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# Liquid Si - MLFF

Generating a machine learning force field for liquid Si. For this tutorial, we expect that the user is already familiar with running conventional ab initio molecular dynamic calculations.

## Input

### POSCAR

In this example we start from a 64 atom super cell of diamond-fcc Si (the same as in Liquid Si - Standard MD):

Si cubic diamond 2x2x2 super cell of conventional cell
5.43090000000000
2.00000000   0.00000000   0.00000000
0.00000000   2.00000000   0.00000000
0.00000000   0.00000000   2.00000000
Si
64
Direct
0.00000000   0.00000000   0.00000000
0.50000000   0.00000000   0.00000000
0.00000000   0.50000000   0.00000000
0.50000000   0.50000000   0.00000000
0.00000000   0.00000000   0.50000000
0.50000000   0.00000000   0.50000000
0.00000000   0.50000000   0.50000000
0.50000000   0.50000000   0.50000000
0.37500000   0.12500000   0.37500000
0.87500000   0.12500000   0.37500000
0.37500000   0.62500000   0.37500000
0.87500000   0.62500000   0.37500000
0.37500000   0.12500000   0.87500000
0.87500000   0.12500000   0.87500000
0.37500000   0.62500000   0.87500000
0.87500000   0.62500000   0.87500000
0.00000000   0.25000000   0.25000000
0.50000000   0.25000000   0.25000000
0.00000000   0.75000000   0.25000000
0.50000000   0.75000000   0.25000000
0.00000000   0.25000000   0.75000000
0.50000000   0.25000000   0.75000000
0.00000000   0.75000000   0.75000000
0.50000000   0.75000000   0.75000000
0.37500000   0.37500000   0.12500000
0.87500000   0.37500000   0.12500000
0.37500000   0.87500000   0.12500000
0.87500000   0.87500000   0.12500000
0.37500000   0.37500000   0.62500000
0.87500000   0.37500000   0.62500000
0.37500000   0.87500000   0.62500000
0.87500000   0.87500000   0.62500000
0.25000000   0.00000000   0.25000000
0.75000000   0.00000000   0.25000000
0.25000000   0.50000000   0.25000000
0.75000000   0.50000000   0.25000000
0.25000000   0.00000000   0.75000000
0.75000000   0.00000000   0.75000000
0.25000000   0.50000000   0.75000000
0.75000000   0.50000000   0.75000000
0.12500000   0.12500000   0.12500000
0.62500000   0.12500000   0.12500000
0.12500000   0.62500000   0.12500000
0.62500000   0.62500000   0.12500000
0.12500000   0.12500000   0.62500000
0.62500000   0.12500000   0.62500000
0.12500000   0.62500000   0.62500000
0.62500000   0.62500000   0.62500000
0.25000000   0.25000000   0.00000000
0.75000000   0.25000000   0.00000000
0.25000000   0.75000000   0.00000000
0.75000000   0.75000000   0.00000000
0.25000000   0.25000000   0.50000000
0.75000000   0.25000000   0.50000000
0.25000000   0.75000000   0.50000000
0.75000000   0.75000000   0.50000000
0.12500000   0.37500000   0.37500000
0.62500000   0.37500000   0.37500000
0.12500000   0.87500000   0.37500000
0.62500000   0.87500000   0.37500000
0.12500000   0.37500000   0.87500000
0.62500000   0.37500000   0.87500000
0.12500000   0.87500000   0.87500000
0.62500000   0.87500000   0.87500000

K-Points
0
Gamma
1  1  1
0  0  0

### INCAR

#Basic parameters
ISMEAR = 0
SIGMA = 0.1
LREAL = Auto
ALGO = FAST
ISYM = -1
NELM = 100
EDIFF = 1E-4
LWAVE = .FALSE.
LCHARG = .FALSE.

