A density functional theory parameterised neural network model of zirconia
Chen Wang, Akshay Tharval and John R. Kitchin, “A density functional theory parameterised neural network model of zirconia”, Molecular Simulation 44(8), 623–630 (2018). https://doi.org/10.1080/08927022.2017.1420185
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Question
Zirconia (ZrO2) matters for ceramics, fuel cells, gas sensors, nuclear fuel cladding and gate dielectrics, and it exists in several polymorphs (monoclinic, tetragonal and cubic at ambient pressure, plus high-pressure orthorhombic forms). Density functional theory (DFT) describes its potential energy surface well but is too expensive for large cells and long times, while physics-based force fields are limited by their fixed functional forms. The paper asks whether a Behler–Parrinello neural network (BPNN) trained on DFT data can reproduce DFT-quality energies and forces for bulk zirconia with oxygen vacancies, how far it transfers to structures outside the training data, and how one can tell when it is not trustworthy (Wang et al. 2018).
Approach
The reference data were 2178 VASP calculations (PBE functional, PAW potentials with Zr semicore p states, 540 eV cutoff, 4 × 4 × 4 k-point grid) on cells of 6 to 96 atoms. Most structures came from systematic distortions of the cubic, tetragonal, monoclinic and orthorhombic (Pnma) polymorphs, from intermediate geometries of relaxations, and from short molecular dynamics runs at several temperatures; supercells with oxygen vacancies and nudged elastic band (NEB) images were added. The BPNN was trained with the Atomistic Machine-learning Package (AMP), which plugs into ASE. It uses symmetry-function fingerprints within a 6.5 Å cutoff, and the best network had two hidden layers of 13 tanh nodes. A random 10% of the data was held out for validation. The trained model was then tested on lattice parameters, equations of state, vacancy formation energies, vacancy migration barriers, molecular dynamics and timing, including the high-pressure orthorhombic Pca21 phase and the cubic vacancy pathway, neither of which was in the training set (Wang et al. 2018).
Key findings
Training and validation RMSEs were 7.90 and 2.80 meV/atom, which the authors take as evidence against overfitting. More than 95% of validation structures had errors under about 17.6 meV/atom, and the largest errors came from the under-sampled tetragonal vacancy diffusion path. For the trained polymorphs, the BPNN lattice parameters, volumes and bulk moduli closely track PBE (for example, cubic a = 5.118 Å from the BPNN against 5.121 Å from PBE). The equations of state had an average absolute residual of about 6.3 meV/atom, and both methods give the stability order m > t > c > o. For the untrained Pca21 phase, volume and bulk modulus errors were around 1.6% and 0.4%, but the equation-of-state residual rose to about 27.8 meV/atom and reached about ±0.1 eV/atom under compression, which reflects the limits of extrapolation.
Oxygen vacancy formation energies (relative to half of a triplet O2) in cubic zirconia fell from 5.89 eV to 5.64 eV as the supercell grew, which suggests repulsive defect interactions, with dilute behavior near 0.01 vacancies/atom. The BPNN matched PBE within 0.012, 0.037 and 0.072 eV for the cubic, tetragonal and Pnma phases. It underestimated the two monoclinic values by 0.31 and 0.23 eV, likely because large monoclinic supercells were poorly represented in training, and it was off by 0.162 eV for the untrained Pca21 phase. For the tetragonal vacancy hop along [100], PBE gave a 2.13 eV barrier and the BPNN reproduced it within 0.033 eV. On the untrained cubic [100] path, the BPNN overestimated the barrier by 0.083 eV. Along cubic [110], however, two separately trained BPNNs disagreed by up to about 2.29 eV. The authors propose this kind of disagreement between independently trained networks as a practical flag for configurations outside the training data.
In a 600 K, 1 ps MD run on a 12-atom cubic cell, energies of snapshots checked against DFT deviated by at most 8 meV/atom (about 3 meV/atom on average). Running 10 ps MD on defective tetragonal zirconia from 1000 to 1800 K, the authors fit mean squared displacements to an Arrhenius law and obtained an effective vacancy diffusion activation energy of 0.272 eV, close to a 0.270 eV literature value. BPNN cost scaled roughly linearly with atom count up to 105 atoms, far better than the cubic scaling of DFT. The authors also note that ReaxFF had earlier been found to be up to ten times faster than this BPNN implementation for Au.
Significance
This work shows that a modestly sized BPNN trained with AMP can stand in for DFT on bulk zirconia properties, including vacancy energetics and diffusion, at a fraction of the cost. Just as important, it makes explicit that such potentials do not extrapolate reliably to configurations unlike the training data, such as surfaces, clusters or unsampled diffusion directions. It also offers a cheap diagnostic, comparing two independently trained networks, with retraining on added structures as the remedy (Wang et al. 2018).