Introduction

Dislocations in both structural and functional ceramics are rapidly gaining attention due to their intriguing potential in property tuning, and thus are now perceived as promising atomic-scale entities for next-generation device applications1,2,3,4,5,6,7,8,9,10. As a type of topological one-dimensional defect, dislocations in ceramics feature non-stoichiometry and excess charge at their cores and local strain field around the cores, making them powerful entities for “defect engineering”. By manipulating dislocations into ceramics or interacting dislocations with other defects, one can harness unprecedented mechanical properties and tailored functional properties. Examples include exceptional plasticity6, anisotropic thermal transport7, controllable ferroelectric properties8,9, and enhanced photoconductivity10. Despite the significant effect and exciting application potential of dislocations in ceramics, the detailed structure and stability of dislocations, as well as the atomic-scale understanding of dislocation behaviors, are still challenging mainly due to their complication of the basic crystalline structures and the technical difficulty of the experiment, e.g., well-prepared samples and high-resolution ultramicroscopy11,12.

Atomistic simulations play increasingly important roles in revealing atomic-scale insights into dislocations in ceramics. Density functional theory (DFT) calculations provide detailed atomic and electronic structures but are limited by the size and time scales it can handle. This restricts its use for many dislocation-related problems in ceramics, e.g., dislocation interaction with grain boundary13, which often needs models with thousands to millions of atoms. In contrast, molecular dynamics (MD) simulations can study large systems of millions of atoms on a nanosecond scale. The challenge in conducting realistic MD simulations lies in the requirement for a natural interatomic potential. A wide array of empirical and semi-empirical potentials have been developed for ceramics, including the rigid ion model14, Stillinger-Weber (SW)15, Vashishta16, ReaxFF17, COMB318, and SMTB-Q19 potentials. However, many of these potentials were not designed to describe dislocations. The verification of their predictive power has been limited to relatively narrow domains of physical conditions directly relevant to their applications in the initial studies. These empirical potentials, based on fixed analytical function forms, lack the sophistication to accurately capture the energy surface of highly distorted dislocation core structures.

In recent years, advances in integrating DFT calculations with machine learning have led to the development of neural network potentials (NNPs)20, which learn the potential energy landscape directly from reference DFT datasets21. They maintain near-DFT accuracy while affording atomic resolution at larger system sizes and time scales, comparable to classical interatomic potentials. NNPs have been successfully developed for several ceramic materials, e.g., TiO222, ZnO23, TiN24, Cu2O25, KNbO326, GaN27, SrTiO328, and HfO229. Most of these potentials focus on near-equilibrium configurations, with few being validated for modeling and simulating dislocations. Several NNPs have been constructed and validated for dislocations in metals and their alloys30,31,32,33. However, given the significant differences between dislocation nature in metals and ceramics, such as the nonstoichiometric and charged dislocation core in ceramics34 and the mixed ionic-covalent bonding in ceramic materials, developing NNPs for dislocation plasticity in ceramics remains particularly challenging.

The primary issue lies in generating a training dataset that includes dislocation structures due to the complexity of their core configurations in ceramics and the computational demands involved, typically requiring the models composed of hundreds of atoms. This issue has also impeded the construction of NNPs for dislocation plasticity in ceramics using the newly developed on-the-fly active learning technology35,36. Active learning employs biased or unbiased MD simulation to generate candidate pools serving as training datasets. Active learning automatically targets regions with inaccurate energy predictions, eliminating the manual trial-and-error process of evaluating potential performance and designing new training configurations. Capturing the highly distorted dislocation core structures within this framework is particularly difficult because these structures exist in high-energy states, making them rare events37. As a result, it is essential to additionally incorporate dislocation structures in active learning. This requires automated generation of these structures along with computation of their DFT energies to achieve the active learning process. Consequently, it remains a significant question whether an alternative training dataset, one that eliminates the need to construct dislocation structures and is straightforward to generate automatically, can be effectively utilized to construct NNPs for dislocation plasticity in ceramics.

In this work, we propose a training dataset easy to compute via high-throughput calculation for the construction of NNPs for dislocation plasticity in ceramics, using three typical functional ceramics ZnO, GaN, and SrTiO3 as prototypes. Wurtzite ZnO and GaN are direct wide bandgap semiconductors38 with significant potential for various applications, while cubic SrTiO3, a typical perovskite oxide, has high dielectric constants, making it attractive for capacitors, varistors, and more39. The mechanical and functional tailoring of dislocations in these three ceramics has been widely studied experimentally10,40,41,42,43,44; however, the atomic-scale mechanism remains largely unexplored. The NNPs developed in this study provide powerful tools for detailed atomistic modeling and simulation of dislocation plasticity in these three ceramics. They also suggest an effective framework to construct NNPs, opening new avenues for exploring the plastic behavior of ceramics. The dataset, easy to compute via high-throughput calculation could also be used in the on-the-fly active learning of NNP for dislocation plasticity in ceramics. For instance, the structures within this dataset can serve as candidate pools, automatically selected for inclusion in the training dataset during the active learning process.

Results

A general recipe of easy-to-compute training datasets

To address the challenge of constructing accurate dislocation core configurations and calculating their DFT energies for a training dataset, this study shifts focus from complex dislocation core structures, which necessitate large models and substantial computational resources, to simpler dislocation-related structures, that enable high-throughput DFT calculation (see Fig. 1). Ultimately, we aim to demonstrate that such simplified model datasets, excluding any dislocation core structures, can effectively describe dislocation properties and prove to be a powerful tool for developing NNPs for dislocation plasticity in ceramics.

Fig. 1: Strategy and workflow of the construction of NNP for dislocation plasticity in ceramics in this work.
figure 1

The dislocation-related structures were generated using our open-source ADAIS code45 together with the VASP code46, enabling high-throughput DFT calculations. The basic structures provide a description of the essential properties of materials, which serve as the foundation of an NNP. For more details on the implementations, workflows, and execution examples of the ADAIS code, one may refer to our previous publication45. In addition to the broad validation across various properties, the effectiveness of the training dataset and NNPs was further confirmed by assessing the similarity to the training dataset and evaluating the uncertainty through the ensemble-based method, using snapshots derived from large-scale simulations.

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The dislocation-related structures include uniformly deformed structures, cleavage structures, and stacking fault structures. The training dataset involving the dislocation-related structures not only eliminates the need to construct complex dislocation configurations but is also readily computable through high-throughput calculation. It is made possible by our open-source ADAIS code45, along with the Vienna ab initio simulation package (VASP) code46. The ADAIS code has been designed specifically to automatically ascertain anisotropic ideal strength utilizing high-throughput DFT calculations.

