Liquid Si - MLFF: Difference between revisions

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In this example we start from a 64 atom super cell of diamond-fcc Si (the same as in {{TAG|Liquid Si - Standard MD}}):
In this example we start from a 64 atom super cell of diamond-fcc Si (the same as in {{TAG|Liquid Si - Standard MD}}):
  Si cubic diamond 2x2x2 super cell of conventional cell
  Si_CD_2x2x2
       5.43090000000000
       5.43090000000000
     2.00000000  0.00000000  0.00000000
     2.00000000  0.00000000  0.00000000
Line 114: Line 114:
  {{TAGBL|NSW}} = 10000
  {{TAGBL|NSW}} = 10000
  {{TAGBL|POTIM}} = 3.0
  {{TAGBL|POTIM}} = 3.0
{{TAGBL|RANDOM_SEED}} =          88951986                0                0
   
   
  #Machine learning paramters
  #Machine learning paramters
  {{TAGBL|ML_FF_LMLFF}} = .TRUE.
  {{TAGBL|ML_LMLFF}} = .TRUE.
  {{TAGBL|ML_FF_ISTART}} = 0
  {{TAGBL|ML_ISTART}} = 0
{{TAGBL|ML_FF_EATOM}} = -.70128086E+00
 
;{{TAG|ML_FF_LMLFF}} = ''.TRUE.'': switches on the machine learning of the force field
;{{TAG|ML_FF_ISTART}} = 0: corresponds to learning from scratch
;{{TAG|ML_FF_EATOM}} = -.70128086E+00: sets the atomic reference energy for each species. We show to obtain this energy below.
;{{TAG|IWAVPR}} = 1: extrapolate the total charge density from the fixed atomic ones. This is more advisable than the default wavefunction-based extrapolation from previous steps, because during the on-the-fly learning, several thousands of molecular dynamics steps can lie between two ab initio calculations.


== Calculation ==
== 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 {{TAG|INCAR}} file
#Basic parameters
{{TAG|ISMEAR}} = 0
{{TAG|SIGMA}} = 0.1
{{TAG|LREAL}} = Auto
{{TAG|ISYM}} = 0
{{TAG|NELM}} = 100
{{TAG|EDIFF}} = 1E-4
{{TAG|LWAVE}} = .FALSE.
{{TAG|LCHARG}} = .FALSE.
{{TAG|ISPIN}} = 2
The {{TAG|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 {{TAG|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 {{TAG|OUTCAR}} or {{TAG|OSZICAR}} file. We use this energy for the {{TAG|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 {{TAG|ML_FF_EATOM}} tag. The ordering of the energies corresponds to the ordering of the elements in the {{TAG|POTCAR}} file.