#Parallelization of ab initio calculations
NCORE = 2

#MD
IBRION = 0
MDALGO = 2
ISIF = 2
SMASS = 1.0
TEBEG = 2000
NSW = 10000
POTIM = 3.0

#Parameters for initialization of wavefunctions and charge density
INIWAV = 1
IWAVPR = 1
ISTART = 0
ICHARG = 0

#Machine learning paramters
ML_FF_LMLFF = .TRUE.
ML_FF_ISTART = 0
ML_FF_NWRITE = 2
ML_FF_EATOM = -.70128086E+00
ML_FF_LMLFF = .TRUE.
switches on the machine learning of the force field
ML_FF_ISTART = 0
corresponds to learning from scratch
ML_FF_NWRITE = 2
leads to a more verbose output where the error on energies, forces and stress are written to the ML_LOGFILE file. This setting is very handy to check the accuracy of the force field.
ML_FF_EATOM = -.70128086E+00
sets the atomic reference energy for each species. We show to obtain this energy below.
IWAVPR = 1
extrapolates the total charge density from the atomic ones. This is more advisable than the default, because during the on-the-fly learning several hundreds or thousands of molecular dynamics steps can lie between two ab initio calculations.

## Calculation

### Reference energy

To obtain the force field for liquid Si, we require the atomic energy of a single Si atom. We create a new working directory for this calculation

mkdir Si_ATOM
cd Si_ATOM
ln -s ../POTCAR

In this directory, we create the following INCAR file

#Basic parameters
ISMEAR = 0
SIGMA = 0.1
LREAL = Auto
ALGO = FAST
ISYM = 0
NELM = 100
EDIFF = 1E-4
LWAVE = .FALSE.
LCHARG = .FALSE.
ISPIN = 2

The POSCAR file contains a single atom in a large enough box (the box should be orthorhombic to have enough degrees of freedom for electronic relaxation)

Si atom
1.00090000000000
12.00000000   0.00000000   0.00000000
0.00000000  12.01000000   0.00000000
0.00000000   0.00000000  12.02000000
Si
1
Direct
0.00000000   0.00000000   0.00000000

Due to the large unit cell, we can use a Gamma point calculation set up in the KPOINTS file

Gamma point only
0 0 0
Gamma
1 1 1
0 0 0

After running the calculation, we obtain the total energy from the OUTCAR or OSZICAR file. We use this energy for the ML_FF_EATOM tag. If multiple atom types are present in the structure, this step has to be repeated for each atom type separately and the reference energies are provided as a list after the ML_FF_EATOM tag. The ordering of the energies corresponds to the ordering of the elements in the POTCAR file.

### Creating the liquid structure

Because we don't have a structure of liquid silicon readily available, we first create that structure by starting from a super cell of crystalline silicon with 64 atoms. The temperature is set to 2000 K so that the crystal melts rapidly in the MD run. To improve the simulation speed drastically, we utilize the on-the-fly machine learning. Most of the ab initio steps will be replaced by very fast force-field ones. Within 10000 steps equivalent to 30 ps, we have obtained a good starting position for the subsequent simulations in the CONTCAR file. You can copy the input files or download them.

After running the calculation, we obtained a force field, but its initial trajectory might be tainted but the unreasonable starting position. Nevertheless, it is instructive to inspect the output to understand how to assess the accuracy of a force field, before refining it in subsequent calculations. The main output files for the machine learning are

ML_ABNCAR
contains the ab initio structure datasets used for the learning. It will be needed for continuation runs as ML_ABCAR.
ML_FFNCAR
contains the regression results (weights, parameters, etc.). It will be needed for continuation runs as ML_FFCAR.
ML_LOGFILE
logging the proceedings of the machine learning. With ML_FF_NWRITE = 2 set, this file lists the error between force-field and ab initio calculation for energy, forces and the stress. The entry for the last step should be similar to the following (the inherent randomness of a molecular dynamics simulation will lead to slightly different results):
====================================================================================================
Information on error estimations
----------------------------------------------------------------------------------------------------
Error in energy (eV atom^-1):     0.016994
Error in force  (eV Angst^-1):     0.247664
Error in stress (kB):     5.368218
Bayesian error (eV Angst-1):     0.064904    0.127195
Spilling factor (-):     0.000690    0.020000
====================================================================================================

