Deep potentials for materials science

doi: 10.1088/2752-5724/ac681d

Tongqi Wen¹,
Linfeng Zhang^{2, 3},
Han Wang^{4, 5
,},
Weinan E^{3, 6, 7},
David J Srolovitz^{1, 8
,}

1.
Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, Hong Kong Special Administrative Region of China
2.
DP Technology, Beijing, People’s Republic of China
3.
AI for Science Institute, Beijing, People’s Republic of China
4.
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, People’s Republic of China
5.
HEDPS, CAPT, College of Engineering, Peking University, Beijing, People’s Republic of China
6.
School of Mathematical Sciences, Peking University, Beijing, People’s Republic of China
7.
Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, United States of America
8.
International Digital Economy Academy (IDEA), Shenzhen, People’s Republic of China

More Information

Corresponding author: E-mail: wang_han@iapcm.ac.cn; E-mail: srol@hku.hk

摘要

Abstract: To fill the gap between accurate (and expensive) ab initio calculations and efficient atomistic simulations based on empirical interatomic potentials, a new class of descriptions of atomic interactions has emerged and been widely applied; i.e. machine learning potentials (MLPs). One recently developed type of MLP is the deep potential (DP) method. In this review, we provide an introduction to DP methods in computational materials science. The theory underlying the DP method is presented along with a step-by-step introduction to their development and use. We also review materials applications of DPs in a wide range of materials systems. The DP Library provides a platform for the development of DPs and a database of extant DPs. We discuss the accuracy and efficiency of DPs compared with ab initio methods and empirical potentials.
- deep potential /
- atomistic simulation /
- machine learning potential /
- neural network
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Figure 1. Schematic of the non-smooth DP descriptor for a water molecule. The red and blue spheres denote oxygen and hydrogen atoms, respectively.

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Figure 2. Workflow for constructing the smooth DP descriptors.

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Figure 3. Schematic depiction of an over-fit ML model (a) and a properly fit ML model (b). The ground truth and ML model are denoted by the red and blue lines. The training data are denoted by red . The over-fit values are close to the ground truth for the training data, while the gradients deviate from the ground truth (red and blue arrows). Both the values and gradients of the properly fit model are close to the ground truth for the training data.

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Figure 4. Schematic illustration of the DP-GEN scheme. (a) The DP-GEN scheduler runs iteratively, performing three steps: exploration, labelling, and training. (b) In the exploration step, different structures are sampled using DP MD and an error indication is applied to choose the candidates for labelling. (c) In the labelling step, DFT calculations are performed on the candidates to obtain the energy, forces, and virial tensor. (d) In the training process, DeePMD-kit package is used to train a new DP based on the initial datasets and candidates from each iteration. Reprinted from [84], Copyright (2020), with permission from Elsevier.

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Figure 5. Visualisation of the model deviation. DL 1 to 4 indicates 4 Deep Learning models trained from the existing data (blue crosses).

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Figure 6. Elements for which DPs are currently available in the DP Library (web link 11 in section S2 of the SM) are indicated in black.

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Figure 7. A comparison of the speed of MD simulations using the compressed Ti DP, an EAM, and/or an MEAM potential on (a) CPU and (b) GPU systems [101].

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Figure 8. The workflow for specialising a general DP. R_i is the atomic coordinate of atom i, E is the total energy of one configuration, V is the virial (stress) tensor of one configuration, f_i is the force on atom i, n is the number of atoms in one configuration, DP¹ is the first ensemble of trial DPs, labels the ^th DP in the ensemble, is the standard deviation, and _lo and _hi are two thresholds in DP-GEN. The specialisation step is shown in the green box. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [Nature] [npj Computational Materials] [101].

