Machine learning enhanced empirical potentials for metals and alloys☆,☆☆
Introduction
Atomistic modeling plays a vital role in materials science. Ab initio calculation or force-field based molecular dynamics simulation (MD) are effective ways to study, understand or predict chemical and physical properties of materials. Ab initio approaches are generally much more precise but they are rarely used on large-scale metallic systems due to their extremely-high computational expenses. Physical model based empirical potentials (force-field), such as the embedded-atom method (EAM) [1], [2], [3], [4], [5], modified embedded-atom method (MEAM) [6], [7], [8], [9], bond-order potential (BoP) [10], [11], [12], [13], or angular-dependent potential (ADP) [14], [15], [16], [17], [18], still play the major role in long-time simulations and these empirical methods can achieve reasonable accuracy with much lower computation costs. Empirical potentials generally have very few learnable parameters and both microscopic observables (energy, forces, virial, etc.) and macroscopic observables (melting point, surface energy, etc.) can be used to tune these parameters. Finding optimal parameters of empirical potentials is always a challenging task. Global optimization (GO) approaches (Basin-Hopping [19], [20], genetic algorithm [21], [22], etc) are traditionally used to find the best possible parameters. However, the gradients of the losses with respect to model parameters are difficult or even impossible to calculate. Hence, global optimizations are generally not that effective.
In the last decade, machine learning (ML) has become one of the hottest topics in many research areas. In the field of materials science, researchers have made great efforts on developing ML models to describe atomic interactions. Such ML models are considered as machine learning interaction potentials (MLIPs). Until now, hundreds of MLIPs have been proposed. Among them, the symmetry-function based atomistic neural network (ANN) model, published by Parinello and Behler in 2007 [23], [24], [25], [26], [27], is still the most popular choice in modeling metallic interactions [28], [29], [30]. The smooth overlap atomic positions descriptor based gaussian approximation potential (SOAP-GAP) [31], [32], [33], [34], developed by Bartòk et al., can give extremely accurate prediction results, although it's a bit computationally expensive. Recently, Thompson and co-workers proposed another quantum-accurate MLIP named the spectral neighbor analysis potential (SNAP) [35] and it has been proven working on a broad range of metals and alloys [36], [37], [38].
In many cases, MILPs can easily outperform state-of-art empirical potentials. Compared with empirical potentials, MLIPs generally have orders of magnitudes more model parameters. The redundant parameter space greatly reduces the difficulty of fitting complicated potential energy surfaces. But, to effectively train a MLIP and avoid overfitting, a large high-quality (versatile) training dataset is probably needed. However, MILPs can really take advantages of “big data” for two reasons. First, MLIPs typically only have basic or simple arithmetic operations. Thus, MILPs can be implemented within modern deep learning frameworks (TensorFlow [39], PyTorch [40], etc) so that the gradients of the total loss with respect to fitting parameters can be obtained with the backpropagation algorithm automatically and efficiently. Second, MLIPs are mostly vectorizable. Hence, graphic processing units (GPUs) can be utilized to significantly accelerate training and using of MILPs.
However, MLIPs also have challenges. The large parameter space and lack of physical background makes the “big data” a requisite. The cost of dataset is non-negligible. Besides, even “big data” can only cover a small portion of real physical environments (temperature, external pressure, etc). Outside the training zone, the performances of MILPs may not that stable. For long-time molecular dynamics (MD) simulations of large-scale (105 or more) atoms, computation efficiency also becomes a major concern. Recent benchmark tests [41] suggest that MILPs are still too expensive. At present, most MLIPs are used to examine small to medium (103 to 104) atoms [37], [28], [29], [30], [34].
In this work, instead of designing new atomic descriptors, we chose a new route to develop MLIP: combining machine learning approach with empirical potentials. We successfully implemented EAM and its variant ADP within TensorFlow so that machine learning approaches can be used directly to tune EAM and ADP potentials. Our results suggest ML-EAM or ML-ADP can be as precise as the SNAP machine learning method.
This paper is organized as follows. Section 2 describes the theoretical background of this work, including the formalism of the embedded atom method and algorithms of the machine learned EAM. Section 3 describes the implementation details. Section 4 summarizes the results of ML-EAM and ML-ADP. Further discussions are given in section 4.3.
All the new algorithms have been implemented in our program TensorAlloy [42].
Section snippets
Theory
In the original EAM formalism [1], the total energy, , is the sum of atomic energies: where is the cutoff radius, a and b represents species of atoms i and j, is energy of the pairwise interaction between i and j, is the embedding energy and is the local electron density of atom i. can be calculated with the following equation: where is the electron density function of specie b. F, ρ and ϕ can be
Implementation
Fig. 2 shows the architecture of the TensorAlloy program. The program has two major steps: the first step is calculating the losses of energy, force and stress terms while the second step is measuring the contributions of the physical constraints.
For the first step, the implementation is almost the same with our previous work [42] except the formula of the potential energy. Briefly, this step starts from checking through the training dataset and determine some key constants, including the
Results
The public Ni-Mo dataset [37] is used to evaluate our program. This dataset is provided by Shyue Ping Ong and co-workers along with their SNAP models. It contains 3973 different Ni-Mo solids. All calculations were done by VASP [55] with the PBE [56] functional and the projector augmented-wave approach [57].
The optimized parameters are listed in the appendix.
Conclusions
To conclude, we have successfully combined the training of empirical potentials with the machine learning approach into our TensorAlloy framework. The machine learning approaches (big data, optimization, etc), together with physical constraints (Rose EOS, elastic constants) can significantly improve the performances of EAM/ADP. For the fcc Ni, bcc Mo and Mo-Ni alloys, ML-EAM and ML-ADP can be as accurate as the SNAP method, while later method is about three orders of magnitude slower. Our work
CRediT authorship contribution statement
D.Y.L and H.F.S contributed the central conceptions of this study, designed the research project, conducted the analyses and revised the manuscript. X.C. designed the research project, wrote the related codes, trained the machine learning interatomic potentials and conducted the analyses. L.F.W performed molecular dynamics simulations and validated our models. X.Y.G., Y.F.Z. and W.D.C: provided valuable comments and suggestions to the work. X.C. and L.F.W. wrote the initial draft of the paper.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by the National Key Research and Development Program of China under Grant No. 2016YFB0201204, the Science Challenge Project under Grant No. TZ2018002, the National Natural Science Foundation of China under Grant No. U1630250, No. 12002064 and No. 12004046.
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The review of this paper was arranged by Prof. D.P. Landau.
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This paper and its associated computer program are available via the Computer Physics Communications homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).