You can see that additional information about the Bayesian error and the spilling factor is reported. The first entry of the Bayesian error is the estimated error from our model. If the estimated error exceeds the threshold listed next to it a new structure dataset is generated. The threshold is automatically determined during the calculations. The first and second entry for the spilling factor are the calculated spilling factor and the threshold, respectively. The threshold for the spilling factor is usually always kept constant during the calculations.

### Structral properties of the force field

To examine the accuracy of structural properties, we compare the deviations between a 3 ps molecular dynamics run using the force field and a full ab initio calculation. For a meaningful comparison, it is best to start from the same initial structure. We will use the liquid structure, we obtained in the previous step and back it up

cp CONTCAR POSCAR.T2000_relaxed

Now, we proceed with the force field calculation and set up the required files

cp POSCAR.T2000_relaxed POSCAR
cp ML_ABNCAR ML_ABCAR
cp ML_FFNCAR ML_FFCAR

To run a shorter simulation using only the force field, we change the following INCAR tags to

ML_FF_ISTART = 2
NSW = 1000

After the calculation finished, we backup the history of the atomic positions

cp XDATCAR XDATCAR.MLFF_3ps

We repeat the same process for the ab initio simulation recovering the same initial structure

cp POSCAR.T2000_relaxed POSCAR

switching the machine learning off in the INCAR file by setting

ML_FF_LMLFF = .FALSE.

We store the trajectory once the calculation completed

cp XDATCAR XDATCAR.AI_3ps

To analyze the pair correlation function, we use the PERL script pair_correlation_function.pl

Click to show/hide pair_correlation_function.pl
#!/usr/bin/perl

use strict;
use warnings;

#configuration for which ensemble average is to be calculated
my \$confmin=1;            #starting index of configurations in XDATCAR file for pair correlation function
my \$confmax=20000;           #last index of configurations in XDATCAR file for pair correlation function
my \$confskip=1;           #stepsize for configuration loop
my \$species_1 = 1;        #species 1 for which pair correlation function is going to be calculated
my \$species_2 = 1;        #species 2 for which pair correlation function is going to be calculated
my \$rmin=0.0;             #minimal value of radial grid
my \$rmax=10.0;            #maximum value of radial grid
my \$nr=300;                #number of equidistant steps in radial grid
my \$dr=(\$rmax-\$rmin)/\$nr; #stepsize in radial grid
my \$tol=0.0000000001;     #tolerance limit for r->0

my \$z=0;                  #counter
my \$numelem;              #number of elements
my @elements;             #number of atoms per element saved in list/array
my \$lattscale;            #scaling factor for lattice
my @b;                    #Bravais matrix
my \$nconf=0;              #number of configurations in XDATCAR file
my @cart;                 #Cartesian coordinates for each atom and configuration
my \$atmin_1=0;            #first index of species one
my \$atmax_1;              #last index of species one
my \$atmin_2=0;            #first index of species two
my \$atmax_2;              #last index of species two
my @vol;                  #volume of cell (determinant of Bravais matrix)
my \$pi=4*atan2(1, 1);     #constant pi
my \$natom=0;              #total number of atoms in cell
my @pcf;                  #pair correlation function (list/array)
my \$mult_x=1;             #periodic repetition of cells in x dimension
my \$mult_y=1;             #periodic repetition of cells in y dimension
my \$mult_z=1;             #periodic repetition of cells in z dimension
my @cart_super;           #Cartesian cells over multiple cells
my @vec_len;              #Length of lattice vectors in 3 spatial coordinates
#my \$ensemble_type="NpT";  #Set Npt or NVT. Needs to be set since both have different XDATCAR file.
my \$ensemble_type="NVT";  #Set Npt or NVT. Needs to be set since both have different XDATCAR file.
my \$av_vol=0;               #Average volume in cell