=== Creating the liquid structure ===
=== Creating the liquid structure ===
Line 168: Line 129:
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.
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
The main output files for the machine learning are
;{{TAG|ML_ABNCAR}}: contains the ab initio structure datasets used for the learning. It will be needed for continuation runs as {{TAG|ML_ABCAR}}.
;{{TAG|ML_ABN}}: contains the ab initio structure datasets used for the learning. It will be needed for continuation runs as {{TAG|ML_AB}}.
;{{TAG|ML_FFNCAR}}: contains the regression results (weights, parameters, etc.). It will be needed for continuation runs as {{TAG|ML_FFCAR}}.
;{{TAG|ML_FFN}}: contains the regression results (weights, parameters, etc.). It will be needed for continuation runs as {{TAG|ML_FF}}.
;{{TAG|ML_LOGFILE}}: logging the proceedings of the machine learning. With {{TAG|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):
;{{TAG|ML_LOGFILE}}: logging the proceedings of the machine learning. This file consists of keywords that are nicely "grepable." The keywords are explained in the in the beginning of the file and upon "grepping". The status of each MD step is given by the keyword "STATUS". Please invoke the following command:
  ====================================================================================================
grep STATUS ML_LOGFILE
      Information on error estimations
The output should look similar to the following:
  ----------------------------------------------------------------------------------------------------
# STATUS ###############################################################
               Error in energy (eV atom^-1):    0.016994
# STATUS This line describes the overall status of each step.
              Error in force (eV Angst^-1):    0.247664
  # STATUS
                      Error in stress (kB):    5.368218
# STATUS nstep ..... MD time step or input structure counter
                Bayesian error (eV Angst-1):    0.064904    0.127195
  # STATUS state ..... One-word description of step action
                        Spilling factor (-):    0.000690    0.020000
# STATUS            - "accurate"  (1) : Errors are low, force field is used
  ====================================================================================================
# STATUS            - "threshold" (2) : Errors exceeded threshold, structure is sampled from ab initio
You can see that additional information about the Bayesian error and the spilling factor is reported.
# STATUS            - "learning"  (3) : Stored configurations are used for training force field
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 kept constant during the calculations.
# STATUS            - "critical"  (4) : Errors are high, ab initio sampling and learning is enforced
# STATUS            - "predict"  (5) : Force field is used in prediction mode only, no error checking
# STATUS is ........ Integer representation of above one-word description (integer in parenthesis)
# STATUS doabin .... Perform ab initio calculation (T/F)
# STATUS iff ....... Force field available (T/F, False after startup hints to possible convergence problems)
# STATUS nsample ... Number of steps since last reference structure collection (sample = T)
# STATUS ngenff .... Number of steps since last force field generation (genff = T)
# STATUS ###############################################################
# STATUS            nstep    state is doabin    iff  nsample    ngenff
# STATUS               2        3  4      5      6        7        8
# STATUS ###############################################################
STATUS                  0 threshold  2      T      F        0        0
STATUS                  1 critical  4      T      F        0        1
STATUS                  2 critical  4      T      F        0        2
STATUS                  3 critical  4      T      T        0        1
STATUS                  4 critical  4      T      T        0        1
STATUS                  5 critical  4      T      T        0        1
      .                  .        .  .      .      .        .        .
      .                  .        .  .      .      .        .        .
      .                  .        .  .      .      .        .        .
STATUS              9997 accurate  1      F      T      945      996
STATUS              9998 accurate  1      F      T      946      997
STATUS              9999 accurate  1      F      T      947      998
STATUS              10000 learning  3      T      T      948      999
 
Another important keyword is "ERR". For this instance we should type the following command:
grep ERR ML_LOGFILE
The output should look like the following:
# ERR ######################################################################
# ERR This line contains the RMSEs of the predictions with respect to ab initio results for the training data.
# ERR
# ERR nstep ......... MD time step or input structure counter
# ERR rmse_energy ... RMSE of energies (eV atom^-1)
  # ERR rmse_force .... RMSE of forces (eV Angst^-1)
# ERR rmse_stress ... RMSE of stress (kB)
# ERR ######################################################################
# ERR              nstep      rmse_energy      rmse_force      rmse_stress
# ERR                  2                3                4                5
# ERR ######################################################################
ERR                    0  0.00000000E+00  0.00000000E+00  0.00000000E+00
ERR                    1   0.00000000E+00  0.00000000E+00  0.00000000E+00
ERR                    2  0.00000000E+00  0.00000000E+00  0.00000000E+00
  ERR                    3  2.84605192E-05  9.82351889E-03  2.40003743E-02
ERR                    4  1.83193349E-05  1.06700600E-02  5.37606479E-02
ERR                    5  4.12132223E-05  1.34123085E-02  1.01588957E-01
ERR                    6  9.51627413E-05  1.90335214E-02  1.31959103E-01
.                      .                .                .                .
.                      .                .                .                .
.                      .                .                .                .
ERR                  9042  1.07159240E-02  2.41283323E-01  4.95695745E+00
ERR                  9052  1.07159240E-02  2.41283323E-01  4.95695745E+00
ERR                10000  1.07159240E-02  2.41283323E-01  4.95695745E+00
 
This tag lists the errors on the energy, forces and stress of the force field compared to the ab initio results on the available training data. The second column shows the MD step. We see that the entry is not output at every MD step. The errors only change if the force field is updated, hence when an ab initio calculation is executed (it should correlate with the doabin column of the STATUS keyword). The other three columns show the errors on the energy (eV/atom), forces (ev/Angstrom) and stress (kB).


