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+ "_Gábor Csányi (gc121@cam.ac.uk), James R. Kermode (j.r.kermode@warwick.ac.uk)_\n", + "\n", + "In this tutorial, we will use Gaussian process regression, GPR (or equivalently, Kernel Ridge Regression, KRR) to train and predict charges of atoms in small organic molecules. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In GPR, we are fitting a function in a moderate dimensional space, using basis functions that are typically symmetric, \"similarity\" functions, they describe how similar we expect the function value to be at two different points in the input space. In its simplest form, when we fit a function $f$ using input data $y_i$ that are function values at selected points $x_i$, we have\n", + "\n", + "$$\n", + "f(x) = \\sum_i^N \\alpha_i K(x_i, x)\n", + "$$\n", + "\n", + "where $K(x,x')$ is the positive definite similarity function, with a value $1$ when $x=x'$, and lower values for different arguments. The $\\alpha$ coefficients are the degrees of freedom in the fit, and we need to determine them from the data. The sum runs through the available data points (but in principle we can choose fewer basis functions than datapoints (this is called sparsification), or even more if we want to, but that goes beyond the scope of this tutorial. )" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The difference between KRR and GPR is in the interpretation, in GPR we construct a probability distribution for the unknown function values, and the $K$ is taken to be the formal covariance between function values,\n", + "\n", + "$$\n", + "K(x,x') = \\rm{Cov}\\left((f(x),f(x')\\right)\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Regardless of the interpretation, when the ${x,y}$ data is plugged in, we get a system of linear equations:\n", + "\n", + "$$\n", + "y_j = \\sum_i \\alpha_i K(x_i,x_j)\n", + "$$\n", + "\n", + "The solution is given by (in vectorial form)\n", + "\n", + "$$\n", + "{\\bf \\alpha} = \\left({\\bf K} + \\lambda^2 {\\bf I}\\right)^{-1} {\\bf y}\n", + "$$\n", + "\n", + "where the matrix $K$ has elements $K(x_i,x_j)$. If the data was perfectly consistent, without any noise, then in principle we could get an fit with $lambda$ set to zero. In practice however, our data might have noise, or we might _choose_ to not want the interpolant (ie the fitted function f) to go through each data point _exactly_, but prefer smoother functions that just go _close_ to the datapoints. We can achieve this by choosing a nonzero (but small) $\\lambda$. In the GPR interpretation, $lambda$ should be set to the standard deviation of the noise our data has. More on this below. " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "%pylab inline\n", + "import numpy as np\n", + "import quippy\n", + "import matplotlib as mpl\n", + "import matplotlib.pyplot as plt\n", + "import os\n", + "from __future__ import print_function\n", + "import sys\n", + "from scripts.Visualise import ViewStructure" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Database\n", + "\n", + "We load a database of small molecule geometries, and precomuputed atomic charges. This file is a subset of 2000 molecules from the GDB9 dataset. The molecules contain H, C, N, O, and F atoms. " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "atAll = quippy.AtomsList(\"data/data_GDB9.xyz\")" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "view = ViewStructure(atAll[2])\n", + "view" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# We can access the atomic numbers of any molecule\n", + "atAll[0].Z" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# similarly, the positions\n", + "atAll[1].positions " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# and the charges\n", + "atAll[1].charge" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## SOAP Kernel and descriptor\n", + "\n", + "Next, we need to define a kernel. There are many ways to define what the atomic charges would be a function of, but somehow we need to describe the environment of the atom, and then construct a similarity function that can serve as the kernel function. \n", + "\n", + "In this tutorial, we are going to make the atomic charge a function of the near-environment of an atom (within a cutoff), and we will describe that environment using the SOAP descriptor and compare them using the SOAP kernel. Note right away that the quantum mechanically computed atomic charge is not fully determined by the near-environment of atoms (far-away atoms can also influence the charge, even if just to a small extent), so this is an early indication that we will be making use of the \"noise\" interpretation of the $\\lambda$ regularization parameter: we don't expect (and don't want) our fitted function to precisely go through each datapoint.