while (<>)
{
chomp;
\$_=~s/^/ /;
my @help=split(/[\t,\s]+/);
\$z++;
if (\$z==2)
{
\$lattscale = \$help[1];
}
if (\$z==3)
{
\$b[\$nconf+1][1][1]=\$help[1]*\$lattscale;
\$b[\$nconf+1][1][2]=\$help[2]*\$lattscale;
\$b[\$nconf+1][1][3]=\$help[3]*\$lattscale;
}
if (\$z==4)
{
\$b[\$nconf+1][2][1]=\$help[1]*\$lattscale;
\$b[\$nconf+1][2][2]=\$help[2]*\$lattscale;
\$b[\$nconf+1][2][3]=\$help[3]*\$lattscale;
}
if (\$z==5)
{
\$b[\$nconf+1][3][1]=\$help[1]*\$lattscale;
\$b[\$nconf+1][3][2]=\$help[2]*\$lattscale;
\$b[\$nconf+1][3][3]=\$help[3]*\$lattscale;
}
if (\$z==7)
{
if (\$nconf==0)
{
\$numelem=@help-1;
for (my \$i=1;\$i<=\$numelem;\$i++)
{
\$elements[\$i]=\$help[\$i];
\$natom=\$natom+\$help[\$i];
}
}
}
if (\$_=~m/Direct/)
{
\$nconf=\$nconf+1;
#for NVT ensemble only one Bravais matrix exists, so it has to be copied
if (\$ensemble_type eq "NVT")
{
for (my \$i=1;\$i<=3;\$i++)
{
for (my \$j=1;\$j<=3;\$j++)
{
\$b[\$nconf][\$i][\$j]=\$b[1][\$i][\$j];
}
}
}
for (my \$i=1;\$i<=\$natom;\$i++)
{
\$_=<>;
chomp;
\$_=~s/^/ /;
my @helpat=split(/[\t,\s]+/);
\$cart[\$nconf][\$i][1]=\$b[1][1][1]*\$helpat[1]+\$b[1][1][2]*\$helpat[2]+\$b[1][1][3]*\$helpat[3];
\$cart[\$nconf][\$i][2]=\$b[1][2][1]*\$helpat[1]+\$b[1][2][2]*\$helpat[2]+\$b[1][2][3]*\$helpat[3];
\$cart[\$nconf][\$i][3]=\$b[1][3][1]*\$helpat[1]+\$b[1][3][2]*\$helpat[2]+\$b[1][3][3]*\$helpat[3];
}
if (\$ensemble_type eq "NpT")
{
\$z=0;
}
}
last if eof;
}

if (\$confmin>\$nconf)
{
print "Error, confmin larger than number of configurations. Exiting...\n";
exit;
}
if (\$confmax>\$nconf)
{
\$confmax=\$nconf;
}

for (my \$i=1;\$i<=\$nconf;\$i++)
{
#calculate lattice vector lengths
\$vec_len[\$i][1]=(\$b[\$i][1][1]*\$b[\$i][1][1]+\$b[\$i][1][2]*\$b[\$i][1][2]+\$b[\$i][1][3]*\$b[\$i][1][3])**0.5;
\$vec_len[\$i][2]=(\$b[\$i][2][1]*\$b[\$i][2][1]+\$b[\$i][2][2]*\$b[\$i][2][2]+\$b[\$i][2][3]*\$b[\$i][2][3])**0.5;
\$vec_len[\$i][3]=(\$b[\$i][3][1]*\$b[\$i][3][1]+\$b[\$i][3][2]*\$b[\$i][3][2]+\$b[\$i][3][3]*\$b[\$i][3][3])**0.5;
#calculate volume of cell
\$vol[\$i]=\$b[\$i][1][1]*\$b[\$i][2][2]*\$b[\$i][3][3]+\$b[\$i][1][2]*\$b[\$i][2][3]*\$b[\$i][3][1]+\$b[\$i][1][3]*\$b[\$i][2][1]*\$b[\$i][3][2]-\$b[\$i][3][1]*\$b[\$i][2][2]*\$b[\$i][1][3]-\$b[\$i][3][2]*\$b[\$i][2][3]*\$b[\$i][1][1]-\$b[\$i][3][3]*\$b[\$i][2][1]*\$b[\$i][1][2];
\$av_vol=\$av_vol+\$vol[\$i];
}
\$av_vol=\$av_vol/\$nconf;