=== Structral properties of the force field ===
=== Structral properties of the force field ===
Line 189: Line 203:
Now, we proceed with the force field calculation and set up the required files
Now, we proceed with the force field calculation and set up the required files
  cp POSCAR.T2000_relaxed {{TAGBL|POSCAR}}
  cp POSCAR.T2000_relaxed {{TAGBL|POSCAR}}
  cp {{TAGBL|ML_ABNCAR}} {{TAGBL|ML_ABCAR}}
  cp {{TAGBL|ML_FFN}} {{TAGBL|ML_FF}}
cp {{TAGBL|ML_FFNCAR}} {{TAGBL|ML_FFCAR}}
To run a shorter simulation using only the force field, we change the following {{TAG|INCAR}} tags to
To run a shorter simulation using only the force field, we change the following {{TAG|INCAR}} tags to
  {{TAGBL|ML_FF_ISTART}} = 2
  {{TAGBL|ML_ISTART}} = 2
  {{TAGBL|NSW}} = 1000
  {{TAGBL|NSW}} = 1000
After the calculation finished, we backup the history of the atomic positions
After the calculation finished, we backup the history of the atomic positions
  cp {{TAGBL|XDATCAR}} XDATCAR.MLFF_3ps
  cp {{TAGBL|XDATCAR}} XDATCAR.MLFF_3ps
To analyze the pair correlation function, we use the PERL script ''pair_correlation_function.pl''
<div class="toccolours mw-customtoggle-script">'''Click to show/hide pair_correlation_function.pl'''</div>
<div class="mw-collapsible mw-collapsed" id="mw-customcollapsible-script">
<pre>
#!/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
#setting radial grid
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
#reading in XDATCAR file
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;
}


We repeat the same process for the ab initio simulation recovering the same initial structure
if ($confmin>$nconf)
cp POSCAR.T2000_relaxed {{TAGBL|POSCAR}}
{
switching the machine learning off in the {{TAG|INCAR}} file by setting
  print "Error, confmin larger than number of configurations. Exiting...\n";
  {{TAGBL|ML_FF_LMLFF}} = .FALSE.
  exit;
We store the trajectory once the calculation completed
}
cp {{TAGBL|XDATCAR}} XDATCAR.AI_3ps
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;


To analyze the pair correlation function, we use the PERL script ''pair_correlation_function.pl''
#choose species 1 for which pair correlation function is going to be calculated
<div class="toccolours mw-customtoggle-script">'''Click to show/hide pair_correlation_function.pl'''</div>
$atmin_1=1;
<div class="mw-collapsible mw-collapsed" id="mw-customcollapsible-script">{{:Pair correlation script}}</div>
if ($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";
}
</pre>
</div>
and process the previously saved {{TAG|XDATCAR}} files
and process the previously saved {{TAG|XDATCAR}} files
  perl pair_correlation_function.pl XDATCAR.MLFF_3ps > pair_MLFF_3ps.dat
  perl pair_correlation_function.pl XDATCAR.MLFF_3ps > pair_MLFF_3ps.dat
  perl pair_correlation_function.pl XDATCAR.AI_3ps > pair_AI_3ps.dat
 
To save time the pair correlation function for 1000 ab initio MD steps is precalculated in the file ''pair_AI_3ps.dat''.
 
The interested user can of course calculate the results of the ab initio MD by rerunning the above steps while switching off machine learning via
  {{TAGBL|ML_LMLFF}} = .FALSE.
 