\n", + "\n", + "The SOAP descriptor of an atomic environment is based on a spherical harmonic expansion of the neighbour density, and truncating this expansion at some maximum numer of radial (n_max) and angular (l_max) indices gives rise to some parameters. We also need to give the cutoff within which we consider the neighbour environment.\n", + "\n", + "Writing the descriptor vector as $p_{ss'nn'l}$, where $s$ and $s'$ are indices that run over the different atomic species in the atom's environment, $n$ and $n'$ are radial and $l$ is an angular index, the kernel between two atomic environments is\n", + "\n", + "$$\n", + "K(p,p') = \\delta^2 \\left| \\sum_{ss'nn'l} p_{ss'nn'l} p'_{ss'nn'l}\\right|^\\zeta \\equiv \\delta^2 \\left| {\\bf p} \\cdot {\\bf p'}\\right|^\\zeta\n", + "$$\n", + "\n", + "The factor of $\\delta^2$ allows the setting of the scale of fitted function, relative to the error specification $\\lambda$. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Below we fit and predict the atomic charge of H atoms" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# First we create the descriptor object\n", + "\n", + "Z = 1 # compute environment around H atoms\n", + "# other parameters\n", + "# atom_sigma : width of Gaussian smearing around each neighbour atom\n", + "# n_max : maximum radial index\n", + "# l_max : maximum angular index\n", + "# cutoff : environment radius in Å\n", + "# Z : environments around this atomic number will be considered\n", + "# species_Z : atomic numbers we wish to consider in the environment around atoms of type Z\n", + "# n_species : length of species_Z list\n", + "desc = quippy.Descriptor(\n", + " \"soap atom_sigma=0.5 n_max=3 l_max=3 cutoff=3.0 Z={:d} n_species=5 species_Z='1 6 7 8 9'\".format(Z))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The ```desc``` object can be used to calculate the desrciptor for all the atoms in an Atoms object:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "Z" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "p=[]\n", + "c=[]\n", + "for at in atAll:\n", + " at.set_cutoff(desc.cutoff())\n", + " at.calc_connect()\n", + " \n", + " descAt = desc.calc(at)[\"descriptor\"] # calc() returns a dictionary, in which we find the descriptor vector\n", + " \n", + " for pp in descAt:\n", + " p.append(pp)\n", + " for i in quippy.frange(at.n):\n", + " if at.Z[i] == Z:\n", + " c.append(at.charge[i]) # pick up the target atomic charge and store it in the same order \n", + "p = np.array(p)\n", + "c = np.array(c)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We now have a list of charges, and correspond list of descriptors, collected into a matrix" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "p.shape" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "c.shape" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "p" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We now fit a function using GPR on a subset of the charge data, using the SOAP descriptor and kernel" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "n_fit = 10 # use the first n_fit data points to fit\n", + "lambda_fit = 0.02 # assumed noise in the data, due to \"model error\", i.e. the limitations of the descriptor/kernel\n", + "zeta = 2 # kernel parameter\n", + "delta = 0.2 # kernel parameter\n", + "\n", + "x_fit = p[:n_fit,:]\n", + "y_fit= c[:n_fit]\n", + "\n", + "# compute the covariance matrix, including regularisation\n", + "K_fit = delta**2*np.dot(x_fit,x_fit.T)**zeta + lambda_fit**2 * np.eye(n_fit)\n", + "\n", + "# solve linear system to obtain fit coefficients\n", + "alpha = np.linalg.solve(K_fit,y_fit)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "x_test = p[n_fit:,:]\n", + "y_test = c[n_fit:]\n", + "k_test = delta**2*np.dot(x_test,x_fit.T)**zeta\n", + "y_test_predict = np.dot(k_test,alpha)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "plt.plot(y_test,y_test_predict,\".\")\n", + "plt.plot([-0.6,0.6],[-0.6,0.6])\n", + "plt.xlabel(\"Test data\")\n", + "plt.ylabel(\"Predicted data\")\n", + "print(\"RMS error:\", np.sqrt(np.mean((y_test-y_test_predict)**2)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Error estimates\n", + "\n", + "The GPR interpretation allows the estimation of the error of the prediction, as the square root of the variance of the posterior distribution of the function. The corresponding formula is\n", + "\n", + "$$\n", + "{\\rm err}(x)^2 = K(x,x) - {\\bf k}(x)^T \\left({\\bf K} + \\lambda^2 {\\bf I}\\right)^{-1} {\\bf k}(x)\\qquad\\qquad [{\\bf k}(x)]_i = K(x,x_i)\n", + "$$\n", + "\n", + "Notice how the error does not actually depend on the data values $\\{y\\}$, only on the locations $\\{x\\}$ and the kernel function. As you see below, this error estimate can on occasion be quite different from the actual error." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "invK = np.linalg.inv(K_fit)\n", + "y_test_error_predict = np.zeros(len(x_test))\n", + "for i in range(len(x_test)):\n", + " kappa = delta**2*np.dot(x_test[i].T,x_test[i])**zeta\n", + " y_test_error_predict[i] = sqrt(kappa - np.dot(k_test[i,:].T, np.dot(invK, k_test[i,:])))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "plt.errorbar(y_test[::100], y_test_predict[::100], yerr=y_test_error_predict[::100], fmt=\".\")\n", + "plt.plot([-0.,0.4],[-0.0,0.4])\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Tasks\n", + "\n", + "Now you are ready to complete the following exercises\n", + "\n", + "1. Increase the radial and angular expansions to try and achieve a better fit. Try to go in small steps, because for large expansions, the calculation takes significantly longer. Notice how the predictions and the errors behave if you reduce the radial cutoff of the environment definition, can you explain what you observe? \n", + "\n", + "2. Fit and predict the charge of other species (you will need to create a new descriptor object).\n", + "\n", + "3. Study how the accuracy of prediction depends on the number of fitting data points.\n", + "\n", + "4. For the low-quality fit above, you see that there are two groups of H atoms that are clearly separated. Try to identify what characterises those groups? Inspect the molecules and H atoms in each group. " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "anaconda-cloud": {}, + "kernelspec": { + "display_name": "Python 2", + "language": "python", + "name": "python2" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 2 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython2", + "version": "2.7.15" + } + }, + "nbformat": 4, + "nbformat_minor": 1 +} diff --git a/quip-gap/soap_si_surface.ipynb b/quip-gap/soap_si_surface.ipynb new file mode 100644 index 0000000000000000000000000000000000000000..6cb7906f61241e8c008a8e8beae1dd8f1812027d --- /dev/null +++ b/quip-gap/soap_si_surface.ipynb @@ -0,0 +1,1073 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# The SOAP descriptor, Gaussian Approximation Potentials (GAP) and machine-learning of force fields " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Adam Fekete, Matrina Stella, Henry Lambert, Alessandro De Vita, Gabor Csanyi (gc121@cam.ac.uk)\n", + "\n", + "## Introduction\n", + "\n", + "In this tutorial we will be using a Gaussaion Approximation Potentials to analyse results of TB DFT calculations of Si surface. Along the way we will learn about different descriptors (2b, 3b, soap) to describe local atomic environment in order to predict energies and forces of Si surface. \n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "from __future__ import print_function\n", + "\n", + "import numpy as np\n", + "\n", + "from quippy import *\n", + "\n", + "from ase import units\n", + "from ase import Atoms as aseAtoms\n", + "from ase.build import add_vacuum\n", + "from ase.lattice.cubic import Diamond\n", + "from ase.io.trajectory import Trajectory\n", + "from ase.io import write\n", + "\n", + "from ase.constraints import FixAtoms\n", + "\n", + "from ase.md.velocitydistribution import MaxwellBoltzmannDistribution\n", + "from ase.md.verlet import VelocityVerlet\n", + "from ase.md.langevin import Langevin\n", + "\n", + "from ase.optimize.precon import PreconLBFGS, Exp\n", + "\n", + "from IPython.display import display\n", + "from scripts.Visualise import ViewStructure, MyASEStructure\n", + "import nglview\n", + "\n", + "import matplotlib.pylab as plt\n", + "\n", + "# from quippy import verbosity_push, PRINT_VERBOSE\n", + "# verbosity_push(PRINT_VERBOSE)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Part I: MD, descriptors\n", + "### Parameters" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can define some parameters here for the MD calculations:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "T = 1000.0 # Temperature [Kelvin]\n", + "timestep = 1.0 * units.fs" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## System" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Defining configuration for Si slab" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": false + }, + "outputs": [], + "source": [ + "def build_slab(size=(1,2,2), vacuum=10.):\n", + " # Build Si lattice.