#choose species 1 for which pair correlation function is going to be calculated
\$atmin_1=1;
if (\$species_1>1)
{
for (my \$i=1;\$i<\$species_1;\$i++)
{
\$atmin_1=\$atmin_1+\$elements[\$i];
}
}
\$atmax_1=\$atmin_1+\$elements[\$species_1]-1;
#choose species 2 to which paircorrelation function is calculated to
\$atmin_2=1;
if (\$species_2>1)
{
for (my \$i=1;\$i<\$species_2;\$i++)
{
\$atmin_2=\$atmin_2+\$elements[\$i];
}
}
\$atmax_2=\$atmin_2+\$elements[\$species_2]-1;
#initialize pair correlation function
for (my \$i=0;\$i<=(\$nr-1);\$i++)
{
\$pcf[\$i]=0.0;
}
# loop over configurations, make histogram of pair correlation function
for (my \$j=\$confmin;\$j<=\$confmax;\$j=\$j+\$confskip)
{
for (my \$k=\$atmin_1;\$k<=\$atmax_1;\$k++)
{
for (my \$l=\$atmin_2;\$l<=\$atmax_2;\$l++)
{
if (\$k==\$l) {next};
for (my \$g_x=-\$mult_x;\$g_x<=\$mult_x;\$g_x++)
{
for (my \$g_y=-\$mult_y;\$g_y<=\$mult_y;\$g_y++)
{
for (my \$g_z=-\$mult_y;\$g_z<=\$mult_z;\$g_z++)
{
my \$at2_x=\$cart[\$j][\$l][1]+\$vec_len[\$j][1]*\$g_x;
my \$at2_y=\$cart[\$j][\$l][2]+\$vec_len[\$j][2]*\$g_y;
my \$at2_z=\$cart[\$j][\$l][3]+\$vec_len[\$j][3]*\$g_z;
my \$dist=(\$cart[\$j][\$k][1]-\$at2_x)**2.0+(\$cart[\$j][\$k][2]-\$at2_y)**2.0+(\$cart[\$j][\$k][3]-\$at2_z)**2.0;
\$dist=\$dist**0.5;
#determine integer multiple
my \$zz=int((\$dist-\$rmin)/\$dr+0.5);
if (\$zz<\$nr)
{
\$pcf[\$zz]=\$pcf[\$zz]+1.0;
}
}
}
}
}
}
}

#make ensemble average, rescale functions and print
for (my \$i=0;\$i<=(\$nr-1);\$i++)
{
my \$r=\$rmin+\$i*\$dr;
if (\$r<\$tol)
{
\$pcf[\$i]=0.0;
}
else
{
\$pcf[\$i]=\$pcf[\$i]*\$av_vol/(4*\$pi*\$r*\$r*\$dr*((\$confmax-\$confmin)/\$confskip)*(\$atmax_2-\$atmin_2+1)*(\$atmax_1-\$atmin_1+1));#*((2.0*\$mult_x+1.0)*(2.0*\$mult_y+1.0)*(2.0*\$mult_z+1.0)));
}
print \$r," ",\$pcf[\$i],"\n";
}

and process the previously saved XDATCAR files

perl pair_correlation_function.pl XDATCAR.MLFF_3ps > pair_MLFF_3ps.dat
perl pair_correlation_function.pl XDATCAR.AI_3ps > pair_AI_3ps.dat