Aside from the unit cell’s structure files and a series of deformation directions, no additional parameters or files were required to yield the ultimate training dataset of dislocation-related structures. It is suitable for target ceramics with any crystal structure or symmetry. For more details on the implementations, workflows, and execution examples of the open-source ADAIS code, one may refer to our previous publication45. In comparison, models that include complex dislocation cores need to be manually constructed, as their core structures and boundary conditions must be thoroughly checked before calculating their DFT energy to serve as training datasets. Furthermore, dislocation-related structures are typically composed of only tens of atoms. These structures are considerably smaller compared to models that incorporate dislocation cores, which are typically composed of hundreds of atoms. As a result, the generation of the training dataset becomes more efficient and less computationally demanding.

These dislocation-related structures represent various modes of deformation along specific deformation directions corresponding to the slip systems of the target ceramics, thereby closely aligning with their dislocation properties. In uniformly deformed structures, all crystal cell layers are displaced uniformly along designated directions47,48. This study employed uniform deformations of uniaxial tension, uniaxial compression, and pure shear. In contrast, cleavage and stacking fault structures involve localized deformations in the given directions, affecting only two neighboring atomic planes while other layers maintain their original positions47,48. The energetics of these highly deformed structures are fundamental in describing not only the crystal instability, which leads to dislocation nucleation and cracking, but also the notably distorted atomic structure near the dislocation core.

The deformation directions were chosen based on the crystal’s symmetry, usually covering all inequivalent low-index crystallographic orientations and planes, along with previously reported slip systems for the target ceramics. In this work, for wurtzite ZnO and GaN, typical deformation directions on the basal {0001}, prismatic I {10\(\overline{1}\)0}, prismatic II {11\(\overline{2}\)0}, pyramidal I {10\(\overline{1}\)1}, and pyramidal II {11\(\overline{2}\)2} planes in the hexagonal close packing (HCP) structure are considered, as shown in Fig. 2a. In the wurtzite structure, there are two different slip planes, i.e., glide-set and shuffle-set, on basal (0001) and prismatic I (10\(\overline{1}\)0) planes (see Fig. 2b). Several of these deformation directions have been reported in wurtzite ZnO and GaN. For instance, the {10\(\overline{11}\)}\(\langle \overline{1}2\overline{1}0\rangle\) and {0001}\(\langle \overline{1}2\overline{1}0\rangle\) slip systems were observed for {2\(\overline{11}\)0} indentation, and that for {10\(\overline{1}\)0} indentation was {10\(\overline{1}\)0}\(\langle \overline{1}2\overline{1}0\rangle\)49. The {0001} basal and {101\(\overline{1}\)} pyramidal dislocations were found under {0001} indentation50,51. For the cubic SrTiO3, the typical deformation directions on the {001}, {011}, and {111} planes were considered, as shown in Fig. 2c, several of which have been observed in cubic SrTiO3. For instance, under uniaxial compression, the {110}\(\langle 1\overline{1}0\rangle\) and {100}〈010〉 slip systems are activated at low and high temperatures, respectively41. The straight and highly 〈111〉-orientated 〈110〉-type 35.26° dislocations were found in the polished SrTiO3 single crystals52.

Fig. 2: Typical slips systems in ZnO, GaN, and SrTiO3.
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a Typical slip systems on the basal {0001}, prismatic I \(\{10\overline{1}0\}\), prismatic II \(\{11\overline{2}0\}\), pyramidal I \(\{10\overline{1}1\}\), and pyramidal II \(\{11\overline{2}2\}\) planes in the hexagonal close packing (HCP) structure. b Basal and prismatic I planes of shuffle-set and glide-set in the wurtzite structure. c Typical slip systems on the {001}, {011}, and {111} planes in the cubic structure.

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In addition to dislocation-related structures, the training dataset includes distorted structures, elastically deformed structures, defect/boundary structures, surface structures, amorphous structures, and isolated atoms, akin to those used in conventional NNPs. These basic structures provide a description of the essential properties of materials, such as lattice constants and elastic constants, which serve as the foundation of an NNP. The distorted structures were generated through ab initio MD simulations at finite temperatures or by applying random distortions to atoms within the structures. Unlike uniformly deformed structures, which mainly concentrate on configurations that are nearing instability, elastically deformed structures are focused on maintaining within the elastic limit. The unrelaxed surface structures are identical to the cleavage structures with a significant separation distance between the deformed neighboring planes. For further details about the training dataset, please refer to Tables S2, S6, S10 in the Supplementary Materials.

Training NNPs with small atomic structures raises concerns about their accuracy when applied to large-scale simulations. The complexity and diversity inherent in larger systems may not be adequately represented by small-scale training datasets, potentially leading to biases and inaccuracies in the simulations. As a result, the reliability of the training dataset in large-scale simulations requires further verification. In this work, beyond a broad validation across various properties, the effectiveness of our training dataset was further confirmed by measuring the similarity to the training dataset and assessing the uncertainty through an ensemble-based method53,54,55,56, using snapshots derived from large-scale simulations (see Fig. 1).

The general recipe proposed in this work for generating easy-to-compute training datasets via high-throughput DFT calculations eliminates the need to create complex dislocation configurations, and is suitable regardless of the crystal structure and symmetry of the target ceramics. Consequently, it can be readily applied to other important ceramics, such as wurtzite AlN, InN, and SiC; cubic MgO; and cubic and tetragonal perovskite oxides like BaTiO3, KNbO3, DyScO3, BiFeO3, MgSiO3, and PbTiO3. Additionally, it is applicable to complex crystal structures, including ZrO2, CeO2, and UO2.

Accuracy and performance of neural network potential

The accuracy and performance of the NNP were first validated across various properties, including lattice constants, elastic constants, elastic moduli, phonon dispersion, surface energy, energy under volumetric strain, cleavage energy, formation energy of planar defects, SFE, dislocation core structure, and dislocation slip barrier. The NNP’s performance concerning the phonon dispersion is particularly noteworthy, as depicted in Fig. 3. The phonon dispersion curves predicted by the NNP exhibit a reasonable agreement with the DFT results. In the following section, we will illustrate and discuss the performance of the NNP on dislocation-related properties. In this work, the dislocation dipole model with periodic quadrupolar arrangements57,58,59,60 was used to determine the dislocation core structure and slip barrier (see Fig. 4). Please refer to Table 1 and the Supplementary Materials for detailed information on other validation results.

