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speeding up ab-initio molecular dynamics with artificial neural networks PDF

192 Pages·2017·10.81 MB·English
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SPEEDING UP AB-INITIO MOLECULAR DYNAMICS WITH ARTIFICIAL NEURAL NETWORKS by Håkon Vikør Treider Thesis for the degree of Master of Science Faculty of Mathematics and Natural Sciences University of Oslo November 2017 Abstract Inthisthesis, wereviewtheprocessofusingartificialneuralnetworksashyperpo- tential energy surfaces in molecular dynamics simulations, with thorough focus on creation, training and validation. We also outline a more general machine learning process, with emphasis on choosing an atomic environment descriptor together with a regression model. ThemoleculardynamicslibraryLAMMPSisextendedwithaminimal-working example of an artificial neural network potential with support for monatomic sys- tems, where long-range interactions are not present. The potential is pre-trained using TensorFlow, a powerful machine learning library from Google, on atomic configurations computed with classical potentials. Our results for bulk-silicon compare nicely with the literature, and encourages future use where the atomic configurations are handled using quantum mechanics, for example by density functional theory. This represents an entirely new way of running ab initio molecular dynamics simulations, with unprecedented computational efficiency. To my girlfriend, friends and family. For their endless support and encouragement. Acknowledgements I clearly remember my first encounter with physics. It was in high school, with Bjørnar Pettersen teaching in an elegant manner; always in a light-hearted mood and with an infectious charisma! You got me hooked on physics! When I first started here at the University of Oslo, I immediately fell in love with the numerical methods that were being taught from the very first semester. There are many people who deserve credit for this, but a select few are named here: Morten Hjorth-Jensen, Anders Malthe-Sørensen and Hans Petter Lang- tangen. Wow, what a vitamin injection you have given the scientific education programmes! Later, when applying for my Master’s degree, I followed the computational path and became a part of the amazing computational physics group; a bunch of way-too-knowledgeable people when it comes to scientific programming. A spe- cial thanks to Sebastian, Alocias and Håkon for making my time here great! The group is led by aforementioned professors, Anders Malthe-Sørensen and Morten Hjorth-Jensen. To you I’d like to say: the inspiration you continually bring to the group is unique and I couldn’t be happier about my choice. You are both outstanding supervisors; down-to-earth, positive and always encouraging! Thank you! I also want to thank my non-official supervisor, Anders Hafreager. You know the ins and outs of molecular dynamics (and LAMMPS) like a taxi-driver knows London! Thanks for all the hours of technical help! Going through this Master with me, - no one else has had the same impact as you, John-Anders Stende. We went through the dark ages, but ended up in the Renaissance. Thanks for your endless spirit! Lastly, a big thanks to my dear Marit for all the patience and love throughout my years of studying! Collaboration details All developed code for this thesis is written from scratch by the author. However, I have been working in close collaboration with John-Anders Stende, and so, the overall structure of our codes have ended up using similar solutions. Håkon Vikør Treider Oslo, November 2017 Contents List of Figures 2 List of Acronyms 4 1 Introduction 5 1.1 Multiscale Modelling . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Bridging Quantum Mechanics and Molecular Dynamics . . 7 1.2 Project Description . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.2 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . 9 2 Machine Learning: A Historical Overview 13 I Theory 19 3 Machine Learning 21 3.1 Learning Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.1 Supervised Learning . . . . . . . . . . . . . . . . . . . . . 21 3.1.2 Unsupervised Learning . . . . . . . . . . . . . . . . . . . . 22 3.1.3 Reinforcement Learning . . . . . . . . . . . . . . . . . . . 23 4 Artificial Neural Networks 25 4.1 Inspiration from Biology . . . . . . . . . . . . . . . . . . . . . . . 25 4.1.1 Extracting a Computational Model from the Biological Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Feed-Forward Neural Networks . . . . . . . . . . . . . . . . . . . 28 4.2.1 Mathematical Description . . . . . . . . . . . . . . . . . . 29 4.2.2 Convolutional Neural Networks . . . . . . . . . . . . . . . 32 4.2.3 Autoencoder . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3 Recurrent Neural Networks . . . . . . . . . . . . . . . . . . . . . 35 4.3.1 Long Short-Term Memory . . . . . . . . . . . . . . . . . . 35 4.4 Deciding on an ANN Architecture . . . . . . . . . . . . . . . . . . 36 ix x Contents 4.5 Activation Functions . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.5.1 Sigmoid Functions . . . . . . . . . . . . . . . . . . . . . . 37 4.5.2 Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . 39 4.5.3 Linear Rectifier . . . . . . . . . . . . . . . . . . . . . . . . 41 4.5.4 Other Activation Functions . . . . . . . . . . . . . . . . . 42 4.6 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.6.1 Hidden Layers . . . . . . . . . . . . . . . . . . . . . . . . 43 4.6.2 Nodes per Hidden Layer . . . . . . . . . . . . . . . . . . . 46 4.6.3 Weights and Biases . . . . . . . . . . . . . . . . . . . . . . 46 4.6.4 Standardizing the Network Inputs . . . . . . . . . . . . . . 47 4.7 Training the Network . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.7.1 Splitting the Available Data . . . . . . . . . . . . . . . . . 49 4.7.2 Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . 50 4.7.3 Extended Cost Functions . . . . . . . . . . . . . . . . . . 51 4.7.4 Error Backpropagation . . . . . . . . . . . . . . . . . . . . 53 4.7.5 Full Training Algorithm using Gradient Descent . . . . . . 59 4.7.6 Learning Rate . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.7.7 Online- and Offline Learning . . . . . . . . . . . . . . . . 62 4.8 Overfitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.8.1 Early Stopping . . . . . . . . . . . . . . . . . . . . . . . . 64 4.8.2 Regularizations Techniques . . . . . . . . . . . . . . . . . 64 5 Molecular Dynamics 67 5.1 Classical Potential Energy Surfaces . . . . . . . . . . . . . . . . . 67 5.1.1 Bonded vs. Non-Bonded Potentials . . . . . . . . . . . . . 68 5.1.2 Extracting a Potential Energy Surface from Quantum Me- chanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.1.3 Force Calculation . . . . . . . . . . . . . . . . . . . . . . . 69 5.2 Time Integration: Velocity-Verlet . . . . . . . . . . . . . . . . . . 69 5.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 71 6 Atomic Environment Descriptors 73 6.1 Image- and Atom-Centered . . . . . . . . . . . . . . . . . . . . . 73 6.1.1 Naïve Approach: Cartesian Coordinates . . . . . . . . . . 74 6.2 Gaussian Descriptor . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.2.1 Radial Symmetry Functions . . . . . . . . . . . . . . . . . 76 6.2.2 Angular Symmetry Functions . . . . . . . . . . . . . . . . 78 6.2.3 Cutoff Functions . . . . . . . . . . . . . . . . . . . . . . . 80 6.2.4 Analytic Derivatives . . . . . . . . . . . . . . . . . . . . . 83 6.2.5 Symbolic Differentiation and Automatic Code Generation using SymPy . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.3 Other Descriptors . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.3.1 Zernike Descriptor . . . . . . . . . . . . . . . . . . . . . . 87

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together with a regression model. The molecular dynamics library LAMMPS is extended with a minimal-working example of an artificial neural network potential with support for monatomic sys- tems, where long-range interactions are not present. The potential is pre-trained using TensorFlow, a powerful
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