Machine Learning of Atomic-Scale Properties Based on Physical Principles

We briefly summarize the kernel regression approach, as used recently in materials modelling, to fitting functions, particularly potential energy surfaces, and highlight how the linear algebra framework can be used to both predict and train from linear functionals of the potential energy, such as the total energy and atomic forces. We then give a detailed account of the Smooth Overlap of Atomic Positions (SOAP) representation and kernel, showing how it arises from an abstract representation of smooth atomic densities, and how it is related to several popular density-based representations of atomic structure. We also discuss recent generalisations that allow fine control of correlations between different atomic species, prediction and fitting of tensorial properties, and also how to construct structural kernels—applicable to comparing entire molecules or periodic systems— that go beyond an additive combination of local environments.

[1]  Andrea Grisafi,et al.  Symmetry-Adapted Machine Learning for Tensorial Properties of Atomistic Systems. , 2017, Physical review letters.

[2]  Volker L. Deringer,et al.  Gaussian approximation potential modeling of lithium intercalation in carbon nanostructures. , 2017, The Journal of chemical physics.

[3]  A. Atiya,et al.  Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond , 2005, IEEE Transactions on Neural Networks.

[4]  Michele Parrinello,et al.  Demonstrating the Transferability and the Descriptive Power of Sketch-Map. , 2013, Journal of chemical theory and computation.

[5]  Galli,et al.  Large scale electronic structure calculations. , 1992, Physical review letters.

[6]  M. Finnis,et al.  Interatomic Forces in Condensed Matter , 2003 .

[7]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[8]  Walter Kohn.  NEARSIGHTEDNESS OF ELECTRONIC MATTER , 2008 .

[9]  Gábor Csányi,et al.  Many-Body Coarse-Grained Interactions Using Gaussian Approximation Potentials. , 2016, The journal of physical chemistry. B.

[10]  Volker L. Deringer,et al.  Growth Mechanism and Origin of High sp^{3} Content in Tetrahedral Amorphous Carbon. , 2018, Physical review letters.

[11]  Carl E. Rasmussen,et al.  A Unifying View of Sparse Approximate Gaussian Process Regression , 2005, J. Mach. Learn. Res..

[12]  Yang,et al.  Direct calculation of electron density in density-functional theory. , 1991, Physical review letters.

[13]  O. A. von Lilienfeld,et al.  Communication: Understanding molecular representations in machine learning: The role of uniqueness and target similarity. , 2016, The Journal of chemical physics.

[14]  Volker L. Deringer,et al.  Machine learning based interatomic potential for amorphous carbon , 2016, 1611.03277.

[15]  S. Goedecker.  Linear scaling electronic structure methods , 1999 .

[16]  R. Kondor,et al.  On representing chemical environments , 2012, 1209.3140.

[17]  Anders S. Christensen,et al.  Alchemical and structural distribution based representation for universal quantum machine learning. , 2017, The Journal of chemical physics.

[18]  R. Kondor,et al.  Gaussian approximation potentials: the accuracy of quantum mechanics, without the electrons. , 2009, Physical review letters.

[19]  Seiji Kajita,et al.  A Universal 3D Voxel Descriptor for Solid-State Material Informatics with Deep Convolutional Neural Networks , 2017, Scientific Reports.

[20]  Volker Roth,et al.  Automatic Model Selection in Archetype Analysis , 2012, DAGM/OAGM Symposium.

[21]  Marco Cuturi,et al.  Sinkhorn Distances: Lightspeed Computation of Optimal Transport , 2013, NIPS.

[22]  Petros Drineas,et al.  CUR matrix decompositions for improved data analysis , 2009, Proceedings of the National Academy of Sciences.

[23]  Joel M. Bowman,et al.  Permutationally invariant potential energy surfaces in high dimensionality , 2009 .

[24]  T. Morawietz,et al.  How van der Waals interactions determine the unique properties of water , 2016, Proceedings of the National Academy of Sciences.

[25]  Jörg Behler,et al.  Nuclear Quantum Effects in Water at the Triple Point: Using Theory as a Link Between Experiments. , 2016, The journal of physical chemistry letters.

[26]  Rustam Z. Khaliullin,et al.  Microscopic origins of the anomalous melting behavior of sodium under high pressure. , 2011, Physical review letters.

[27]  Teofilo F. GONZALEZ,et al.  Clustering to Minimize the Maximum Intercluster Distance , 1985, Theor. Comput. Sci..

[28]  Aldo Glielmo,et al.  Efficient nonparametric n -body force fields from machine learning , 2018, 1801.04823.

[29]  Jörg Behler,et al.  Comparison of permutationally invariant polynomials, neural networks, and Gaussian approximation potentials in representing water interactions through many-body expansions. , 2018, The Journal of chemical physics.

[30]  A Data-Driven Construction of the Periodic Table of the Elements , 2018 .

[31]  K. Müller,et al.  Fast and accurate modeling of molecular atomization energies with machine learning. , 2011, Physical review letters.

[32]  Jörg Behler,et al.  Automatic selection of atomic fingerprints and reference configurations for machine-learning potentials. , 2018, The Journal of chemical physics.

[33]  Gabor Csanyi,et al.  Achieving DFT accuracy with a machine-learning interatomic potential: thermomechanics and defects in bcc ferromagnetic iron , 2017, 1706.10229.

[34]  Gábor Csányi,et al.  Development of a machine learning potential for graphene , 2017, 1710.04187.

[35]  Peter Sollich,et al.  Accurate interatomic force fields via machine learning with covariant kernels , 2016, 1611.03877.

[36]  Volker L. Deringer,et al.  Data-Driven Learning of Total and Local Energies in Elemental Boron. , 2017, Physical review letters.

[37]  A. Haar.  Der Massbegriff in der Theorie der Kontinuierlichen Gruppen , 1933 .

[38]  Noam Bernstein,et al.  Hybrid atomistic simulation methods for materials systems , 2009 .

[39]  Zoubin Ghahramani,et al.  Sparse Gaussian Processes using Pseudo-inputs , 2005, NIPS.

[40]  Gábor Csányi,et al.  Accuracy and transferability of Gaussian approximation potential models for tungsten , 2014 .

[41]  E Weinan,et al.  Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics , 2017, Physical review letters.

[42]  Klaus-Robert Müller,et al.  Machine learning of accurate energy-conserving molecular force fields , 2016, Science Advances.

[43]  Gábor Csányi,et al.  Comparing molecules and solids across structural and alchemical space. , 2015, Physical chemistry chemical physics : PCCP.

[44]  Donald W. Brenner,et al.  The Art and Science of an Analytic Potential , 2000 .

[45]  Frederick R. Manby,et al.  Machine-learning approach for one- and two-body corrections to density functional theory: Applications to molecular and condensed water , 2013 .

[46]  Michele Parrinello,et al.  Generalized neural-network representation of high-dimensional potential-energy surfaces. , 2007, Physical review letters.

[47]  George E. Dahl,et al.  Prediction Errors of Molecular Machine Learning Models Lower than Hybrid DFT Error. , 2017, Journal of chemical theory and computation.

[48]  Carl E. Rasmussen,et al.  Derivative Observations in Gaussian Process Models of Dynamic Systems , 2002, NIPS.

[49]  Kristof T. Schütt,et al.  How to represent crystal structures for machine learning: Towards fast prediction of electronic properties , 2013, 1307.1266.

[50]  Gábor Csányi,et al.  Gaussian approximation potentials: A brief tutorial introduction , 2015, 1502.01366.

[51]  Felix A Faber,et al.  Crystal structure representations for machine learning models of formation energies , 2015, 1503.07406.

[52]  Christian Trott,et al.  Spectral neighbor analysis method for automated generation of quantum-accurate interatomic potentials , 2014, J. Comput. Phys..

[53]  Noam Bernstein,et al.  Machine learning unifies the modeling of materials and molecules , 2017, Science Advances.

[54]  J S Smith,et al.  ANI-1: an extensible neural network potential with DFT accuracy at force field computational cost , 2016, Chemical science.

[55]  Theory and Practice of Atom-Density Representations for Machine Learning , 2018 .