above and the completing square method[4] is obtained as follows

Since the prior and the likelood function are described by multivariate Gaussian distributions the posterior describes a multivariate Gaussian written as

where we use the following definitions for the center of the Guassian distribution and the covariance matrix

Finally using the relation

and the completing square method[4] the distribution of is written as

The mean vector contains the results of the predictions on the dimensionless energy per atom, forces and stress tensor. The diagonal elements of , which correspond to the variances of the predicted results, are used as the uncertainty in the prediction.