Fig. 3: NNP performance for the phonon dispersion compared with the DFT results.
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a wurtzite ZnO, b wurtzite GaN, and c cubic SrTiO3.

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Fig. 4: Dislocation dipole model with periodic quadrupolar arrangements used for MD simulation of dislocation.
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The periodic cell of the arrangement is shown by the green-shaded area, with Ci representing its periodicity vectors. The cut vector of the dislocation dipole is denoted by V.

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Table 1 NNP and DFT predicted lattice constants (a and c in Å), elastic constants (cij in GPa), polycrystalline shear (μ) and bulk (κ) moduli (in GPa) of wurtzite ZnO, wurtzite GaN and cubic SrTiO3
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Stacking fault energy

Figure 5 illustrates the performance of the NNP in predicting the SFEs of pyramidal planes in wurtzite ZnO and GaN, compared with the results from DFT and empirical potential. Additional NNP-determined SFEs are presented in Figs. S13 and S25 in the Supplementary Materials. The NNP-predicted SFEs are in good agreement with the DFT results, providing a robust foundation for atomistic simulations of dislocation plasticity in wurtzite ZnO and GaN using these NNPs, given that the SFE is a fundamental property for dislocation nucleation and slip61,62. The empirical potentials considerably overestimate the SFE (see Figs. 5c, d). For instance, the predicted unstable SFE of ZnO (10\(\overline{1}\)1)[1\(\overline{2}\)10] via the analytic bond-order63 and Buckingham-type64 empirical potentials are 2.68 and 1.68 J/m2, respectively, significantly larger than those of DFT and NNP, which are 1.30 and 1.28 J/m2, respectively.

Fig. 5: Accuracy and performance of NNP on the SFEs of ZnO and GaN.
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a, b Diagrams depicting a pyramidal I and b pyramidal II planes in the wurtzite structure. The in-plane lattices are accentuated by red lines, and the slip planes are underscored by black dashed lines. c, d The calculated SFEs of the pyramidal planes using NNP, DFT, and empirical potentials for c ZnO (10\(\overline{1}\)1)[1\(\overline{2}\)10], and d GaN (11\(\overline{2}\)2)[11\(\overline{23}\)]. The analytic bond-order63 and Buckingham-type64 empirical potentials of ZnO, as well as the analytic bond-order121 and SW122 empirical potentials of GaN, were employed for comparison.

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Figure 6a, b illustrate the performance of the SrTiO3 NNP in predicting the (110)[1\(\overline{1}\)0] SFE, compared with the results from DFT and the pair empirical potential65. The NNP results align well with the DFT results, whereas the pair empirical potential65 overestimates the SFE. The formation of a nonstoichiometric stacking fault in cubic SrTiO3 through the dissociation of a [012] dislocation, where two (001) Ti-O layers end up neighboring across the stacking fault plane due to the absence of a (001) Sr-O layer, has been reported66, as depicted in Fig. 6c. Figure 6d presents the relative energies of the stacking faults with respect to the most stable configuration that features a 1/2[010] displacement component (see Fig. 6c). The NNP results agree well with the DFT results, even though these results were not included in the training dataset. The pair empirical potential65 fails to predict this nonstoichiometric model. Other SFE results predicted by the NNP are presented in Fig. S34 in the Supplementary Materials, demonstrating good agreement with the DFT results.

Fig. 6: Accuracy and performance of NNP on the SFEs of SrTiO3.
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a Structures and b energy profile of the (110)[1\(\overline{1}\)0] stacking fault in cubic SrTiO3, as calculated by NNP, DFT, and pair empirical potential. c Nonstoichiometric stacking fault formed with the dissociation of [012] dislocation in cubic SrTiO366, where two (001) Ti-O layers are positioned as neighbors across the stacking fault plane due to the absence of a (001) Sr-O layer. d Relative energies of the stacking faults compared to the most stable configuration, which includes a 1/2[010] displacement component (i.e., u = 0.5). The results of the pair empirical potential65, NNP in ref. 28, and DFT in ref. 66 are shown for comparison. Black dashed lines in (a, c) highlight the slip planes.

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Dislocation in wurtzite ZnO and GaN

Figure 7a, b depict the atomic structure of a basal glide-set 60° partial dislocation dipole in wurtzite ZnO, with a stacking fault situated between two partials. Each partial has an extra Zn or O atom at the dislocation core, referred to as the Zn core and O core, respectively. These nonstoichiometric dislocation cores are charged. To demonstrate the performance of the NNP for the charged dislocation cores and the long-range electrostatic interaction between them, one partial dislocation in the dipole model was displaced by a distance d (see Fig. 7c). The resulting energy change, ΔEtotal, originates from three sources: (1) the elastic interaction between dislocations, ΔEelastic; (2) the SFE due to the change in the area of the stacking fault, ΔESFE; and (3) the long-range electrostatic interaction between the charged dislocations, ΔEelectro.

Fig. 7: Accuracy and performance of NNP on the dislocation in ZnO.
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a, b Atomic structure of the basal glide-set 60° partial dislocation dipole with Zn core and O core: a viewed along the dislocation line, and b viewed perpendicular to the (0001) plane of the region outlined by the black dashed rectangle in (a). The black broken line in (b) represents dislocation lines along the cores. c Dislocation dipole model with periodic quadrupolar arrangements composed of positively charged (Zn core) and negatively charged (O core) basal glide-set 60° partial dislocations, where one dislocation is displaced by a distance d from its initial position. The periodic cell of the arrangement is shown by the green-shaded area. d Energy change ΔE of the dislocation dipole model as a function of the displacement d, with a model size of (n, m) = (12,4).

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In this study, we determined the extra charge at the dislocation core through Bader charge analysis67 after DFT static calculation, yielding values of +0.88e and −0.88e for the Zn and O cores, respectively. We then calculated ΔEelectro using the Ewald summation68 with setting two virtual charged atoms (±0.88e) at the positions of the dislocation core. Figure 7d presents the energy change ΔEtotal calculated by the NNP and DFT using a dipole model of (n, m) = (12,4). The NNP results align well with the DFT results. It is important to note that the training dataset did not include these dislocation structures. The discrepancy between the NNP and DFT results is significantly smaller than ΔEelectro (see Fig. 7d), suggesting that the NNP can accurately account for the long-range electrostatic interaction, which is not truncated at the cutoff radius, even though the charge information is not included in the NNP. It’s worth noting that the minimum distance between two charged partial dislocations is ~12 Å at d = ±4(a/2), which is significantly larger than the cutoff radius of NNP (i.e., 7.0 Å).