\n", + " # lattice = Diamond('Si', directions=([1, 0, 0], [0, 1, 0], [0, 0, 1]), latticeconstant=5.44, size=size)\n", + " lattice = Diamond('Si', latticeconstant=5.44, size=size)\n", + " atoms = Atoms(lattice)\n", + " \n", + "\n", + " # Fixing the bottom layer \n", + " bottom = atoms.positions[:,2].min()\n", + " fixed_mask = (abs(atoms.positions[:,2] - bottom) < 2.0)\n", + " atoms.set_constraint(FixAtoms(mask=fixed_mask))\n", + " \n", + " # build surface by adding space to z direction\n", + " add_vacuum(atoms, vacuum)\n", + " # atoms.center(vacuum=10.0, axis=2)\n", + " \n", + " return atoms\n", + "\n", + "\n", + "atoms = build_slab()\n", + "print('Number of atoms:', len(atoms))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's visualise the initial configuration:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": false + }, + "outputs": [], + "source": [ + "def visualise(atoms, repetition=(1,1,1)):\n", + "# visualisation\n", + " view = nglview.NGLWidget(gui=False)\n", + " view.stage.set_parameters(clip_dist=0)\n", + " view.add_structure(MyASEStructure(atoms))\n", + " view.add_unitcell()\n", + " view.add_structure(MyASEStructure(supercell(atoms, *repetition)))\n", + " return view\n", + "\n", + "view = visualise(atoms)\n", + "display(view)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## MD dynamics with TB DFTB" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We use TB DFTB potential to calculate the potential energies and forces for the each timestep:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# attach tight binding calculator\n", + "qm_pot = Potential('TB DFTB', param_filename='/opt/quip/doc/params.xml')\n", + "atoms.set_calculator(qm_pot)\n", + "\n", + "# Thermalize atoms\n", + "MaxwellBoltzmannDistribution(atoms, 2.0 * T * units.kB)\n", + "# dynamics = VelocityVerlet(atoms, timestep)\n", + "dynamics = Langevin(atoms, timestep, T * units.kB, 0.002)\n", + " \n", + "#attach observer to dynamics to track quantity of interest (temperature)\n", + "def print_status(): \n", + " print('Step = {}, time = {} [fs], T = {} [K]'.format(\n", + " dynamics.nsteps,\n", + " dynamics.nsteps * dynamics.dt / units.fs,\n", + " atoms.get_kinetic_energy() / (1.5 * units.kB * len(atoms))\n", + " ))\n", + "\n", + "def print_energy(a=atoms): # store a reference to atoms in the definition.\n", + " \"\"\"Function to print the potential, kinetic and total energy.\"\"\"\n", + " epot = a.get_potential_energy() / len(a)\n", + " ekin = a.get_kinetic_energy() / len(a)\n", + " print('Energy per atom: Epot = %.3feV Ekin = %.3feV (T=%3.0fK) '\n", + " 'Etot = %.3feV' % (epot, ekin, ekin / (1.5 * units.kB), epot + ekin))\n", + " \n", + "dynamics.attach(print_status, interval=10)\n", + "dynamics.attach(print_energy, interval=1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's do the dynamics:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "dynamics.run(steps=100)\n", + "\n", + "# for _ in range(150):\n", + "# dynamics.run(steps=1)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "view = visualise(atoms)\n", + "display(view)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Using descriptors\n", + "\n", + "In this section we will intorduce some descriptors which could be useful for describing local atomic enviroments\n", + "\n", + "You can find more details on the following webpage:\n", + "- https://libatoms.github.io/QUIP/Tutorials/quippy-descriptor-tutorial.html" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Pairwise\n", + "\n", + "Here we use a simple pair distance between Silicon atoms, with a cutoff of 4.0 Angstrom. There are several descriptors that can do this, one is distance_2b, which takes a cutoff argument. Alternatively, the distance_Nb descriptor could also do this, with order=2. This is more general, order=3 is a triangle-like three-body descriptor of the three sides of a triangle of 3 atoms." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "desc = Descriptor(\"distance_Nb order=2 cutoff=4.0 n_Z=1 Zs={14}\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This descriptor is very simple: it is scalar (dimension 1), and hence only has a two (but equivalent) permutations." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "desc.n_perm # number of permutations" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "desc.permutations() # array of permutation arrays\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Many descriptors rely on the neighbour connectivity, so we need to call calc_connect, after setting the Atoms cutoff:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "atoms.set_cutoff(desc.cutoff())\n", + "atoms.calc_connect()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can now calculate how many instances of this descriptor are found in an Atoms (or ASEAtoms) object:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "desc.count(atoms)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This also works transparently for iterables (such as lists), returning a list of the counts.