Difference between revisions of "Two-body DeepPot-SE"
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\mathcal{D}^i = (\mathcal{G}^i)^T \mathcal{R}^i (\mathcal{R}^i)^T \mathcal{G}^i_< | \mathcal{D}^i = (\mathcal{G}^i)^T \mathcal{R}^i (\mathcal{R}^i)^T \mathcal{G}^i_< | ||
</math> | </math> | ||
where <math>\mathcal{R}^i \in \mathbb{R}^{N \times \{1,4\}}</math> is the coordinate matrix, and each row of <math>\mathcal{R}^i</math> can be constructed as one of the following forms | |||
<math> | |||
(\mathcal{R}^i)_j = | |||
\begin{cases} | |||
\{ | |||
\begin{array}{cccc} | |||
s(r_{ji}) & \frac{s(r_{ji})x_{ji}}{r_{ji}} & \frac{s(r_{ji})y_{ji}}{r_{ji}} & \frac{s(r_{ji})z_{ji}}{r_{ji}} | |||
\end{array} | |||
\}, &\text{full information} \\ | |||
\{ | |||
\begin{array}{c} | |||
s(r_{ji}) | |||
\end{array} | |||
\}, &\text{radial-only information} | |||
\end{cases} | |||
\label{eq:rij} | |||
</math> | |||
where <math>\mathbf{R}_{ji}=\mathbf{R}_j-\mathbf{R}_i = (x_{ji}, y_{ji}, z_{ji})</math> is the relative coordinate and <math>r_{ji}=\lVert \mathbf{R}_{ji} \lVert</math> is its norm. The switching function $s(r)$ is defined as: | |||
<math> | |||
s(r)= | |||
\begin{cases} | |||
\frac{1}{r}, & r<r_s \\ | |||
\frac{1}{r} \{ {(\frac{r - r_s}{ r_c - r_s})}^3 (-6 {(\frac{r - r_s}{ r_c - r_s})}^2 +15 \frac{r - r_s}{ r_c - r_s} -10) +1 \}, & r_s \leq r<r_c \\ | |||
0, & r \geq r_c | |||
\end{cases} | |||
</math> | |||
Each row of the embedding matrix <math>\mathcal{G}^i \in \mathbb{R}^{N \times M}</math> consists of outputs of a neural network function <math>\mathcal{N}</math> of <math>s(r_{ji})</math>: | |||
<math> | |||
(\mathcal{G}^i)_j = \mathcal{N}(s(r_{ji})) | |||
\label{eq:G2} | |||
</math> | |||
where the neural network function $\mathcal{N}$ will be introduced in Section \ref{section:NN}. | |||
<math>\mathcal{G}^i_< \in \mathbb{R}^{N \times M_<}</math> takes first $M_<$ columns of $\mathcal{G}^i$. | |||
Both <math>M</math> and <math>M_<</math> are pre-defined parameters. | |||
== Usage == | == Usage == | ||
Revision as of 11:17, 4 December 2021
Two-body DeepPot-SE (se_e2_a or se_e2_r) is a descriptor of the Deep Potential Smooth Edition (DeepPot-SE) constructed from two-body information of atomic configurations.
Theory
The descriptor is given by [math]\displaystyle{ \mathcal{D}^i = (\mathcal{G}^i)^T \mathcal{R}^i (\mathcal{R}^i)^T \mathcal{G}^i_\lt }[/math]
where [math]\displaystyle{ \mathcal{R}^i \in \mathbb{R}^{N \times \{1,4\}} }[/math] is the coordinate matrix, and each row of [math]\displaystyle{ \mathcal{R}^i }[/math] can be constructed as one of the following forms [math]\displaystyle{ (\mathcal{R}^i)_j = \begin{cases} \{ \begin{array}{cccc} s(r_{ji}) & \frac{s(r_{ji})x_{ji}}{r_{ji}} & \frac{s(r_{ji})y_{ji}}{r_{ji}} & \frac{s(r_{ji})z_{ji}}{r_{ji}} \end{array} \}, &\text{full information} \\ \{ \begin{array}{c} s(r_{ji}) \end{array} \}, &\text{radial-only information} \end{cases} \label{eq:rij} }[/math] where [math]\displaystyle{ \mathbf{R}_{ji}=\mathbf{R}_j-\mathbf{R}_i = (x_{ji}, y_{ji}, z_{ji}) }[/math] is the relative coordinate and [math]\displaystyle{ r_{ji}=\lVert \mathbf{R}_{ji} \lVert }[/math] is its norm. The switching function $s(r)$ is defined as: [math]\displaystyle{ s(r)= \begin{cases} \frac{1}{r}, & r\lt r_s \\ \frac{1}{r} \{ {(\frac{r - r_s}{ r_c - r_s})}^3 (-6 {(\frac{r - r_s}{ r_c - r_s})}^2 +15 \frac{r - r_s}{ r_c - r_s} -10) +1 \}, & r_s \leq r\lt r_c \\ 0, & r \geq r_c \end{cases} }[/math] Each row of the embedding matrix [math]\displaystyle{ \mathcal{G}^i \in \mathbb{R}^{N \times M} }[/math] consists of outputs of a neural network function [math]\displaystyle{ \mathcal{N} }[/math] of [math]\displaystyle{ s(r_{ji}) }[/math]: [math]\displaystyle{ (\mathcal{G}^i)_j = \mathcal{N}(s(r_{ji})) \label{eq:G2} }[/math] where the neural network function $\mathcal{N}$ will be introduced in Section \ref{section:NN}. [math]\displaystyle{ \mathcal{G}^i_\lt \in \mathbb{R}^{N \times M_\lt } }[/math] takes first $M_<$ columns of $\mathcal{G}^i$. Both [math]\displaystyle{ M }[/math] and [math]\displaystyle{ M_\lt }[/math] are pre-defined parameters.