The accuracy of the NNP has recently been further enhanced by employing additional neural networks to construct environment-dependent atomic charges, thereby explicitly accounting for long-range electrostatic interactions23,69. As demonstrated in this work, even for systems with substantial charge transfer between atoms, such as nonstoichiometric dislocation core structures, long-range electrostatic interactions can be captured with high accuracy in an effectively screened manner by short-ranged NNPs. Moreover, explicitly including electrostatic interactions using the atomic charges increases the computational costs of NNP because a second set of neural networks is introduced23,69, and an Ewald summation68 or PPPM solver70 has to be used to calculate the electrostatic energies and forces. Therefore, in this study, we employed short-ranged NNPs and achieved excellent accuracy compared to DFT results, even for nonstoichiometric and charged dislocation core structures.

The Peierls potential of the basal glide-set 60° partial dislocation in ZnO, as calculated through NNP and DFT, is illustrated in Fig. 8a. The results presented herein are the average values for the dislocations with O core and Zn core. The Peierls barrier, as determined through ZnO NNP, is 0.018 eV/Å, a value that closely aligns with the DFT result of 0.017 eV/Å via a simulation cell of (n, m) = (12,4). Figure 8c shows the average value of the Peierls potential for the basal glide-set 60° partial dislocations with N core and Ga core in GaN. The Peierls barrier of GaN, at 0.054 eV/Å (NNP) and 0.062 eV/Å (DFT) with a model size of (n, m) = (12,4), significantly exceeds that of ZnO.

Fig. 8: Accuracy and performance of NNP on the dislocation in ZnO and GaN.
figure 8

Peierl potential of the basal glide-set 60° partial dislocation for a, b ZnO and c, d GaN, calculated using the dislocation dipole models and the NEB method. a, c Average values of the Peierls potential for the dislocations with O core and Zn core in ZnO (a), as well as those with N core and Ga core in GaN (c). b, d The Peierls potential of the dislocation with either an O core, Zn core, N core, or Ga core, which was calculated by moving only one dislocation within a large dipole model of (n, m) = (48,16). The DFT results are provided for comparison.

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The Peierls potential of the dislocation with either an O core, Zn core, N core, or Ga core was also calculated within a large dipole model of (n, m) = (48,16), where only one dislocation was moved. In a model of this size, the variations in elastic and electrostatic interactions, as the distance between two partial dislocations shifts, are negligible. Along the directions of stacking fault growth and decline, there is a difference in the Peierls barrier and Peierls stress of the partial dislocation. Owing to the positive SFE, the Peierls barrier and Peierls stress are more pronounced along the stacking fault growth directions. As shown in Fig. 8b, d, the basal glide-set 60° partial dislocation with O core or N core, i.e., 0.011 eV/Å for O core and 0.044 eV/Å for N core, has a smaller Peierls barrier compared to the dislocation with Zn core or Ga core (0.029 eV/Å for Zn core and 0.061 eV/Å for Ga core) along the stacking fault growth direction. Utilizing Eq. (2), the Peierls stresses along the direction of stacking fault growth are determined to be 1.2 and 3.8 GPa for O core and Zn core in ZnO, and 6.4 and 9.4 GPa for N core and Ga core in GaN, respectively.

Dislocation in cubic SrTiO3

The 〈001〉 edge dislocation with a 〈001〉 Burgers vector, {110} glide plane, and \(\langle 1\overline{1}0\rangle\) line vector has been investigated in experiment71. Upon inserting the 〈001〉 edge dislocation dipole, as shown in Fig. 9a–c, the dislocation cores are found to be polar. They consist of alternating (SrTiO2)2+ layers (see Fig. 9a) and (SrTiO4)2− layers (see Fig. 9b). A charge balance was achieved by moving one of the two extra oxygen atoms along the [1\(\overline{1}\)0] column from the negatively charged core to the positively charged core, resulting in two reconstructed and nonpolar dislocations.

Fig. 9: Accuracy and performance of NNP on the dislocation in SrTiO3.
figure 9

a Positively and b negatively charged 〈001〉 edge dislocations in c the dislocation dipole model, which has a \(\langle 1\overline{1}0\rangle\) line vector and a {110} slip plane. For the charge-balanced dislocation core, one of the two extra oxygen atoms along the [1\(\overline{1}\)0] column was relocated from the position marked by the solid circle in (b) to the position indicated by the dashed circle in (a). d Dislocation dipole model with periodic quadrupolar arrangements, comprising positively and negatively charged dislocations, which result in long-range electrostatic interaction. The periodic cell of the arrangement is delineated by the green-shaded area. e Energy difference ΔE between the dislocation dipole models before and after charge balance as a function of the distance x defined in (d). The results of DFT and pair empirical potential65 are also shown for comparison.

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The energy difference between the dislocation dipole models before and after charge balance originates from two factors: (1) the core energy, resulting from the reconstruction of the dislocation core structure after charge balance (\(\Delta {E}^{{\rm{core}}}\)), which is model-size independent; and (2) the long-range electrostatic interaction between charged dislocations (ΔEelectro). Upon achieving charge balance, the long-range electrostatic interaction significantly diminishes. As the model size increases, ΔEelectro decreases, and with an infinite model, ΔEelectro → 0. The elastic interaction of the dislocation dipole models before and after charge balance, which is a function of the Burgers vector and the distance between dislocations72, is assumed to be equal (i.e., ΔEelastic = 0) as the Burgers vector and the distance between dislocations remain unchanged.

Figure 9e presents the results of the energy difference between the dislocation dipole models before and after charge balance as a function of the distance x defined in Fig. 9d. The energy difference ΔE, as determined by NNP, is −0.17 eV/Å at x = 29 Å. This aligns closely with the DFT result of −0.18 eV/Å and converges towards the \(\Delta {E}^{{\rm{core}}}\), i.e., −0.37 eV/Å, as the model size increases. The negative value of \(\Delta {E}^{{\rm{core}}}\) suggests that the charge-balanced 〈100〉 edge dislocation is more stable compared to its counterpart without charge balance. The \(\Delta {E}^{{\rm{electro}}}=\Delta {E}^{{\rm{total}}}-\Delta {E}^{{\rm{core}}}\) decreases from 0.20 eV/Å at the distance x = 29 Å to ~0 at the distance x = 91 Å. The energy difference predicted by the pair empirical potential does not converge at a distance x greater than 400 Å (see Fig. 9e), as the ΔEelectro encompasses not only the electrostatic interaction between the charged dislocation cores, but also the electrostatic interaction between the charged dislocation core and each atom with a fixed charge.