We can compare the pair correlation functions, e.g. with gnuplot using the following command
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
  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
Line 216: Line 437:
[[File:PC MLFF vs AI 3ps.jpg|400px]]
[[File:PC MLFF vs AI 3ps.jpg|400px]]


We see that pair correlation is quite well reproduced although the error on the force of 0.247664 eV/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.
We see that pair correlation is quite well reproduced although the error in the force of ~0.242 eV/<math>\AA</math> shown above is a little bit too large. This error is maybe too large for accurate production calculations (usually an accuracy of approximately 0.1 eV/<math>\AA</math> is targeted), but since the pair correlation function is well reproduced it is perfectly fine to use this on-the-fly force field in the time-consuming melting of the crystal.


=== Obtaining a more accurate force field ===
=== Obtaining a more accurate force field ===
Line 226: Line 447:
and change the following {{TAG|INCAR}} tags
and change the following {{TAG|INCAR}} tags
  {{TAGBL|ALGO}} = Normal
  {{TAGBL|ALGO}} = Normal
  {{TAGBL|ML_FF_LMLFF}} = .TRUE.
  {{TAGBL|ML_LMLFF}} = .TRUE.
  {{TAGBL|ML_FF_ISTART}} = 0
  {{TAGBL|ML_ISTART}} = 0
  {{TAGBL|NSW}} = 10000
  {{TAGBL|NSW}} = 1000
If you run have resources to run in parallel, you can reduce the computation time further by adding k point parallelization with the {{TAG|KPAR}} tag.
If you run have resources to run in parallel, you can reduce the computation time further by adding k point parallelization with the {{TAG|KPAR}} tag.
We use a denser k-point mesh in the {{TAG|KPOINTS}} file
We use a denser k-point mesh in the {{TAG|KPOINTS}} file
Line 236: Line 457:
   2 2 2
   2 2 2
   0 0 0
   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'').
We will learn a new force field with 1000 MD steps (each of 3 fs). 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 {{TAG|ML_LOGFILE}}, where the obtained values should be close to
After running the calculation, we examine the error "ERR" in the {{TAG|ML_LOGFILE}} by typing:
  ====================================================================================================
grep "ERR" ML_LOGFILE
      Information on error estimations
where the last entries should be close to
  ----------------------------------------------------------------------------------------------------
  ERR                  886  5.98467749E-03  1.48190308E-01  2.38264786E+00
              Error in energy (eV atom^-1):    0.006079
  ERR                  908  5.98467749E-03  1.48190308E-01  2.38264786E+00
              Error in force (eV Angst^-1):    0.165678
ERR                  925  5.98467749E-03  1.48190308E-01  2.38264786E+00
                      Error in stress (kB):    2.553148
ERR                  959  5.98467749E-03  1.48190308E-01  2.38264786E+00
                Bayesian error (eV Angst-1):    0.044695    '''0.101357'''
  ERR                  980  5.98467749E-03  1.48190308E-01  2.38264786E+00
                        Spilling factor (-):    0.002285    0.020000
ERR                  990  5.99559653E-03  1.50261779E-01  2.40349561E+00
====================================================================================================
ERR                  1000  5.99559653E-03  1.50261779E-01  2.40349561E+00


We immediately see that the errors are significantly lower than in the previous calculation with only one k point. This is due to the less noisy ab initio data which is easier to learn.
We immediately see that the errors for the forces are significantly lower than in the previous calculation with only one k point. This is due to the less noisy ab initio data which is easier to learn.


To understand how the force field is learned, we inspect the {{TAG|ML_ABNCAR}} file containing the training data. In the beginning of this file, you will find information about the number of reference structures
To understand how the force field is learned, we inspect the {{TAG|ML_ABN}} file containing the training data. In the beginning of this file, you will find information about the number of reference structures for training
  1.0 Version
  **************************************************
  **************************************************
       The number of configurations
       The number of configurations
  --------------------------------------------------
  --------------------------------------------------
           55
           48
and the size of the basis set
and the number of local reference configurations (size of the basis set)
  **************************************************
  **************************************************
       The numbers of basis sets per atom type
       The numbers of basis sets per atom type
  --------------------------------------------------
  --------------------------------------------------
         396
         382


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
We will further continue the training a 1000 MD steps and see how the number of training structures and the number of local reference configurations change.  
  cp {{TAGBL|ML_ABNCAR}} {{TAGBL|ML_ABCAR}}
Do the following steps:
  cp {{TAGBL|ML_ABN}} {{TAGBL|ML_AB}}
  cp {{TAGBL|CONTCAR}} {{TAGBL|POSCAR}}
  cp {{TAGBL|CONTCAR}} {{TAGBL|POSCAR}}
and set the following {{TAG|INCAR}} tags
and set the following {{TAG|INCAR}} tag
  {{TAGBL|ML_FF_ISTART}} = 1
  {{TAGBL|ML_ISTART}} = 1
{{TAGBL|ML_FF_CTIFOR}} = ''x''
where ''x'' is the threshold of the Bayesian error obtained from the {{TAG|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 {{TAG|ML_FF_CTIFOR}} = <math>10^{-16}</math>.


After running the calculation, we inspect the last instance of the errors in the {{TAG|ML_LOGFILE}}, which should look similar to
After running the calculation, we inspect the last instance of the errors in the {{TAG|ML_LOGFILE}} by typing:
  ====================================================================================================
  grep ERR ML_LOGFILE
      Information on error estimations
The last few lines should have values close to:
  ----------------------------------------------------------------------------------------------------
  ERR                  675  5.10937061E-03  1.46895065E-01  2.50094941E+00
              Error in energy (eV atom^-1):    0.006252
ERR                  861  5.10937061E-03  1.46895065E-01  2.50094941E+00
              Error in force (eV Angst^-1):    0.168549
  ERR                  924  5.10937061E-03  1.46895065E-01  2.50094941E+00
                      Error in stress (kB):    2.606332
ERR                  942  5.01460989E-03  1.47816836E-01  2.47329693E+00
                Bayesian error (eV Angst-1):    0.037569    0.101357
ERR                  1000  5.01460989E-03  1.47816836E-01  2.47329693E+00
                        Spilling factor (-):    0.001114    0.020000
We see that the accuracy has changed slightly. We also look at the ML_ABN file and the number of reference structures for training should increase compared to the run before
====================================================================================================
  1.0 Version
We see that the accuracy has not changed significantly and the threshold for the Bayesian error is set to the one specified in the {{TAG|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 {{TAG|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 {{TAG|ML_FF_CTIFOR}}. We continue from the existing structure and force field
cp {{TAGBL|CONTCAR}} {{TAGBL|POSCAR}}
cp {{TAGBL|ML_ABNCAR}} {{TAG|ML_ABCAR}}
Remove the {{TAG|ML_FF_CTIFOR}} tag from the {{TAG|INCAR}} file.
The default value for {{TAG|ML_FF_CTIFOR}} is practically zero, so we expect more structures to be added. We inspect the {{TAG|ML_ABNCAR}} to investigate how the number of structures and the basis set size changed
  **************************************************
       The number of configurations
       The number of configurations
  --------------------------------------------------
  --------------------------------------------------
        160
          99
...
Also the number of local reference configurations (basis sets) increases compared to the previous calculation
  **************************************************
  **************************************************
       The numbers of basis sets per atom type
       The numbers of basis sets per atom type
  --------------------------------------------------
  --------------------------------------------------
         726
         665
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.
 
;Exercise: Continue learning and observe the convergence of the number of reference structures. Most likely the maximum allowed basis set size {{TAG|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 {{TAG|ML_FF_WTOTEN}}, forces {{TAG|ML_FF_WTIFOR}} and stress {{TAG|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. {{TAG|Interface pinning calculations}}, {{TAG|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 {{TAGBL|INCAR}} INCAR.run
cp {{TAGBL|CONTCAR}} POSCAR.run
The ''INCAR.run'' file  will serve as a template for several short calculations with different weights for the energy. Set
{{TAG|ML_FF_ISTART}} = 1
{{TAG|NSW}} = 1
to generate a single new test structure.
 
