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 [math]\displaystyle{ N_\text{R}^l }[/math] (for all [math]\displaystyle{ l }[/math]) of radial basis functions used to expand the angular descriptor within the machine learning force field method.

The angular descriptor is constructed from

[math]\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) }[/math]

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

[math]\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). }[/math]

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

Mind: The number of angular descriptor expansion coefficients [math]\displaystyle{ p_{n\nu l}^{i} }[/math] scales quadratically with [math]\displaystyle{ N_\text{R}^l }[/math] 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]|10}}
+{{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 withtin 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 angular descriptor is constructed from
-== Related Tags and Sections ==
+<math>
-{{TAG|ML_FF_LMLFF}}, {{TAG|ML_FF_MRB1_MB}}, {{TAG|ML_FF_W1_MB}}, {{TAG|ML_FF_W2_MB}}
+\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>
-{{sc|ML_FF_MRB2_MB|Examples|Examples that use this tag}}
+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>
+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>.
+{{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.}}
+== Related tags and articles ==
+{{TAG|ML_LMLFF}}, {{TAG|ML_LMAX2}}, {{TAG|ML_MRB1}}, {{TAG|ML_W1}}, {{TAG|ML_RCUT2}}, {{TAG|ML_SION2}}
+{{sc|ML_MRB2|Examples|Examples that use this tag}}
 ----
-[[Category:INCAR]][[Category:Machine Learning]][[Category:Machine Learned Force Fields]][[Category:VASP6]]
+[[Category:INCAR tag]][[Category:Machine-learned force fields]]