We can compare the pair correlation functions, e.g. with gnuplot using the following command

gnuplot -e "set terminal jpeg; set xlabel 'r(Ang)'; set ylabel 'PCF'; set style data lines; plot 'pair_MLFF_3ps.dat', 'pair_AI_3ps.dat' " > PC_MLFF_vs_AI_3ps.jpg

The pair correlation functions obtained that way should look similar to this figure

We see that pair correlation is quite well reproduced although the error on the force is a little bit large. This demonstrates a drastic improvement of the time consuming melting a crystal by machine learning and maybe useful even in situations where full ab initio accuracy for the liquid MD is required, because the initial equilibration phase is accelerated.

### Obtaining a more accurate force field

Including the melting phase in the force field may impact the accuracy of the force field. To improve it is usually advisable to learn on the pure structures, which in our case this means to use the CONTCAR file obtained after the melting. Furthermore, the force field was learned using only a single k point so that we can improve the accuracy of the reference data by including more k points. In most calculations, it is important to conduct accurate ab initio calculations since bad reference data can limit the accuracy or even inhibit the learning of a force field.

We restart from the liquid structure obtained before

cp POSCAR.T2000_relaxed POSCAR

and change the following INCAR tags

ALGO = Normal
ML_FF_LMLFF = .TRUE.
ML_FF_ISTART = 0
NSW = 10000

If you run have resources to run in parallel, you can reduce the computation time further by adding k point parallelization with the KPAR tag. We use a denser k-point mesh in the KPOINTS file 2x2x2 Gamma-centered mesh

0 0 0
Gamma
2 2 2
0 0 0

We will learn a new force field on a run of 30 ps. Keep in mind to run the calculation using the standard version of VASP (usually vasp_std). After running the calculation, we examine the error in the ML_LOGFILE, where the obtained values should be close to

====================================================================================================
Information on error estimations
----------------------------------------------------------------------------------------------------
Error in energy (eV atom^-1):     0.006079
Error in force  (eV Angst^-1):     0.165678
Error in stress (kB):     2.553148
Bayesian error (eV Angst-1):     0.044695    0.101357
Spilling factor (-):     0.002285    0.020000
====================================================================================================

We immediately see that the errors are significantly lower than in the previous calculation with only one k point.

To understand how the force field is learned, we inspect the ML_ABNCAR file containing the training data. In the beginning of this file, you will find information about the number of reference structures

**************************************************
The number of configurations
--------------------------------------------------
55

and the size of the basis set

**************************************************
The numbers of basis sets per atom type
--------------------------------------------------
396

We will monitor how much training data is added to a structure after each learning cycle and what impact this has on the accuracy of the force field. First we will perform a continuation run keeping the Bayesian threshold fixed

cp ML_ABNCAR ML_ABCAR
cp CONTCAR POSCAR

and set the following INCAR tags

ML_FF_ISTART = 1
ML_FF_CTIFOR = x

where x is the threshold of the Bayesian error obtained from the ML_LOGFILE (highlighted bold above; please use the value obtained in your calculation). By setting a good estimate for the threshold several ab initio steps are skipped in the first few steps, which would be required to determine the threshold automatically. Mind: When continuation runs are performed for different crystal structures, the last previous threshold for the Bayesian error might be too large leading to premature skipping of ab initio steps. This is particular relevant when studying the liquid phase, first, and applying its threshold to the solid phase. In that case it is safer to use the default value for ML_FF_CTIFOR = ${\displaystyle 10^{-16}}$.