We can now use a script to set {{TAG|ML_FF_WTOTEN}} to different values and examine how the average error changes
#!/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
To run this script you need to
export executable_path=''path to the vasp executable''
export np=''number of processes you want to use''
This script sets the weight of the energy {{TAG|ML_FF_WTOTEN}} to values between 1.0 and 5.0. It calculates a single structure and compares the error of energy, forces and stress between ab initio calculation and force field. These errors are normalized with respect to the standard deviations of these quantities. The sum of the normalized errors is plotted against the weight of the energy with the following command
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
 
[[File:Error vs eweight.jpg|400px]]
 
From this plot we see that an ideal value is around {{TAG|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 significant increase in the precision.
Compare these values to the errors in ''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 even more accurate, but at the expense of the forces and the stress. Consequently the normalized error increases so that the trajectories during the MD simulation may suffer. We advise not to push the weight of the energy to these extreme values and instead use the value optimizing the overall error.
 
If needed this weighting can be done in a similar way for the forces and stress.
 
=== Production runs using the optimized parameters ===


In the production run, we rely on the structure dataset and the corresponding accurate force field
cp {{TAGBL|CONTCAR}} {{TAG|POSCAR}}
cp {{TAGBL|ML_ABNCAR}} {{TAG|ML_ABCAR}}.
cp {{TAGBL|ML_FFNCAR}} {{TAG|ML_FFCAR}}.
In the {{TAG|INCAR}} file, we switch of the updating of the force field and use the optimized energy
{{TAGBL|ML_FF_ISTART}} = 2
{{TAGBL|ML_FF_WTOTEN}} = 2.5
and adjust the necessary molecular dynamics parameters (like e.g. {{TAG|NSW}} etc.).


;Exercise: Calculate the pair correlation function as above with the machine learned force field and compare to results obtained ab initio when using the same {{TAG|POSCAR}}.
Ideally, one should continue learning until no structures need to be added to the training data and basis set anymore. Very often this can take up to 100ps depending on the material and conditions. In practice, the prediction of the Bayesian error exhibits numerical inaccuracies so that an ab initio calculation is conducted from time to time even if the force field is accurate enough. So measuring only the decreasing frequency of addition of new data is not sufficient to know when to finish. One should also look at the accuracy of the force on the training data and more importantly on the accuracy on some test data that is outside of the training sets.


== Download ==
== Download ==
[http://www.vasp.at/vasp-workshop/examples/MLFF_Liquid_Si.tgz MLFF_Liquid_Si.tgz]
[[Media:MLFF_Liquid_Si_tutorial.tgz| MLFF_Liquid_Si_tutorial.tgz]]


{{Template:Machine learning force field - Tutorial}}
{{Template:Machine learning force field - Tutorial}}


[[Category:Examples]]
[[Category:Examples]]

Latest revision as of 15:13, 15 April 2022

Task

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_CD_2x2x2
     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

KPOINTS

We will start with a single k point in this example: 

K-Points
 0
Gamma
 1  1  1
 0  0  0

INCAR

#Basic parameters
ISMEAR = 0
SIGMA = 0.1
LREAL = Auto
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
RANDOM_SEED =          88951986                0                0

#Machine learning paramters
ML_LMLFF = .TRUE.
ML_ISTART = 0

Calculation

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_ABN
contains the ab initio structure datasets used for the learning. It will be needed for continuation runs as ML_AB.
ML_FFN
contains the regression results (weights, parameters, etc.). It will be needed for continuation runs as ML_FF.
ML_LOGFILE
logging the proceedings of the machine learning. This file consists of keywords that are nicely "grepable." The keywords are explained in the in the beginning of the file and upon "grepping". The status of each MD step is given by the keyword "STATUS". Please invoke the following command:
grep STATUS ML_LOGFILE

