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Updated equitation indexing
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cwl160030 committed Nov 1, 2023
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36 changes: 18 additions & 18 deletions Lesson4_DeepPot.ipynb
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"s(R_{1i}) & s(R_{1i}) \\frac{x_{1i}}{R_{1i}} & s(R_{1i}) \\frac{y_{1i}}{R_{1i}} & s(R_{1i}) \\frac{z_{1i}}{R_{1i}} \\\\ \\\\\n",
"s(R_{2i}) & s(R_{2i}) \\frac{x_{2i}}{R_{2i}} & s(R_{2i}) \\frac{y_{i1}}{R_{2i}} & s(R_{2i}) \\frac{z_{2i}}{R_{2i}} \\\\ \\\\\n",
"\\cdots & \\cdots & \\cdots & \\cdots \\\\ \\\\\n",
"s(R_{ni}) & s(R_{ni}) \\frac{x_{ni}}{R_{ni}} & s(R_{ni}) \\frac{y_{ni}}{R_{ni}} & s(R_{ni}) \\frac{z_{ni}}{R_{ni}}\n",
"s(R_{n_ii}) & s(R_{n_ii}) \\frac{x_{n_ii}}{R_{n_ii}} & s(R_{n_ii}) \\frac{y_{n_ii}}{R_{n_ii}} & s(R_{n_ii}) \\frac{z_{n_ii}}{R_{n_ii}}\n",
"\\end{pmatrix}\n",
"\\end{eqnarray}$$\n",
"<br/>\n",
"\n",
"where $n$ is the number of neighbors for the atom. $s$ is the weighting function.\n",
"where $n_i$ is the number of neighbors for atom $i$ and $s$ is the weighting function.\n",
"\n",
"$$\\begin{eqnarray}\n",
"s(R_{ji}) =\n",
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"Now we use embedding neural networks to transform each of the $s$ values into $M_1$ numbers. This gives us the local embedding matrix $\\mathcal{G}$. Note the embedding neural network parameters depend on the chemical species of atom $i$ and atom $j$. \n",
"\n",
"$$\\begin{eqnarray}\n",
"{\\mathcal{G}}_i\n",
"g_i\n",
"= \\begin{pmatrix}\n",
"\\left( G[s(R_{1i})] \\right)_1 &\n",
"\\left( G[s(R_{1i})] \\right)_2 & \\cdots &\n",
"\\left( G[s(R_{1i})] \\right)_{m_1} \\\\\n",
"\\left( G[s(R_{2i})] \\right)_1 &\n",
"\\left( G[s(R_{2i})] \\right)_2 & \\cdots & \\left( G[s(R_{2i})] \\right)_{m_1} \\\\\n",
"\\cdots & \\cdots & \\cdots & \\cdots \\\\\n",
"\\left( G[s(R_{ni})] \\right)_1 &\n",
"\\left( G[s(R_{ni})] \\right)_2 & \\cdots & \\left( G [s(R_{ni})] \\right)_{m_1}\n",
"\\left( G[s(R_{n_ii})] \\right)_1 &\n",
"\\left( G[s(R_{n_ii})] \\right)_2 & \\cdots & \\left( G [s(R_{n_ii})] \\right)_{m_1}\n",
"\\end{pmatrix}\n",
"\\end{eqnarray}$$\n",
"\n",
"Two local embedding matrices are used: $\\mathcal{G}_{i1}$ is $n\\times M_1$ dimensions, while $\\mathcal{G}_{i2}$ is $n\\times M_2$ dimensions. The dimensions $M_1$ and $M_2$ represent the number of neural network parameters, where $M_1$ is larger than $M_2$.\n",
"Two local embedding matrices are used: $g_{i}^{1}$ is $n\\times M_1$ dimensions, while $g_{i}^{2}$ is $n\\times M_2$ dimensions. The dimensions $M_1$ and $M_2$ represent the number of neural network parameters, where $M_1$ is larger than $M_2$.\n",
"\n",
"By multiplying our local embedding matrices and environment matrices, we can preserve the translational, rotational, and permutational symmetry in the form of the encoded feature matrix $\\mathcal{D_i}$ \n",
"By multiplying our local embedding matrices and environment matrices, we can preserve the translational, rotational, and permutational symmetry in the form of the encoded feature matrix $D_i$ \n",
"\n",
"$$\\begin{eqnarray}\n",
"{\\mathcal{D}}_i = \\left( {\\mathcal{G}}_i^1 \\right)^T {\\mathcal{R}}_i {\\mathcal{R}}_i^T {\\mathcal{G}}_i^2\n",
"D_i = \\left( g_i^1 \\right)^T {\\mathcal{R}}_i {\\mathcal{R}}_i^T g_i^2\n",
"\\end{eqnarray}$$\n",
"\n",
"The local feature matrix is then mapped to the atomic energy using the fitting neural network. Finally, the atomic energies are summed to yield the total energy of the molecule."
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"source": [
"### The Feature Class\n",
"\n",
"Here we define the feature class, which uses the local environment matrix ($\\mathcal{R}_i$) and local embedding matrices ($\\mathcal{G}^1_i$ and $\\mathcal{G}^2_i$) to construct the encoded feature matrix ($\\mathcal{D}_i$)."
"Here we define the feature class, which uses the local environment matrix ($\\mathcal{R}_i$) and local embedding matrices ($g^1_i$ and $g^2_i$) to construct the encoded feature matrix ($D_i$)."
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