The dislocation with a 〈110〉 Burgers vector, {1\(\overline{1}\)0} glide plane and 〈111〉 line vector, and thus, of mixed character, has been reported in experimental studies52. The Burgers vector can be decomposed into its edge (1/3\(\langle 11\overline{2}\rangle\)) and screw (2/3〈111〉) components. As depicted in Fig. 10a–c, the charged 〈110〉 mixed dislocation dipole was initially inserted, followed by a charge balance operation moving one of the extra oxygen atoms from the negatively charged core to the positively charged core.

Fig. 10: Accuracy and performance of NNP on the dislocation in SrTiO3.
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a Positively and b negatively charged 〈110〉 mixed full dislocations in c the dislocation dipole model, which has a 〈111〉 line vector and a {1\(\overline{1}\)0} slip plane. Black circles highlight the oxygen atoms at the dislocation core. For the charge-balanced dislocation core, one of the extra oxygen atoms at the dislocation core was moved from (b) to (a). d Charge-balanced 〈110〉 mixed dislocation in its glide dissociated form, cut from a larger dislocation dipole model. The locations of partial dislocations, determined based on the Sr-O FCC sublattice via the dislocation extraction algorithm (DXA)95, are highlighted by blue ⊥ symbols.

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Owing to the presence of the stable SFE (see Fig. 6b), the 〈110〉 mixed dislocation tends to split into two partials with a 1/2〈110〉 Burgers vector on a {1\(\overline{1}\)0} plane, thereby creating a stacking fault on this plane. Fig. 10d illustrates the optimal configuration of 〈110〉 mixed dislocations. The splitting distance is ~24 Å, which aligns closely with the experimental results of 25 Å52. The stable SFE (850 mJ/m2) predicted by NNP and DFT is comparable to those estimated in the experiment, i.e., 606.087 ± 76.66 mJ/m2. In comparison, due to the larger stable SFE of 1280 mJ/m2 predicted by the pair empirical potential65, the splitting distance is 12 Å.

Plastic deformation tests and experimental comparisons

Compression of c-oriented GaN nanopillar

Figure 11 illustrates the snapshots of the atomic configuration of c-oriented GaN nanopillar under a compressive engineering strain ε = −0.1164, projected along [1\(\overline{1}\)00] and [0\(\overline{1}\)10]. The 1/3\(\langle 1\overline{2}13\rangle\) dislocations are activated, and their slip planes have an angle of ~58° with respect to the basal plane (highlighted by black dashed lines in Fig. 11c, e). It is indicated that the primary slip system is on the pyramidal II plane. This is in good agreement with experimental results73, which found that the first formed dislocations in microcompressed c-oriented GaN micropillars are 1/3\(\langle 11\overline{23}\rangle\){11\(\overline{2}\)2} dislocations. A secondary slip band oriented at an angle of ~72° relative to the basal plane was also observed experimentally, with a Burgers vector of 1/3\(\langle 1\overline{2}13\rangle\)73. It doesn’t match any major slip system in GaN. As shown in Fig. 11e, f, similar secondary slip planes were also observed, highlighted by the red dashed lines. It is found the secondary slip planes are also the pyramidal II planes but not parallel with the view direction [0\(\overline{1}\)10], showing an angle of ~70° from the basal plane. Our simulation results suggest a plausible interpretation for the experimentally observed secondary slip band73. The compression simulation of the c-oriented GaN nanopillar with the same parameters was also performed using the SW empirical potential for comparison. The results are shown in Fig. S1 in the Supplementary Materials. No obvious pyramidal dislocation is observed due to the overestimated SFE by this empirical potential, as shown in Fig. 5d.

Fig. 11: Compression of c-oriented GaN nanopillar.
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a, b Model of a c-oriented GaN nanopillar. c–f Snapshots of atomic configurations under a compressive engineering strain of ε = −0.1164, projected along c, d [1\(\overline{1}\)00] and e, f [0\(\overline{1}\)10]. They were analyzed via (c, e) the identify diamond structure (IDS) method94, and (d, f) the dislocation extraction algorithm (DXA)95 based on the Ga HCP sublattice. The slip planes are highlighted by black and red dashed lines, with angles of approximately 58° and 70° from the basal plane, respectively. Atoms within perfect wurtzite structures are not shown in (a–c, e).

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Nanoindentation of (0001)-oriented ZnO

Figure 12 displays snapshots of dislocation structures of wurtzite ZnO under (0001) nanoindentation, indicating that basal and pyramidal dislocations nucleate and slip. It is in good agreement with experimental results50,74,75,76. In detail, the basal dislocation includes both the basal shuffle-set 1/3\(\langle 11\overline{2}0\rangle\) full dislocation and the basal glide-set 1/3\(\langle 1\overline{1}00\rangle\) partial dislocation. The primary pyramidal dislocation is the 1/3\(\langle 1\overline{2}13\rangle\) dislocation on the pyramidal I plane, and it aligns ~60° to the (0001) surface. A few dislocations on the pyramidal II plane are also observed, as shown in Fig. 12d. The nanoindentation simulation with the same parameters was also performed using the Buckingham-type empirical potential for comparison. The results are shown in Fig. S2 in the Supplementary Materials. There is little pyramidal dislocation observed due to the overestimated SFE by this empirical potential (see Fig. 5c).

Fig. 12: Nanoindentation of (0001)-oriented ZnO.
figure 12

a (0001)-oriented ZnO slab model used in the nanoindentation simulation. b–d Snapshot of dislocation structures at an indentation depth h = 5.0 nm, projected along b [11\(\overline{2}\)0], c [1\(\overline{2}\)10], and d [1\(\overline{1}\)00] directions. The slip planes that are parallel to the projection direction are emphasized with red rectangles.

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Atomic environment similarity and atomic force uncertainty

As discussed in section “A general recipe of easy-to-compute training datasets”, training NNPs with small atomic structures often prompts questions about their accuracy in large-scale simulations. Therefore, in this work, we further validated the effectiveness of the training dataset and NNPs through similarity measurement and uncertainty quantification. The similarity measures the minimum distance Di of the local atomic environment between the atom i in the lager-scale simulation and any atoms in the training dataset (i.e., Eq. (4)). NNPs usually provide more reliable predictions for structures that exhibit higher similarity to the training dataset.