The output should look similar to the following: 

# STATUS ###############################################################
# STATUS This line describes the overall status of each step.
# STATUS 
# STATUS nstep ..... MD time step or input structure counter
# STATUS state ..... One-word description of step action
# STATUS             - "accurate"  (1) : Errors are low, force field is used
# STATUS             - "threshold" (2) : Errors exceeded threshold, structure is sampled from ab initio
# STATUS             - "learning"  (3) : Stored configurations are used for training force field
# STATUS             - "critical"  (4) : Errors are high, ab initio sampling and learning is enforced
# STATUS             - "predict"   (5) : Force field is used in prediction mode only, no error checking
# STATUS is ........ Integer representation of above one-word description (integer in parenthesis)
# STATUS doabin .... Perform ab initio calculation (T/F)
# STATUS iff ....... Force field available (T/F, False after startup hints to possible convergence problems)
# STATUS nsample ... Number of steps since last reference structure collection (sample = T)
# STATUS ngenff .... Number of steps since last force field generation (genff = T)
# STATUS ###############################################################
# STATUS            nstep     state is doabin    iff   nsample    ngenff
# STATUS                2         3  4      5      6         7         8
# STATUS ###############################################################
STATUS                  0 threshold  2      T      F         0         0
STATUS                  1 critical   4      T      F         0         1
STATUS                  2 critical   4      T      F         0         2
STATUS                  3 critical   4      T      T         0         1
STATUS                  4 critical   4      T      T         0         1
STATUS                  5 critical   4      T      T         0         1
     .                  .        .   .      .      .         .         .
     .                  .        .   .      .      .         .         .
     .                  .        .   .      .      .         .         .
STATUS               9997 accurate   1      F      T       945       996
STATUS               9998 accurate   1      F      T       946       997
STATUS               9999 accurate   1      F      T       947       998
STATUS              10000 learning   3      T      T       948       999

Another important keyword is "ERR". For this instance we should type the following command: 

grep ERR ML_LOGFILE

The output should look like the following: 

# ERR ######################################################################
# ERR This line contains the RMSEs of the predictions with respect to ab initio results for the training data.
# ERR 
# ERR nstep ......... MD time step or input structure counter
# ERR rmse_energy ... RMSE of energies (eV atom^-1)
# ERR rmse_force .... RMSE of forces (eV Angst^-1)
# ERR rmse_stress ... RMSE of stress (kB)
# ERR ######################################################################
# ERR               nstep      rmse_energy       rmse_force      rmse_stress
# ERR                   2                3                4                5
# ERR ######################################################################
ERR                     0   0.00000000E+00   0.00000000E+00   0.00000000E+00
ERR                     1   0.00000000E+00   0.00000000E+00   0.00000000E+00
ERR                     2   0.00000000E+00   0.00000000E+00   0.00000000E+00
ERR                     3   2.84605192E-05   9.82351889E-03   2.40003743E-02
ERR                     4   1.83193349E-05   1.06700600E-02   5.37606479E-02
ERR                     5   4.12132223E-05   1.34123085E-02   1.01588957E-01
ERR                     6   9.51627413E-05   1.90335214E-02   1.31959103E-01
.                       .                .                .                .
.                       .                .                .                .
.                       .                .                .                .
ERR                  9042   1.07159240E-02   2.41283323E-01   4.95695745E+00
ERR                  9052   1.07159240E-02   2.41283323E-01   4.95695745E+00
ERR                 10000   1.07159240E-02   2.41283323E-01   4.95695745E+00

This tag lists the errors on the energy, forces and stress of the force field compared to the ab initio results on the available training data. The second column shows the MD step. We see that the entry is not output at every MD step. The errors only change if the force field is updated, hence when an ab initio calculation is executed (it should correlate with the doabin column of the STATUS keyword). The other three columns show the errors on the energy (eV/atom), forces (ev/Angstrom) and stress (kB).