After running the calculation, we inspect the last instance of the errors in the ML_LOGFILE, which should look similar to

====================================================================================================
Information on error estimations
----------------------------------------------------------------------------------------------------
Error in energy (eV atom^-1):     0.006252
Error in force  (eV Angst^-1):     0.168549
Error in stress (kB):     2.606332
Bayesian error (eV Angst-1):     0.037569    0.101357
Spilling factor (-):     0.001114    0.020000
====================================================================================================

We see that the accuracy has not changed significantly and the threshold for the Bayesian error is set to the one specified in the INCAR file. Looking at the actual errors we observe that the threshold was exceeded only a few times. You can find all these instances with the command

grep "Bayesian error (eV Angst-1):" ML_LOGFILE | awk '{if (\$5 > \$6) { print \$5 }}'

Consequently only very few structures were added to the ab initio data in the ML_ABNCAR file. The entry for the reference structures and basis sets should look similar to

**************************************************
The number of configurations
--------------------------------------------------
65
...
**************************************************
The numbers of basis sets per atom type
--------------------------------------------------
407

From this observation, we conclude either that the force field is already accurate enough or that the threshold is set too high. In the latter case, the machine would not conduct ab initio calculations because the force field is judged accurate enough according to the threshold.

We compare this behavior to the case of using the default value for ML_FF_CTIFOR. We continue from the existing structure and force field

cp CONTCAR POSCAR
cp ML_ABNCAR ML_ABCAR

Remove the ML_FF_CTIFOR tag from the INCAR file.

The default value for ML_FF_CTIFOR is practically zero, so we expect more structures to be added. We inspect the ML_ABNCAR to investigate how the number of structures and the basis set size changed

**************************************************
The number of configurations
--------------------------------------------------
160
...
**************************************************
The numbers of basis sets per atom type
--------------------------------------------------
726

Even though the number of structure datasets and the basis set size increased, the accuracy of the force field did not improve significantly

====================================================================================================
Information on error estimations
----------------------------------------------------------------------------------------------------
Error in energy (eV atom^-1):     0.005950
Error in force  (eV Angst^-1):     0.162064
Error in stress (kB):     2.628567
Bayesian error (eV Angst-1):     0.029723    0.061348
Spilling factor (-):     0.000406    0.020000
====================================================================================================

Ideally, one should continue learning until no structures need to be added to the training data and basis set anymore. In practice, the prediction of the Bayesian error exhibits numerical inaccuracies so that an ab initio calculation is conducted even if the force field is accurate enough. Hence, in this tutorial, we stop at this point because further addition of training data does not improve the accuracy on the test structures compared to the ab initio data. For high quality production calculations suited for publications, we advise to learn until almost no data is added to the training structures and basis set. In many cases this involves a runtime of at least 100 ps for learning.

Continue learning and observe the convergence of the number of reference structures. Most likely the maximum allowed basis set size ML_FF_MB_MB needs to be increased. How does the pair correlation function compare to ab initio data?

### Optimizing weights of energy, forces and stress

Even after a force field has been learned, there is still the possibility to refine it by changing the weights for energy ML_FF_WTOTEN, forces ML_FF_WTIFOR and stress ML_FF_WTSIF. By default all weights are set to 1.0, so energy, force and stress are equally important. Typically the error for a component decreases with increasing weight of that component. For methods where the free energy is important (such as e.g. Interface pinning calculations, Metadynamics etc.) introducing a larger weight for the energy can reduce the error of the energy significantly while only slightly increasing the errors of the forces and the stress. Note that the weights are set relative to each other, i.e., by increasing one the other ones decrease automatically.