\n", + "\n", + "We can also calculate the actual descriptor values – in this case, the list of pairwise distances:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "d = desc.calc(atoms)\n", + "# for k,v in d.items():\n", + "# print(k)\n", + "# print(type(v))\n", + "d" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Notice that the first array is called cutoff, and it supplies the value of a cutoff function, implicit in all descriptors, which takes the descriptor value to zero as the atoms approach the cutoff, i.e. in this case as the distance between the two atoms approaches the cutoff. It is more complicated for three-body and higher-body descriptors, but the final result is always a descriptor which changes smoothly with atomic positions.\n", + "\n", + "Here is a histogram of the resulting descriptor array, i.e. of the interatomic distances" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "plt.hist(d.descriptor, bins=40)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Calculate size of descriptor data:\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "n_desc, n_cross = desc.descriptor_sizes(atoms)\n", + "print(\"n_desc=%d n_cross=%d\" % (n_desc, n_cross))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "plt.matshow(d['descriptor'])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## SOAP descriptor\n", + "\n", + "We will describe the environment using the SOAP descriptor. The SOAP descriptor of an atomic environment is based on a spherical harmonic expansion of the neighbour density, and truncating this expansion at some maximum numer of radial (n_max) and angular (l_max) indices gives rise to some parameters. We also need to give the cutoff within which we consider the neighbour environment.\n", + "\n", + "Writing the descriptor vector as $p_{ss'nn'l}$, where $s$ and $s'$ are indices that run over the different atomic species in the atom's environment, $n$ and $n'$ are radial and $l$ is an angular index." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "desc = Descriptor(\"soap cutoff=4 l_max=3 n_max=4 normalize=T atom_sigma=0.5 n_Z=1 Z={14} \")" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "atoms.set_cutoff(desc.cutoff())\n", + "atoms.calc_connect()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "There are now only 32 descriptors, because SOAP produces one for each atom in the structure" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "desc.descriptor_sizes(atoms)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "But each descriptor now is a long vector, because it encodes the entire environment of the atom up to the cutoff. The length of the vector depends on l_max and n_max and also on the number of atom types." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "d = desc.calc(atoms)\n", + "d" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can visualise the values of the descriptor to have an idea of how vector looks like:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "plt.matshow(d.descriptor)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Part II: GAP: fitting, comparing different descriptors \n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Database\n", + "\n", + "Let's run the MD to generate configurations for the training:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "db = []\n", + "def collect_data():\n", + " db.append(atoms.copy())\n", + "\n", + "dynamics.attach(collect_data, interval=10)\n", + "dynamics.run(steps=500)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# The size of the database:\n", + "len(db)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Save the results into a file for using it as an input for the fitting:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "\n", + "write(\"tmp.xyz\", db)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Fitting the 2-body potential" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We need to calculate the atomic energy and use it as an offset during the fitting:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "isolated_atom = aseAtoms(\"Si\", positions=[[0,0,0]])\n", + "isolated_atom.set_calculator(qm_pot)\n", + "E0 = isolated_atom.get_potential_energy()\n", + "E0" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We are going to use teach_sparse tool to build the GAP potential. For a detailed description of the available options you can run the following command." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": false + }, + "outputs": [], + "source": [ + "!teach_sparse --help" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now we have to do actual fitting by running the following command:\n", + "\n", + "Relevant parameters:\n", + "- covariance_type: Type of covariance function to use. Available: ARD_SE, DOT_PRODUCT, BOND_REAL_SP-\n", + "ACE, PP (piecewise polynomial)\n", + "- theta_uniform: Set the width of Gaussians for the ARD_SE and PP kernel, same in each dimension\n", + "- n_sparse: Number of sparse points to use in the sparsification of the Gaussian process\n", + "- delta: Set the standard deviation of the Gaussian process. Typically this would be set to the standard deviation (i.e. root mean square) of the function that is approximated with the Gaussian process.\n", + "- default_sigma: error in [energies, forces, virials, hessians]\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": false + }, + "outputs": [], + "source": [ + "# !teach_sparse at_file=dummy default_sigma={0 0 0 0} gap={distance_Nb delta=0.0 covariance_type=ARD_SE --help}" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": false + }, + "outputs": [], + "source": [ + "!teach_sparse at_file=tmp.xyz \\\n", + "gap={distance_Nb order=2 \\\n", + " cutoff=5.0 \\\n", + " covariance_type=ARD_SE \\\n", + " theta_uniform=1.0 \\\n", + " n_sparse=15 \\\n", + " delta=1.0} \\\n", + "e0=-29.716948405885105 \\\n", + "default_sigma={0.01 0.5 0.0 0.0} \\\n", + "do_copy_at_file=F sparse_separate_file=F \\\n", + "gp_file=gap_2b.xml" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can load and use the fitted potential:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "gap2b = Potential(param_filename=\"gap_2b.xml\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can calculate the potential energies using the new potential and compare them with the TB DFTB ones" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "qm_energies = [dba.energy for dba in db]\n", + "gap2b_energies = []\n", + "for dba in db:\n", + " a = dba.copy()\n", + " a.set_calculator(gap2b)\n", + " gap2b_energies.append(a.get_potential_energy())\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "Natoms = len(db[0])\n", + "plt.scatter(np.array(qm_energies)/Natoms, np.array(gap2b_energies)/Natoms)\n", + "plt.plot([E0-5.0, E0-5.0+0.05], [E0-5.0, E0-5.0+0.05], \"k-\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can also calculate the RMSE (in meV) as follows:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "np.sqrt(sum((np.array(qm_energies)/Natoms - np.array(gap2b_energies)/Natoms)**2)/len(gap2b_energies))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can generate dimers for visualising the pairwise potential: " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "dimers = [aseAtoms(\"2Si\", positions=[[0,0,0], [x, 0,0]]) for x in np.linspace(1.6,6,100)] " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "dimer_curve = []\n", + "for dim in dimers:\n", + " dim.set_calculator(gap2b)\n", + " dimer_curve.append(dim.get_potential_energy())" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "plt.plot([dim.positions[1,0] for dim in dimers], np.array(dimer_curve)/2.0)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Fitting the 2 and 3-body potential" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": false + }, + "outputs": [], + "source": [ + "!teach_sparse at_file=tmp.xyz \\\n", + "gap={distance_Nb order=2 \\\n", + " cutoff=5.0 \\\n", + " covariance_type=ARD_SE \\\n", + " theta_uniform=1.0 \\\n", + " n_sparse=15 \\\n", + " delta=1.0:\\\n", + " distance_Nb order=3 \\\n", + " cutoff=4.0 \\\n", + " covariance_type=ARD_SE \\\n", + " theta_uniform=1.0 \\\n", + " n_sparse=50 \\\n", + " delta=0.004} \\\n", + "e0=-29.716948405885105 \\\n", + "default_sigma={0.005 0.5 0.0 0.0} \\\n", + "do_copy_at_file=F sparse_separate_file=F \\\n", + "gp_file=gap_3b.xml" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can load and use the fitted potential:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "gap3b = Potential(param_filename=\"gap_3b.xml\")\n", + "gap3b_energies = []\n", + "for dba in db:\n", + " a = dba.copy()\n", + " a.set_calculator(gap3b)\n", + " gap3b_energies.append(a.get_potential_energy())\n", + " " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can also calculate the RMSE (in meV) as follows:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "np.sqrt(sum((np.array(qm_energies)/Natoms - np.array(gap3b_energies)/Natoms)**2)/len(gap3b_energies))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "plt.scatter(np.array(qm_energies)/Natoms, np.array(gap3b_energies)/Natoms)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can generate dimers for visualising the pairwise potential: " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "dimer_curve = []\n", + "for dim in dimers:\n", + " dim.set_calculator(gap3b)\n", + " dimer_curve.append(dim.get_potential_energy())\n", + " \n", + "plt.plot([dim.positions[1,0] for dim in dimers], np.array(dimer_curve)/2.0)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Fitting the 2 and 3-body and SOAP potential" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We will describe that environment using the SOAP descriptor and compare them using the SOAP kernel. Note right away that the quantum mechanically computed energies are not fully determined by the near-environment of atoms, so this is an early indication that we will be making use of the \"noise\" interpretation of the $\\lambda$ regularization parameter: we don't expect (and don't want) our fitted function to precisely go through each datapoint.