Figure 13a shows the maximum value of Di, denoted as \(\mathop{\max }\limits_{i}\{{D}^{i}\}\), in the dipole model of basal shuffle-set 1/3\(\langle 11\overline{2}0\rangle\) edge dislocation with reference to different sets of structures within the training dataset. The dipole models were relaxed by DFT at initial and saddle configurations as the dislocation slipped along the [11\(\overline{2}\)0] directions. The inclusion of stacking fault structures significantly increases the similarity between the training dataset and the dislocation core structures, compared to using only basic structures. It is evident as the value of \(\mathop{\max }\limits_{i}\{{D}^{i}\}\) decreases from 1.67 to 1.04 for the initial configuration. Furthermore, the similarity quantified by Di for atoms in the snapshot of the ZnO (0001) nanoindentation simulation is displayed in Fig. 13c, d. Corresponding results for GaN NNP and SrTiO3 NNP can be found in Figs. S3 and S4 in the Supplementary Materials. The inclusion of dislocation-related structures, particularly the uniformly deformed structures, effectively enhances the similarity, indicated by a decrease in the value of Di, between the training dataset and the defected structures where dislocations nucleate and slip. The inclusion of cleavage structures in the training dataset has little effect on the similarity of the dislocation core structures and nanoindentation snapshots shown in Fig. 13. Nonetheless, it could be beneficial for other forms of ceramic dislocation plasticity simulations, e.g., dislocation emission from a notch or crack tip. Therefore, our training dataset is highly optimized and effective, maintaining a minimal yet sufficient size, to construct the NNP for dislocation plasticity in ceramics.

Fig. 13: Atomic environment similarity and atomic force uncertainty.
figure 13

a Maximum value of similarity measurement Di, i.e., \(\mathop{\max }\limits_{i}\{{D}^{i}\}\), in the dipole model of basal shuffle-set 1/3\(\langle 11\overline{2}0\rangle\) edge dislocation in reference to different sets of structures within the training dataset. b The dipole models, color-coded by the uncertainty σi, were relaxed by DFT at (Upper) initial and (Lower) saddle configuration as the dislocation slipped along [11\(\overline{2}\)0] directions. c–f Similarity and uncertainty for atoms in the snapshot shown in Fig. 12d, where only the results of the atoms within the defected structures are used. c Average value and d distribution of Di with reference to different sets of structures within the training dataset. The scatter plot of the σi with e the magnitude of atomic force \({\left\Vert {{\boldsymbol{F}}}_{* }^{i}\right\Vert }_{2}\) and f the Di. \({{\boldsymbol{F}}}_{* }^{i}\) represents the atomic force vector predicted by the NNP used to perform the nanoindentation simulation. Points in the graph are color-coded as value frequencies in each bin. Bin sizes are e [5,5] and f [0.0030,5].

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Given a specific training set, multiple well-trained NNPs typically provide similar predictions near existing samples. However, their predictions can diverge significantly in areas far from the existing data, serving as a basis for approximate, yet useful, uncertainty quantification of NNPs53,54,55,56. This uncertainty offers an estimation of the reliability of NNPs for new and unseen configurations in large-scale systems. In this work, the uncertainty σi of atom i was determined through the variances of the predicted atomic forces by the NNPs in an ensemble (see Eq. (7)).

As shown in Fig. 13b, the NNPs constructed using a training dataset that includes both basic and dislocation-related structures effectively describe the core structure with an uncertainty smaller than 40 meV/Å. For the atoms in the defected structures in the snapshot of ZnO (0001) nanoindentation simulation, the value of uncertainty σi ranging from 101 to 102 meV/Å is one to two orders of magnitude smaller than the magnitude of the atomic force \({\left\Vert {{\boldsymbol{F}}}_{* }^{i}\right\Vert }_{2}\), which ranges from 102 to 103 meV/Å (see Fig. 13e). Here, \({{\boldsymbol{F}}}_{* }^{i}\) represents the atomic force vector predicted by the NNP used to perform the nanoindentation simulation. All of these results reveal that the developed NNPs show reliable prediction (i.e., less uncertainty) for new and unseen configurations in large-scale systems.

Furthermore, the scatter plot of σi and Di is shown in Fig. 13f for ZnO NNP and in Fig. S3d in the Supplementary Materials for GaN NNP, suggesting that there is not a strong correlation between similarity and uncertainty, with considerable deviation observed. While NNPs typically provide more reliable predictions (i.e., lower uncertainty) for structures with higher similarity to the training dataset, as indicated by a lower Di value, both similarity and uncertainty metrics are necessary to validate the reliability of NNPs in large-scale simulations.

Discussion

In this work, we demonstrated the construction of NNP for dislocation plasticity in ceramics, using the dislocation-related structures as a training dataset, which eliminates the need to construct complex dislocation core structures and are readily computable through high-throughput calculation. The complex nature of dislocation cores in ceramics, along with the need for expansive models often composed of hundreds of atoms for dislocation modeling, make the creation of dislocation structures and the computation of DFT energy for training datasets a demanding task. Our proposed dataset effectively addresses these challenges.

The effectiveness of our dataset was validated by the accuracy and performance of the developed NNPs across various properties, especially those not included in the training dataset. Examples include the SFE of pyramidal I and II planes, and the charged core structures and slip barriers of dislocations in wurtzite ZnO and GaN. Furthermore, Fig. 6b shows the SFE results calculated using the NNP from ref. 28, developed without dislocation-related structures in the training dataset. While this NNP accurately describes the structural phase transitions in SrTiO3, its failure to predict SFE highlights the importance of including dislocation-related structures in the training dataset, as proposed in this study. By initializing with this NNP and adding dislocation-related structures as a training dataset, a new NNP was constructed, demonstrating excellent performance for the nonstoichiometric stacking fault and dislocation core structures, which were not included in the training dataset. The effectiveness of our dataset was further validated by measuring the similarity of snapshots derived from large-scale simulations to the training dataset and assessing the uncertainty of the developed NNPs for new and unseen configurations in large-scale systems. The dislocation-related structures effectively enhance the similarity between the training dataset and the defected structures where dislocations nucleate and slip. The developed NNPs show reliable prediction (i.e., less uncertainty) for new and unseen configurations in large-scale systems.

Overall, the training dataset proposed in this work accurately represents the structures under dislocation-dominated plastic deformation in ceramics, and provides an effective framework for constructing NNPs that enable the detailed atomistic modeling of ceramic plasticity, opening new avenues for exploring the plastic behavior of ceramics. This straightforward dataset can be easily created using our open-source code ADAIS, regardless of the crystal structure and symmetry of the target ceramics. The dataset easy to compute via high-throughput calculation could also be used in the on-the-fly active learning of NNP for dislocation plasticity in ceramics.

