ML MRB2: Difference between revisions

Latest revision as of 11:56, 31 March 2023

ML_MRB2 = [integer]
Default: ML_MRB2 = 8

Description: This tag sets the number $N_\text{R}^l$ (for all $l$ ) of radial basis functions used to expand the angular descriptor within the machine learning force field method.

The angular descriptor is constructed from

${\displaystyle \rho_{i}^{(3)}\left(r,s,\theta\right) = \iint d\hat{\mathbf{r}} d\hat{\mathbf{s}} \delta\left(\hat{\mathbf{r}}\cdot\hat{\mathbf{s}} - \mathrm{cos}\theta\right) \sum\limits_{j=1}^{N_{a}} \sum\limits_{k \ne j}^{N_{a}} \rho_{ik} \left(r\hat{\mathbf{r}}\right) \rho_{ij} \left(s\hat{\mathbf{s}}\right), \quad \text{where} \quad \rho_{ij}\left(\mathbf{r}\right) = f_{\mathrm{cut}}\left(r_{ij}\right) g\left(\mathbf{r}-\mathbf{r}_{ij}\right) }$

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 $p_{n\nu l}^{i}$ by expanding it into a set of radial basis functions $\chi_{nl}(r)$ and Legendre polynomials $P_{l}\left(\mathrm{cos}\theta\right)$ (see this section for more details):

${\displaystyle \rho_{i}^{(3)}\left(r,s,\theta\right) = \sum\limits_{l=1}^{L_{\mathrm{max}}} \sum\limits_{n=1}^{N^{l}_{\mathrm{R}}}\sum\limits_{\nu=1}^{N^{l}_{\mathrm{R}}} \sqrt{\frac{2l+1}{2}} p_{n\nu l}^{i}\chi_{nl}\left(r\right)\chi_{\nu l}\left(s\right)P_{l}\left(\mathrm{cos}\theta\right). }$

The tag ML_MRB2 sets the number $N_\text{R}^l$ of radial basis functions to use in this expansion. The same number is used for all $l$ .

Mind: The number of angular descriptor expansion coefficients

p_{n\nu l}^{i}

scales quadratically with

N_\text{R}^l

set by this tag. It also depends on ML_LMAX2 and the number of elements.

@@ Line 1: / Line 1: @@
-{{TAGDEF|ML_FF_MRB2_MB|[integer]|{{TAG|ML_FF_MRB1_MB}}/1.5}}
+{{DISPLAYTITLE:ML_MRB2}}
+{{TAGDEF|ML_MRB2|[integer]|8}}
-Description: This tag sets the number of radial basis sets used to expand the atomic distribution for the angular descriptor within the machine learning force field method.
+Description: This tag sets the number <math>N_\text{R}^l</math> (for all <math>l</math>) of radial basis functions used to expand the angular descriptor 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 angular 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}^{(3)}\left(r,s,\theta\right) = \iint d\hat{\mathbf{r}} d\hat{\mathbf{s}}  \delta\left(\hat{\mathbf{r}}\cdot\hat{\mathbf{s}} - \mathrm{cos}\theta\right) \sum\limits_{j=1}^{N_{a}} \sum\limits_{k \ne j}^{N_{a}} \rho_{ik} \left(r\hat{\mathbf{r}}\right) \rho_{ij} \left(s\hat{\mathbf{s}}\right), \quad \text{where} \quad
+\rho_{ij}\left(\mathbf{r}\right) = 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 <math>p_{n\nu l}^{i}</math> by expanding it into a set of radial basis functions <math>\chi_{nl}(r)</math> and Legendre polynomials <math>P_{l}\left(\mathrm{cos}\theta\right)</math> (see [[Machine learning force field: Theory#Basis set expansion|this section]] for more details):
+<math>
+\rho_{i}^{(3)}\left(r,s,\theta\right) = \sum\limits_{l=1}^{L_{\mathrm{max}}} \sum\limits_{n=1}^{N^{l}_{\mathrm{R}}}\sum\limits_{\nu=1}^{N^{l}_{\mathrm{R}}} \sqrt{\frac{2l+1}{2}} p_{n\nu l}^{i}\chi_{nl}\left(r\right)\chi_{\nu l}\left(s\right)P_{l}\left(\mathrm{cos}\theta\right).
+</math>
-== References ==
+The tag {{TAG|ML_MRB2}} sets the number <math>N_\text{R}^l</math> of radial basis functions to use in this expansion. The same number is used for all <math>l</math>.
-<references/>
+{{NB|mind|The number of angular descriptor expansion coefficients <math>p_{n\nu l}^{i}</math> scales '''quadratically''' with <math>N_\text{R}^l</math> set by this tag. It also depends on {{TAG|ML_LMAX2}} and the number of elements.}}
-<noinclude>
+== Related tags and articles ==
-== Related Tags and Sections ==
+{{TAG|ML_LMLFF}}, {{TAG|ML_LMAX2}}, {{TAG|ML_MRB1}}, {{TAG|ML_W1}}, {{TAG|ML_RCUT2}}, {{TAG|ML_SION2}}
-{{TAG|ML_FF_LMLFF}}, {{TAG|ML_FF_MRB1_MB}}, {{TAG|ML_FF_W1_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_MRB2_MB|Examples|Examples that use this tag}}
+{{sc|ML_MRB2|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]]