\n", + "\n", + "The SOAP descriptor of an atomic environment is based on a spherical harmonic expansion of the neighbour density, and truncating this expansion at some maximum numer of radial (n_max) and angular (l_max) indices gives rise to some parameters. We also need to give the cutoff within which we consider the neighbour environment.\n", + "\n", + "Writing the descriptor vector as $p_{ss'nn'l}$, where $s$ and $s'$ are indices that run over the different atomic species in the atom's environment, $n$ and $n'$ are radial and $l$ is an angular index, the kernel between two atomic environments is\n", + "\n", + "$$\n", + "K(p,p') = \\delta^2 \\left| \\sum_{ss'nn'l} p_{ss'nn'l} p'_{ss'nn'l}\\right|^\\zeta \\equiv \\delta^2 \\left| {\\bf p} \\cdot {\\bf p'}\\right|^\\zeta\n", + "$$\n", + "\n", + "The factor of $\\delta^2$ allows the setting of the scale of fitted function, relative to the error specification $\\lambda$. " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": false + }, + "outputs": [], + "source": [ + "!teach_sparse at_file=tmp.xyz \\\n", + "gap={distance_Nb order=2 \\\n", + " cutoff=5.0 \\\n", + " covariance_type=ARD_SE \\\n", + " theta_uniform=1.0 \\\n", + " n_sparse=15 \\\n", + " delta=1.0:\\\n", + " distance_Nb order=3 \\\n", + " cutoff=4.0 \\\n", + " covariance_type=ARD_SE \\\n", + " theta_uniform=1.0 \\\n", + " n_sparse=50 \\\n", + " delta=0.004:\\\n", + " soap cutoff=4.0 \\\n", + " covariance_type=dot_product \\\n", + " zeta=2 \\\n", + " delta=0.016 \\\n", + " atom_sigma=0.7 \\\n", + " l_max=6 \\\n", + " n_max=6 \\\n", + " n_sparse=200 \\\n", + " sparse_method=cur_points} \\\n", + "e0=-29.716948405885105 \\\n", + "default_sigma={0.001 0.5 0.0 0.0} \\\n", + "do_copy_at_file=F sparse_separate_file=F \\\n", + "gp_file=gap_2b3bsoap.xml 2>&1 | grep -v FoX" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can load and use the fitted potential:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "gap_soap = Potential(param_filename=\"gap_2b3bsoap.xml\")\n", + "gap_energies = []\n", + "for dba in db:\n", + " a = dba.copy()\n", + " a.set_calculator(gap_soap)\n", + " gap_energies.append(a.get_potential_energy())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can also calculate the RMSE (in meV) as follows:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "np.sqrt(sum((np.array(qm_energies)/Natoms - np.array(gap_energies)/Natoms)**2)/len(gap_energies))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "plt.scatter(np.array(qm_energies)/Natoms, np.array(gap_energies)/Natoms)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Using GAP for MD\n", + "We can use now use the fitted potential to do MD or geometry relaxation:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "#attach tight binding calculator\n", + "atoms.set_calculator(gap_soap)\n", + "\n", + "# Thermalize atoms\n", + "# MaxwellBoltzmannDistribution(atoms, 2.0* T * units.kB)\n", + "\n", + "# dynamics = VelocityVerlet(atoms, timestep)\n", + "dynamics = Langevin(atoms, timestep, T * units.kB, 0.002)\n", + " \n", + "dynamics.attach(print_status, interval=10)\n", + "dynamics.attach(print_energy, interval=10)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Dynamics:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "dynamics.run(steps=100)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Geometry optimisation:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "optatoms = db[0].copy()\n", + "optatoms.set_calculator(gap_soap)\n", + "opt = PreconLBFGS(optatoms, precon=Exp(3.0))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "opt.run(fmax=0.01)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's visualise the relaxed structure:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "view = visualise(optatoms, (2,2,1))\n", + "display(view)\n", + "view.camera= 'orthographic'" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# I hope you enjoyed it! Have a nice day!" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 2", + "language": "python", + "name": "python2" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 2 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython2", + "version": "2.7.15" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +}