handson.ipynb

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    "![Logo](figures/GOE_Logo_Quer_IPC_Farbe_RGB.png)\n",
    "# High-Dimensional Neural Network Potentials\n",
    "**[Alexander L. M. Knoll](mailto:aknoll@chemie.uni-goettingen.de), and [Moritz R. Schäfer](mailto:moritzrichard.schaefer@uni-goettingen.de)** \n",
    "\n",
    "Behler Group, Theoretische Chemie, Institut für Physikalische Chemie\n"
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    "For this tutorial it is intended to use the RuNNer release version 1.2, available in conda-forge. \n",
    "The most recent version of RuNNer is [hosted on Gitlab](https://gitlab.com/TheochemGoettingen/RuNNer). For access please contact Prof. Dr. Jörg Behler (joerg.behler@uni-goettingen.de)."
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      "/home/kalm/.pyenv/versions/pyiron/lib/python3.10/site-packages/paramiko/transport.py:236: CryptographyDeprecationWarning: Blowfish has been deprecated\n",
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   "source": [
    "import numpy as np\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import ipympl\n",
    "import ipywidgets as widgets\n",
    "\n",
    "from pyiron_atomistics import Project\n",
    "from pyiron_contrib.atomistics.runner.job import RunnerFit\n",
    "from pyiron_contrib.atomistics.runner.utils import container_to_ase\n",
    "\n",
    "from ase.geometry import get_distances\n",
    "\n",
    "from runnerase import generate_symmetryfunctions\n",
    "\n",
    "%matplotlib inline"
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    "## Background\n",
    "### Architecture of an HDNNP"
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    "**RuNNer** is a stand-alone Fortran program for the construction of high-dimensional neural network potentials (HDNNPs), written mainly by Jörg Behler. The central assumption made in constructing a HDNNP is that the total energy of the system $E_{\\mathrm{tot}}$ [can be separated into atomic contributions $E_i$](https://www.doi.org/10.1103/PhysRevLett.98.146401). HDNNP relates the local environment of the atoms to their atomic energies $E_i$, which contribute to the sum of all $N$ atomic energies, resulting in the total energy of the system $E_\\mathrm{tot}$."
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    "\\begin{align}\n",
    "E_\\mathrm{tot} = \\sum_{i}^{N}E_i\\notag\n",
    "\\end{align}"
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    "Every atomic energy is described by an atomic neural network (NN), which is element-specific. The entirety of all atomic NNs composes a HDNNP, whose general architecture is shown below for a binary system."
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    "<img src=\"figures/2g.png\" class=\"center\" width=\"500\"/>"
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    "As you can see, the atomic contributions in this model are predicted independently from each other. Therefore, the model can easily describe systems with differering numbers of atoms: adding or removing an atom corresponds to adding or removing a row in the figure shown above. This ability is what puts the \"high-dimensional\" into the name \"HDNNP\". \n",
    "\n",
    "Each atomic neural networks receives input information about the local atomic environment up to a certain cutoff radius $R_{\\mathrm{c}}$. This information is encoded based on the Cartesian coordinates in many-body descriptors, so-called [atom-centered symmetry functions (ACSF or just SF)](https://www.doi.org/10.1063/1.3553717). More details about this are shown below. For each atom, the values of multiple SFs compose a SF vector $G$ which is the input layer of the atomic NNs.\n",
    "\n",
    "Atomic NNs look like this:"
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    "<img src=\"figures/ann.png\" width=\"500\"/>"
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    "Every value in the SF vector $G$ serves as one piece of input information to the atomic NN. We refer to the circles in the figure as _nodes_ (from graph theory) or _neurons_ (from neural science). The information from the input nodes flows through the atomic NN from left to right: the input layer is followed by a configurable number of hidden layers which consist, in turn, of an arbitrary number of _hidden nodes_. At the end, all information is collected in the output layer, which in our case is interpreted as the atomic energy contribution of the atom under consideration. The input nodes and the hidden nodes in the first layer are connected by weights. Moreover, the hidden and output nodes carry a bias value.\n",
    "\n",
    "During training, the weights and biases are optimized using backpropagation to represent best the data in the training data set."
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   "source": [
    "### Symmetry Functions (SFs)"
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    "SFs provide the input for the NN and describe the local atomic environment of each atom. In principle, one could also use Cartesian coordinates to capture the atomic positions in a structure. As Cartesian coordinates describe the absolute positions of atoms the numerical input to the atomic NNs would change with translation or rotation of the system. However, these actions do not influence the energy of the system and different numerical inputs belonging to the same NN output lead to large training errors.\n",
    "\n",
    "In contrast, SFs describe the relative positions of the atoms to each other and are hence translationally and rotationally invariant. We differentiate two types of SFs: radial SF depend on the distance between atom pairs and serve as a measure for their bond order. Angular SFs additionally depend on the interatomic angles."
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    "#### Cutoff Functions"
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