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_FFN ML_FF

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

ML_ISTART = 2
NSW = 1000

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

cp XDATCAR XDATCAR.MLFF_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
#setting radial grid 
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

#reading in XDATCAR file
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

To save time the pair correlation function for 1000 ab initio MD steps is precalculated in the file pair_AI_3ps.dat.

The interested user can of course calculate the results of the ab initio MD by rerunning the above steps while switching off machine learning via 

ML_LMLFF = .FALSE.

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 in the force of ~0.242 eV/ shown above is a little bit too large. This error is maybe too large for accurate production calculations (usually an accuracy of approximately 0.1 eV/ is targeted), but since the pair correlation function is well reproduced it is perfectly fine to use this on-the-fly force field in the time-consuming melting of the crystal.

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_LMLFF = .TRUE.
ML_ISTART = 0
NSW = 1000

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 with 1000 MD steps (each of 3 fs). Keep in mind to run the calculation using the standard version of VASP (usually vasp_std). After running the calculation, we examine the error "ERR" in the ML_LOGFILE by typing: 

grep "ERR" ML_LOGFILE

where the last entries should be close to 

ERR                   886   5.98467749E-03   1.48190308E-01   2.38264786E+00
ERR                   908   5.98467749E-03   1.48190308E-01   2.38264786E+00
ERR                   925   5.98467749E-03   1.48190308E-01   2.38264786E+00
ERR                   959   5.98467749E-03   1.48190308E-01   2.38264786E+00
ERR                   980   5.98467749E-03   1.48190308E-01   2.38264786E+00
ERR                   990   5.99559653E-03   1.50261779E-01   2.40349561E+00
ERR                  1000   5.99559653E-03   1.50261779E-01   2.40349561E+00

We immediately see that the errors for the forces are significantly lower than in the previous calculation with only one k point. This is due to the less noisy ab initio data which is easier to learn.

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

 1.0 Version
**************************************************
     The number of configurations
--------------------------------------------------
         48

and the number of local reference configurations (size of the basis set) 

**************************************************
     The numbers of basis sets per atom type
--------------------------------------------------
       382

We will further continue the training a 1000 MD steps and see how the number of training structures and the number of local reference configurations change. Do the following steps: 

cp ML_ABN ML_AB
cp CONTCAR POSCAR

and set the following INCAR tag 

ML_ISTART = 1

After running the calculation, we inspect the last instance of the errors in the ML_LOGFILE by typing: 

grep ERR ML_LOGFILE

The last few lines should have values close to: 

ERR                   675   5.10937061E-03   1.46895065E-01   2.50094941E+00
ERR                   861   5.10937061E-03   1.46895065E-01   2.50094941E+00
ERR                   924   5.10937061E-03   1.46895065E-01   2.50094941E+00
ERR                   942   5.01460989E-03   1.47816836E-01   2.47329693E+00
ERR                  1000   5.01460989E-03   1.47816836E-01   2.47329693E+00

We see that the accuracy has changed slightly. We also look at the ML_ABN file and the number of reference structures for training should increase compared to the run before 

 1.0 Version
**************************************************
     The number of configurations
--------------------------------------------------
         99

Also the number of local reference configurations (basis sets) increases compared to the previous calculation 

**************************************************
     The numbers of basis sets per atom type
--------------------------------------------------
       665


Ideally, one should continue learning until no structures need to be added to the training data and basis set anymore. Very often this can take up to 100ps depending on the material and conditions. In practice, the prediction of the Bayesian error exhibits numerical inaccuracies so that an ab initio calculation is conducted from time to time even if the force field is accurate enough. So measuring only the decreasing frequency of addition of new data is not sufficient to know when to finish. One should also look at the accuracy of the force on the training data and more importantly on the accuracy on some test data that is outside of the training sets.

Download

MLFF_Liquid_Si_tutorial.tgz

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