Active learning of uniformly accurate interatomic potentials for materials simulation

Linfeng Zhang, De-Ye Lin, Han Wang, Roberto Car, and Weinan E

Phys. Rev. Materials 3, 023804 – Published 25 February 2019

Abstract

An active learning procedure called deep potential generator (DP-GEN) is proposed for the construction of accurate and transferable machine learning-based models of the potential energy surface (PES) for the molecular modeling of materials. This procedure consists of three main components: exploration, generation of accurate reference data, and training. Application to the sample systems of Al, Mg, and Al-Mg alloys demonstrates that DP-GEN can produce uniformly accurate PES models with a minimal number of reference data.

Received 28 October 2018

DOI:https://doi.org/10.1103/PhysRevMaterials.3.023804

Physics Subject Headings (PhySH)

Atomic & molecular structure Potential energy surfaces

Alloys Amorphous materials Liquids

Density functional theory First-principles calculations Molecular dynamics

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Linfeng Zhang

Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA

De-Ye Lin

Institute of Applied Physics and Computational Mathematics, Huayuan Road 6, Beijing 100088, People's Republic of China and CAEP Software Center for High Performance Numerical Simulation, Huayuan Road 6, Beijing 100088, People's Republic of China

Han Wang^*

Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Huayuan Road 6, Beijing 100088, People's Republic of China

Roberto Car

Department of Chemistry, Department of Physics, Program in Applied and Computational Mathematics, Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, New Jersey 08544, USA

Weinan E^†

Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA and Beijing Institute of Big Data Research, Beijing 100871, People's Republic of China

^*wang_han@iapcm.ac.cn
^†weinan@math.princeton.edu

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Vol. 3, Iss. 2 — February 2019

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Images

Figure 1
Schematic plot of one iteration of the DP-GEN scheme, taking the Al-Mg system as an example. (a) Exploration with DPMD. (a.1) Preparation of initial structures. (I) For bulk structures: start from stable crystalline structures of pure Al and Mg. In this work, we use face-centered-cubic (fcc), hexagonal-closed-packed (hcp), simple cubic (sc), and diamond structures. (II) Compress and dilate the stable structures uniformly to allow for a larger range of number densities. We use $α$ to denote the scale factor of the compression and dilation operations. Here $α$ ranges in the interval 0.96–1.04. (III) Randomly perturb the atomic positions and cell vectors of all the initial crystalline structures. The magnitude of perturbations on the atomic coordinates is $σ_{a} = 0.01 Å$ . The magnitude of perturbation on each cell vector is $σ_{c} = 0.03$ times the length of the cell vector. (IV) Generate random alloy structures: starting from all the structures prepared for pure systems, randomly place Al or Mg at different sites. (V) Generate structures with rigid displacement: starting from stable fcc and hcp structures, rigidly displace two crystalline halves along specific crystallographic directions. We only use (100), (110), (111), and (0001), $(10 \bar{1} 0)$ , $(11 \bar{2} 0)$ , respectively, for fcc and hcp, as the displacement directions. The magnitudes $d$ of the displacements range in the interval 0.2–10.0 Å. Based on all the displaced structures, perform dilation $α$ and perturbation $σ_{a}$ and $σ_{c}$ , and generate random alloy structures. (a.2) Canonical simulation at a given temperature. The temperature increases with the iteration index within the range 50–2000 K. (b) Labeling with electronic structure calculations. (c) Training with the DP model.
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Figure 2
Phonon dispersion relations of Al at $T = 80 K$ and $P = 1 bar$ . Here $q$ denotes the wave number and $ν$ the frequency. The experimental data is taken from [59].
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Figure 3
RDFs $g (r)$ of liquid Al at $P = 1$ bar and temperatures $T = 943$ K. The DP and MEAM predictions are compared with the experimental data taken from [60]. The inserted plot is the zoom-in of RDFs in the range $3.5 Å \leq r \leq 7 Å$ .
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Figure 4
EOSs of Al. Solid lines, dashed lines, and cross points denote DP, MEAM, and DFT results, respectively. The energies of MEAM are shifted so that the MEAM energy of a stable fcc structure equals that given by DFT. DFT based relaxations fail in some hcp and dhcp structures with a volume per atom larger 30 $Å^{3}$ ; thus the corresponding DFT predictions are not shown. Yellow bars denote volume ranges in the training data. The bcc and dhcp structures were not explicitly added to the training data.
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Figure 5
Surface formation energies of Al and Mg. All the nonequivalent surfaces with Miller index values smaller than 4 and 3 are investigated for Al and Mg, respectively.
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Figure 6
Comparisons of Al-Mg alloy properties predicted by DFT, DP, and MEAM, based on 28 structures in the MP database. (a) 28 formation energies. (b) 28 equilibrium volumes per atom. (c) 225 elastic constants. Three structures, mp-1039192, mp-1094664, and mp-12766, are excluded because they resulted unstable under small displacements within DFT. (d) 125 relaxed vacancy formation energies. Three structures, mp-1039192, mp-1094664, and mp-1038818, are excluded because they resulted unstable under vacancy relaxations within DFT. Two DP predictions and three MEAM predictions are outside the range of the plot due to large errors. (e) $86 199$ energies per atom along the interstitial relaxation pathways within DFT. (f) 903 unrelaxed surface energies.
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