We will start from the force field obtained above, but you can use a more refined one if you completed the user task cp INCAR INCAR.run cp CONTCAR POSCAR.run The INCAR.run file will serve as a template for several short calculations with different weights for the energy. Set

• Copy CONTCAR to POSCAR.run. For each calculation the same POSCAR file needs to be used as a starting point.
• Copy ML_ABNCAR to ML_ABCAR.
• Modify INCAR.run as follows:
• Use the bash script run_param_opt.sh (or use your own) which contains the following lines:
#!/bin/bash

if [ -f error_vs_eweight.dat ]; then
rm error_vs_eweight.dat
fi

touch error_vs_eweight.dat

for i in 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
do
cat INCAR.run > INCAR
echo "ML_FF_WTOTEN = \$i" >> INCAR
cp POSCAR.run POSCAR
mpirun -np \$np \$executable_path/vasp_std
cp ML_LOGFILE ML_LOGFILE.\$i
awk -v var="\$i" '/Error in energy \(eV atom\^-1\)\:/{energy=\$6} /Error in force  \(eV Angst\^-1\)\:/ {force=\$6} /Error in stress \(kB\)\:/{stress=\$5} /Energy SD. \(eV atom\^-1\)\:/{energy_var=\$5}  /Force SD. \(eV Angst\^-1\)\:/{force_var=\$5} /Stress SD. \(kB\)\:/ {stress_var=\$4} END{print var,energy/energy_var+force/force_var+stress/stress_var}' ML_LOGFILE.\$i >> error_vs_eweight.dat
done

The user needs to set the executable path \$executable_path in this directory (and maybe the executable name). This script adds the weight for the energy ML_FF_WTOTEN = \$i to the INCAR file, where \$i runs from 1.0 to 5.0 with a step size of 0.5. After that a single step calculation is performed. The errors compared to ab initio and standard deviations for energies, forces and stress are output to the ML_LOGFILE and from that file the sum of the normalized errors is plotted against the weight of the energy. The normalization is done by dividing the errors by the standard deviation for each component.

The normalized total error vs. weight for the energy is plotted from the file error_vs_eweight.dat:

gnuplot -e "set terminal jpeg; set xlabel 'Energy weight'; set ylabel 'Error'; set style data lines; plot 'error_vs_eweight.dat'" > error_vs_eweight.jpg

It should look similar to:

From this plot we see that an ideal value is around ML_FF_WTOTEN=2.5. Next we take a look at the actual errors for that weighting parameter in the file ML_LOGFILE.2.5:

====================================================================================================
Information on error estimations
----------------------------------------------------------------------------------------------------
Error in energy (eV atom^-1):     0.005322
Error in force  (eV Angst^-1):     0.169356
Error in stress (kB):     2.849817
Bayesian error (eV Angst-1):     0.025548    0.000000
Spilling factor (-):     0.000368    0.020000
====================================================================================================

We see that the error on energy decreased while it increased for forces and stress compared to ML_LOGFILE.1.0:

====================================================================================================
Information on error estimations
----------------------------------------------------------------------------------------------------
Error in energy (eV atom^-1):     0.005954
Error in force  (eV Angst^-1):     0.162006
Error in stress (kB):     2.628128
Bayesian error (eV Angst-1):     0.028542    0.000000
Spilling factor (-):     0.000368    0.020000
====================================================================================================

In this example the decrease of the energy is only small but in many examples this can lead to signifcant increase in the precision.

Here are the errors for ML_LOGFILE_5.0:

====================================================================================================
Information on error estimations
----------------------------------------------------------------------------------------------------
Error in energy (eV atom^-1):     0.004692
Error in force  (eV Angst^-1):     0.179448
Error in stress (kB):     3.180711
Bayesian error (eV Angst-1):     0.023225    0.000000
Spilling factor (-):     0.000368    0.020000
====================================================================================================

We see that for higher values of the weight the energy is still getting more accurate but the forces and stress are getting loosing accuracy at a higher, such that the overall error gets worse. We advise to use the optimal value.

If needed this waiting can be done in a similar way for the forces and stress.

### Production runs using the optimized parameters

The production runs using the optimized parameters are run as normal calculations only using an additional line for the weight on energy. We repeat here which steps have to be performed:

User task: Calculate the pair correlation function as above with and without machine learning starting from the same POSCAR and compare the results.