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Building Nonparametric n-Body Force Fields Using Gaussian Process Regression

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Machine Learning Meets Quantum Physics

Part of the book series: Lecture Notes in Physics ((LNP,volume 968))

Abstract

Constructing a classical potential suited to simulate a given atomic system is a remarkably difficult task. This chapter presents a framework under which this problem can be tackled, based on the Bayesian construction of nonparametric force fields of a given order using Gaussian process (GP) priors. The formalism of GP regression is first reviewed, particularly in relation to its application in learning local atomic energies and forces. For accurate regression, it is fundamental to incorporate prior knowledge into the GP kernel function. To this end, this chapter details how properties of smoothness, invariance and interaction order of a force field can be encoded into corresponding kernel properties. A range of kernels is then proposed, possessing all the required properties and an adjustable parameter n governing the interaction order modelled. The order n best suited to describe a given system can be found automatically within the Bayesian framework by maximisation of the marginal likelihood. The procedure is first tested on a toy model of known interaction and later applied to two real materials described at the DFT level of accuracy. The models automatically selected for the two materials were found to be in agreement with physical intuition. More in general, it was found that lower order (simpler) models should be chosen when the data are not sufficient to resolve more complex interactions. Low n GPs can be further sped up by orders of magnitude by constructing the corresponding tabulated force field, here named “MFF”.

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Notes

  1. 1.

    Choosing a squared error function \(L=(\bar {\epsilon }(\rho )- \epsilon )^2\), the expected error under the posterior distribution reads \( \langle L \rangle = \int d\epsilon \, p(\epsilon \mid \rho , \mathcal {D}) (\bar {\epsilon }(\rho )- \epsilon )^2. \) Minimising this quantity with respect to the unknown optimal prediction \(\bar {\epsilon }(\rho )\) can be done by equating the functional derivative \(\delta \langle L \rangle / \delta \bar {\epsilon }(\rho )\) to zero, yielding the condition \( (\bar {\epsilon }(\rho ) - \langle \epsilon \rangle ) = 0, \) proving that the optimal estimate corresponds to the mean \(\hat {\epsilon }(\rho )\) of the predictive distribution in Eq. (5.5). One can show that choosing an absolute error function \(L=\left | \bar {\epsilon }(\rho )- \epsilon \right |\) makes the mode of the predictive distribution the optimal estimate, this however coincides with the mean in the case of Gaussian distributions.

  2. 2.

    The model evidence is conventionally defined as the integral over the hyperparameter space of the marginal likelihood times the hyperprior (cf. [35]). We here simplify the analysis by jointly considering the model and its hyperparameters.

  3. 3.

    The n-body toy model used was set up as a hierarchy of two-body interactions defined via the negative Gaussian function \( \epsilon ^g(d) = - e^{-\frac {(d-1)^2}{2}}. \) This pairwise interaction, depending only on the distance d between two particles, was then used to generate n-body local energies as \(\epsilon _n(\rho ) = \sum _{i_1\neq \dots \neq i_{n-1}} \epsilon ^g(x_{i_1}) \epsilon ^g(x_{i_{2}}-x_{i_{1}}) \dots \epsilon ^g(x_{i_{n-2}}-x_{i_{n-1}})\) where \(x_{i_1},\dots , x_{i_n-1}\) are the positions, relative to the central atom, of n − 1 surrounding neighbours.

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Acknowledgements

The authors acknowledge funding by the Engineering and Physical Sciences Research Council (EPSRC) through the Centre for Doctoral Training “Cross Disciplinary Approaches to Non-Equilibrium Systems” (CANES, Grant No. EP/L015854/1) and by the Office of Naval Research Global (ONRG Award No. N62909-15-1-N079). The authors thank the UK Materials and Molecular Modelling Hub for computational resources, which is partially funded by EPSRC (EP/P020194/1). ADV acknowledges further support by the EPSRC HEmS Grant No. EP/L014742/1 and by the European Union’s Horizon 2020 research and innovation program (Grant No. 676580, The NOMAD Laboratory, a European Centre of Excellence). We, AG, CZ and AF, are immensely grateful to Alessandro De Vita for having devoted, with inexhaustible energy and passion, an extra-ordinary amount of his time and brilliance towards our personal and professional growth.

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Authors and Affiliations

  1. Department of Physics, King’s College London, London, UK

    Aldo Glielmo, Claudio Zeni, Ádám Fekete & Alessandro De Vita

  2. Dipartimento di Ingegneria e Architettura, University of Trieste, Trieste, Italy

    Alessandro De Vita

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  1. Aldo Glielmo
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  2. Claudio Zeni
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  3. Ádám Fekete
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  4. Alessandro De Vita
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Correspondence to Aldo Glielmo .

Editor information

Editors and Affiliations

  1. Machine Learning, Technical University of Berlin, Berlin, Germany

    Kristof T. Schütt

  2. Machine Learning Group, Technical University of Berlin, Berlin, Germany

    Stefan Chmiela

  3. Institute of Physical Chemistry and MARVEL, University of Basel, Basel, Switzerland

    O. Anatole von Lilienfeld

  4. Department of Physics and Materials Science, University of Luxembourg, Luxembourg, Luxembourg

    Alexandre Tkatchenko

  5. Graduate School of Frontier Sciences, University of Tokyo, Kashiwa, Japan

    Koji Tsuda

  6. Computer Science, Technical University of Berlin, Berlin, Germany

    Klaus-Robert Müller

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Glielmo, A., Zeni, C., Fekete, Á., De Vita, A. (2020). Building Nonparametric n-Body Force Fields Using Gaussian Process Regression. In: Schütt, K., Chmiela, S., von Lilienfeld, O., Tkatchenko, A., Tsuda, K., Müller, KR. (eds) Machine Learning Meets Quantum Physics. Lecture Notes in Physics, vol 968. Springer, Cham. https://doi.org/10.1007/978-3-030-40245-7_5

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