Methods

Details of NNP training

The DeepMD-kit software package77,78 was employed to construct the NNP. 20% training dataset was randomly selected as a testing set to determine the occurrence of overfitting. The ZnO and GaN NNPs were entirely newly trained, whereas the training of SrTiO3 NNP was initialized with the existing SrTiO3 NNP reported in ref. 28. The training parameters are detailed in Table S1 in the Supplementary Materials. The root mean square errors (RMSEs) of energy and atomic force are 3.5 meV/atom and 57.2 meV/Å for the ZnO NNP, 5.5 meV/atom and 84.7 meV/Å for the GaN NNP, and 2.5 meV/atom and 89.6 meV/Å for the SrTiO3 NNP, respectively.

Tables S2, S6, and S10 in the Supplementary Materials list the training dataset used to construct the NNPs of ZnO, GaN, and SrTiO3, all of which were calculated via DFT calculations. Three typical ZnO polymorphs were considered, including cubic rocksalt (B1), cubic zinc-blende (B3), and hexagonal wurtzite (B4), as shown in Fig. S5 in the Supplementary Materials. At ambient conditions, the thermodynamically stable phase is wurtzite79. Zinc-blende ZnO can be stabilized only by growth on cubic substrates, and rocksalt ZnO may be obtained at relatively high pressures79. Two typical GaN polymorphs were considered, including cubic zinc-blende (B3) and hexagonal wurtzite (B4), as shown in Fig. S19 in the Supplementary Materials. GaN usually crystallizes in the wurtzite structure at ambient conditions, but GaN thin films grown epitaxially on various substrates have been reported to be in the zinc-blende structure80,81. Two typical SrTiO3 polymorphs were considered, including cubic (Pm\(\overline{3}\) m) and tetragonal (I4/mcm) phases, as shown in Fig. S28 in the Supplementary Materials. At room temperature, SrTiO3 has a cubic structure. Below 105 K, it undergoes an antiferrodistortive structural transition, transforming to the tetragonal phase82,83.

Density functional theory calculation

Our DFT calculations were performed using the Vienna ab initio simulation package (VASP) code46. We employed the projector augmented wave (PAW) method84 with the Perdew–Burke–Ernzerhof (PBE) version85 of the generalized gradient approximation (GGA) as the exchange-correlation functional for ZnO and GaN. In the case of SrTiO3, we utilized the Perdew–Berke–Ernzerhof revised for solids (PBEsol)86. An energy cutoff of 520 eV (500 eV) was applied for ZnO and GaN (SrTiO3). We chose an energy convergence criterion of 10−6 eV/cell and the Gaussian method87 with a smearing width of 0.01 eV for the electronic self-consistency calculation. A force convergence criterion of 10−3 eV/Å was implemented for ionic relaxation. The Monkhorst–Pack k-mesh88 was used with 6000 k-points per reciprocal atom (KPPRA)89,90 for ZnO and GaN, and with KSPACING of 0.15 for SrTiO3. We determined the elastic constants via the AELAS code90, and calculated the polycrystalline bulk and shear moduli using the Voigt–Reuss–Hill approximation91,92. Table 1 presents the lattice constant, elastic constant, and elastic moduli, all of which align excellently with previously published theoretical values and experimental data, thereby validating the DFT parameters used in this work.

Molecular dynamics simulation

All MD simulations were performed using the large-scale atomic/molecular massively parallel simulator (LAMMPS) code93. We analyzed and visualized atomic configurations of dislocation with the identify diamond structure (IDS)94 and dislocation extraction algorithm (DXA)95 tools of the OVITO program96. The phonon dispersion curve was calculated via a combination of phonoLAMMPS97 and PHONOPY98.

The dislocation core structure and slip barrier were studied via the dislocation dipole model with periodic quadrupolar arrangements57,58,59,60 as shown in Fig. 4. The periodicity vectors C1, C2, and C3 are defined from the unit vectors a1, a2, and a3, i.e., C1 = a1, C2 = na2, and \({{\boldsymbol{C}}}_{3}=\frac{n}{2}{{\boldsymbol{a}}}_{2}+m{{\boldsymbol{a}}}_{3}\), where the dislocation line is parallel with C1. In addition, a homogeneous strain εpl needs to be applied to the simulation cell to accommodate the plastic strain created by the dislocation dipole99,100 as

$${{\boldsymbol{\varepsilon }}}_{{\rm{pl}}}=\frac{1}{2\Omega }({\boldsymbol{b}}\otimes {\boldsymbol{V}}+{\boldsymbol{V}}\otimes {\boldsymbol{b}})$$
(1)

where ⊗ is the tensor product, Ω = (C1 × C2) ⋅ C3 is the supercell volume, b is the Burgers vector, and V is the cut vector of the dislocation dipole (see Fig. 4), whose magnitude is the area of the cut and whose direction is normal to the cut plane.

We constructed the initial atomic structure of the dislocation using the elastic displacements of the Volterra dislocation, which are exact at large distances from the dislocation core but are only approximate near the core. The structures of the dislocation dipole are then relaxed within a fixed cell.

By applying the nudged elastic band (NEB) method101,102,103, we can derive the Peierls potential, which is the dislocation energy Edisl(x) as a function of its core center position x along the slip direction. From this Peierls potential, we can then estimate the Peierls barrier EP and Peierls stress τP104,105 associated with dislocation slip:

$$\begin{array}{l}{E}_{{\rm{P}}}=\mathop{\max }\limits_{x}\{{E}_{{\rm{disl}}}(x)\}-\mathop{\min }\limits_{x}\{{E}_{{\rm{disl}}}(x)\}\\ {\tau }_{{\rm{P}}}=\mathop{\max }\limits_{x}\left\{\frac{1}{b}\frac{\partial {E}_{{\rm{disl}}}(x)}{\partial x}\right\}\end{array}$$
(2)

where b is the magnitude of the Burgers vector b of dislocation. Additionally, we investigated the effect of the cell geometry (i.e., the values of m and n) on the Peierls potential.

Nanopillar compression simulation

A c-oriented GaN nanopillar model with a diameter of 29 nm and a length of 53 nm was prepared, consisting of ~2.3 million atoms (see Fig. 11a, b). The models are periodic along the [0001] direction. The model was relaxed using the NPT ensemble at a temperature of 300 K and a timestep of 0.001 ps for 50 ps to release the stress along the [0001] direction. Subsequently, a uniaxial compressive load was applied by linearly rescaling the system length in the [0001] direction at a constant strain rate of 5 × 107 s−1 in an NVT ensemble at 300 K.

Nanoindentation simulation

An atomistic slab model of wurtzite ZnO, measuring 30.0 nm × 30.0 nm × 30.0 nm and consisting of ~2.1 million atoms, was constructed. The orientation of the model was set as x: [1\(\overline{1}\)00], y: [11\(\overline{2}\)0], and z: [0001] (see Fig. 12a). Before the nanoindentation simulations, the model was relaxed in an NPT ensemble at a temperature of 5 K and a timestep of 0.001 ps for 50 ps to release the in-plane stresses. To perform the displacement-controlled nanoindentation simulation, the z position of the spherical indenter with a radius Rind = 10 nm was controlled to move along the z-axis at a speed of 5.0 m s−1 at a temperature of 5 K.

The repulsive force between the indenter and the slab model was assumed as follows:

$$F({r_{{\rm{ind}}}}) = \left\{ {\begin{array}{*{20}{l}}{ - \Lambda {{({r_{{\rm{ind}}}} - {R_{{\rm{ind}}}})}^2}}&{{r_{{\rm{ind}}}} \le {R_{{\rm{ind}}}}}\\0&{{r_{{\rm{ind}}}} > {R_{{\rm{ind}}}}}\end{array}} \right.$$
(3)

where rind represents the distance of the atoms in the slab model to the centroid of the spherical indenter tip; Λ denotes a force constant related to the effective stiffness of the indenter106,107, which was typically on the order of several eV ⋅ Å−3 and was set to 1.0 eV Å−3 in this work. A nanoindentation simulation with Λ = 10.0 eV Å−3 was also performed, and the results are identical to those of Λ = 1.0 eV Å−3. During the indentation simulation, the center of mass of the atomic slab model was fixed, and the x and y dimensions of the slab model were relaxed such that the normal stress was 0 Pa in these directions.

Similarity measurement

The similarity of the atom i in the simulation snapshot to the training dataset is quantified as the minimum “distance" Di of atom i from any atom j in the training dataset108:

$${D}^{i}=\mathop{\min }\limits_{j}{\Vert {{\boldsymbol{G}}}_{i}^{{\rm{sim}}}-{{\boldsymbol{G}}}_{j}^{{\rm{dataset}}}\Vert }_{2}$$
(4)

where G is a fixed-size vector of atoms to describe their local environment up to the cutoff radius rcut20,109; \({{\boldsymbol{G}}}_{i}^{{\rm{sim}}}\) and \({{\boldsymbol{G}}}_{j}^{{\rm{dataset}}}\) are the G vectors of the atom i in the simulation snapshot and the atom j in the training dataset, respectively. A smaller value of Di indicates a greater similarity. In terms of the distances rij and rik of the atom i from its neighbors j and k and the angle subtended by those distances θijk, the radial and angular components of G are given by:

$$\begin{array}{l}{G}_{i}^{{\rm{rad}}}=\sum\limits_{j\ne i}{e}^{-\eta {({r}_{ij}-{r}_{s})}^{2}}f({r}_{ij})\\ {G}_{i}^{{\rm{ang}}}={2}^{1-\zeta }\sum\limits_{j,k\ne i}{[1+\cos ({\theta }_{ijk}-{\theta }_{s})]}^{\zeta }{e}^{-\eta {[\frac{1}{2}({r}_{ij}+{r}_{ik})-{r}_{s}]}^{2}}f({r}_{ij})f({r}_{ik})\end{array}$$
(5)

where the smooth cutoff function f is given by

$$f({r}_{ij})=\left\{\begin{array}{cc}\frac{1}{2}\left[\cos \left(\frac{\pi {r}_{ij}}{{r}_{{\rm{cut}}}}\right)+1\right]&{r}_{ij}\le {r}_{{\rm{cut}}}\\ 0&{r}_{ij} \,>\, {r}_{{\rm{cut}}}\end{array}\right.$$
(6)

and η, ζ, rs, and θs are parameters, different for the radial and angular components. The vector G consists of multiple \({G}_{i}^{{\rm{rad}}}\) and \({G}_{i}^{{\rm{ang}}}\) elements, each of which is generated by varying the parameters η, ζ, rs, and θs. Table S1 in the Supplementary Materials shows all values of there parameters selected in this study. The cutoff values rcut are the same as those in NNP, i.e., 7.0 Å for ZnO and GaN, and 6.0 Å for SrTiO3.

Uncertainty quantification

Ensemble-based models are among the most widely used and reliable methods for uncertainty quantification53,54,55,56. As proposed in ref. 53, ensemble-based methods surpass single deterministic models in terms of generalization and robustness, including methods like mean-variance estimation, deep evidential regression, and Gaussian mixture models. An ensemble involves training multiple neural networks using various sources of randomness during the training process, such as different initializations, stochastic optimization, or hyperparameter choices. Uncertainty is estimated by the variance of predictions from these NNPs within the ensemble, with greater prediction variation indicating higher uncertainty.

In the present work, we utilized ensemble-based models. Ten distinct NNPs were trained, which were independently initialized using different random seeds, influencing both the parameter initialization and the random selection of the testing dataset (which is 20% of the entire dataset). Since forces are derivatives of energy, they tend to display higher variability when overfitting occurs or when predictions are made outside the training domain. Therefore, forces are more suitable descriptors of epistemic uncertainty compared to energy-derived measures. For atom i in a system, the mth NNP in the ensemble, which consists of a total of M NNPs (M = 10 in this work), predicts the atomic forces \({{\boldsymbol{F}}}_{m}^{i}\). The uncertainty estimates σi for atom i are provided by the variances of the predicted atomic forces, which can be computed as

$${\sigma }^{i}=\sqrt{\frac{1}{M}\mathop{\sum }\limits_{m=1}^{M}{\left\Vert {{\boldsymbol{F}}}_{m}^{i}-{\hat{{\boldsymbol{F}}}}^{i}\right\Vert }_{2}^{2}}$$
(7)

where \({\hat{{\boldsymbol{F}}}}^{i}\) is the arithmetic mean of the predictions from each NNP in the ensemble, i.e., \({\hat{{\boldsymbol{F}}}}^{i}=\frac{1}{M}\mathop{\sum }\nolimits_{m=1}^{M}{{\boldsymbol{F}}}_{m}^{i}\).