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Figure 9. The generalised stacking fault energy (-lines) on the (a), (c) Basal, (b), (d) Prism, and (e) Pyramidal I plane of HCP Ti calculated using DFT, empirical potentials (Hennig [109], Ko [111], Dickel [112], and Mendelev (Ti3’ in [108])), and a specialised DP. The configurations in the dashed red box and at zero slip (origin) are included in the training sets. All configurations on the Pyramidal I narrow -line (e) are included in the training dataset. (f) Each point denotes the energy of a simulation cell containing a screw $⟨ a ⟩$ dislocation core from a 600 K MD simulation after quenches to zero temperature (each point is one picosecond apart in the MD simulation). Two types of screw dislocation cores are observed: one delocalised onto a Prism plane (above the blue line) and one on Pyramidal I plane (below the blue line). Some of the cores are slightly distorted leading to small variations in the energy. See [101] for details.

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Figure 10. The screw $⟨ c + a ⟩$ dislocation core structure on the Pyramidal I plane of HCP Ti determined via DFT, DP [101], and the Hennig MEAM potential [109]. The red, blue, and white shaded atoms denote local HCP, BCC, and indeterminate atomic environments.

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Figure 11. Phase diagram of water. (a₁) DP model (red solid lines) and experiment (gray solid lines) for $T <$ 420 K. (a₂) Phase diagram at high T and P. (b) Phase diagram of TIP4P/2005 [204] water. Reprinted figure with permission from [95], Copyright (2021) by the American Physical Society.

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Table 1. Trust levels _lo and _hi employed in DP-GEN for several systems.

System	[_lo,_hi]
Mg [83]	[0.03,0.13]
Al [83] & Al-Mg [83]	[0.05,0.15]
Cu [84]	[0.05,0.20]
Mg-Al-Cu [100]	[0.05,0.20]
Ti [101]	[0.10,0.25] at $T < 1.5 T_{m}$ ^a for bulk
	exploration and [0.15,0.30] elsewhere
W [102]	[0.20,0.35]
Ag-Au [103]	[0.05,0.20]
water [95]	[0.15,0.25] in first 24 iterations
	[0.18,0.32] in iterations 25 to 32
	[0.20,0.35] in iterations 33 to 36
SiC [104]	[0.15,0.30]
Li₁₀(Ge,Si, or Sn)P₂S₁₂ [105]	[0.12,0.25]
T_m is 1941 K, which is the experimental melting point for Ti.

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Table 2. Example applications of DP in materials science.

System	Reference
Elemental bulk systems
Al	[83, 114-118]
Mg	[83]
Cu	[84]
Ti, W	[101, 102]
Ag, Au	[103, 119, 120]
Li	[121]
Be	[122]
Ga	[123]
Sb	[124]
C	[125]
Si	[126, 127]
P	[98]

Multi-element bulk systems
Al-Mg, Al-Cu-Mg	[83, 100, 128, 129]
Al-Cu, Al-Zn-Mg	[130, 131]
Al-Cu-Ni	[132]
Ag-Au	[103, 119]
Pd-Si, Nb₅Si₃, Zr₇₇Rh₂₃, Bi₂Te₃	[133-136]
Al₉₀X₁₀ (X = Tb, Cr, or Ce)	[137-140]
(Pd, Pt) $_{x}$ (Ge, Sn, Pb) $_{y}$	[141]
P₂Sn₅	[142]
Silica, silicate	[143-146]
SiC	[104, 147]
B₄C	[148]
Molten salt LiF, FLiBe, and chloride	[149-157]
Li or Na-based battery materials	[105, 158-162]
TiO₂	[163]
-Ga₂O₃	[164]
Ferroelectrics HfO₂	[165]
Ag₂S	[166]
MoS₂	[167]
SnSe	[168]
Zr $_{1 - x}$ W_xB₂	[169]
(Hf_0.2Zr_0.2Ta_0.2Nb_0.2Ti_0.2)X (X = C or B₂)	[170, 171]

Aqueous systems
Water	[95, 172-183]
Zinc ion in water	[184]
Water-vapour interface	[185, 186]
Water-TiO₂ interface	[187]
Ice	[188, 189]

Molecular systems and clusters
Organic molecules	[99, 190-195]
Metal and alloy clusters	[119, 196]

Surfaces and low-dimensional systems
Metal and alloy surfaces	[103, 119, 129]
Graphane	[125, 197]
Monolayer In₂Se₃	[198]
2D Co-Fe-B	[199]

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