ML MRB1: Difference between revisions

Latest revision as of 08:06, 9 May 2023

ML_MRB1 = [integer]
Default: ML_MRB1 = 12

Description: This tag sets the number $N_\text{R}^0$ of radial basis functions used to expand the radial descriptor $\rho^{(2)}_i(r)$ within the machine learning force field method.

The radial descriptor is constructed from

${\displaystyle \rho_{i}^{(2)}\left(r\right) = \frac{1}{4\pi} \int \rho_{i}\left(r\hat{\mathbf{r}}\right) d\hat{\mathbf{r}}, \quad \text{where} \quad \rho_{i}\left(\mathbf{r}\right) = \sum\limits_{j=1}^{N_{\mathrm{a}}} f_{\mathrm{cut}}\left(r_{ij}\right) g\left(\mathbf{r}-\mathbf{r}_{ij}\right) }$

and $g\left(\mathbf{r}\right)$ is an approximation of the delta function. In practice, the continuous function above is transformed into a discrete set of numbers by expanding it into a set of radial basis functions $\chi_{n0}(r)$ (see this section for more details):

${\displaystyle \rho_{i}^{(2)}\left(r\right) = \frac{1}{\sqrt{4\pi}} \sum\limits_{n=1}^{N^{0}_{\mathrm{R}}} c_{n00}^{i} \chi_{n0}\left(r\right). }$

The tag ML_MRB1 sets the number $N_\text{R}^0$ of radial basis functions to use in this expansion.

@@ Line 1: / Line 1: @@
-{{TAGDEF|ML_FF_MRB1_MB|[integer]|NINT({{TAG|ML_FF_RCUT1_MB}}/{{TAG|ML_FF_SION1_MB}}*1.5)}}
+{{DISPLAYTITLE:ML_MRB1}}
+{{TAGDEF|ML_MRB1|[integer]|12}}
-Description: This tag sets the number of radial basis sets used to expand the atomic distribution for the radial descriptor within the machine learning force field method.
+Description: This tag sets the number <math>N_\text{R}^0</math> of radial basis functions used to expand the radial descriptor <math>\rho^{(2)}_i(r)</math> within the machine learning force field method.
 ----
-The values for {{TAG|ML_FF_MRB1_MB}} and {{TAG|ML_FF_MRB2_MB}} are automatically set by the code (empirically) and usually need not to be set by the user. Only in very few cases if the error in the radial expansion (see later in the text) is not sufficiently low these values need to be adjusted manually.
+The radial descriptor is constructed from
-The tags {{TAG|ML_FF_MRB1_MB}} and {{TAG|ML_FF_MRB2_MB}} set the number of radial basis sets used to expand the atomic distribution of the radial and angular density. These tags depend very sensitively on the cut-off radius of the descriptor ({{TAG|ML_FF_RCUT1_MB}} and {{TAG|ML_FF_RCUT2_MB}}) and the width of the Gaussian functions used in the broadening of the atomic distributions ({{TAG|ML_FF_SION1_MB}} and {{TAG|ML_FF_SION2_MB}}). The error occuring due to the expansion of the radial basis functions is monitored in the {{TAG|ML_LOGFILE}} file by searching for the following line "''Error in radial expansion: ...''". A typical reasonable value for the error threshold that was empirically determined (by us and in reference {{cite|szlachta:prb:2014}}) is <math>\pm 0.02</math>. Hence, the number of basis functions should be adjusted until the error written in the {{TAG|ML_LOGFILE}} is smaller than this value. A more detailed description of the basis sets is given in appendix A of reference {{cite|jinnouchi2:arx:2019}}.
+<math>
+\rho_{i}^{(2)}\left(r\right) = \frac{1}{4\pi} \int \rho_{i}\left(r\hat{\mathbf{r}}\right) d\hat{\mathbf{r}}, \quad \text{where} \quad
+\rho_{i}\left(\mathbf{r}\right) = \sum\limits_{j=1}^{N_{\mathrm{a}}} f_{\mathrm{cut}}\left(r_{ij}\right) g\left(\mathbf{r}-\mathbf{r}_{ij}\right)
+</math>
+and <math>g\left(\mathbf{r}\right)</math> is an approximation of the delta function. In practice, the continuous function above is transformed into a discrete set of numbers by expanding it into a set of radial basis functions <math>\chi_{n0}(r)</math> (see [[Machine learning force field: Theory#Basis set expansion|this section]] for more details):
+<math>
+\rho_{i}^{(2)}\left(r\right) = \frac{1}{\sqrt{4\pi}} \sum\limits_{n=1}^{N^{0}_{\mathrm{R}}} c_{n00}^{i} \chi_{n0}\left(r\right).
+</math>
-== References ==
+The tag {{TAG|ML_MRB1}} sets the number <math>N_\text{R}^0</math> of radial basis functions to use in this expansion.
-<references/>
-<noinclude>
+== Related tags and articles ==
+{{TAG|ML_LMLFF}}, {{TAG|ML_MRB2}}, {{TAG|ML_W1}}, {{TAG|ML_RCUT1}}, {{TAG|ML_SION1}}
-== Related Tags and Sections ==
+{{sc|ML_MRB1|Examples|Examples that use this tag}}
-{{TAG|ML_FF_LMLFF}}, {{TAG|ML_FF_MRB2_MB}}, {{TAG|ML_FF_W1_MB}}, {{TAG|ML_FF_RCUT1_MB}}, {{TAG|ML_FF_RCUT2_MB}}, {{TAG|ML_FF_SION1_MB}}, {{TAG|ML_FF_SION2_MB}}
-{{sc|ML_FF_MRB1_MB|Examples|Examples that use this tag}}
 ----
-[[Category:INCAR]][[Category:Machine Learning]][[Category:Machine Learned Force Fields]][[Category: Alpha]]
+[[Category:INCAR tag]][[Category:Machine-learned force fields]]