Towards large-scale and spatiotemporally resolved diagnosis of electronic density of states by deep learning

Modern laboratory techniques such as ultrafast laser excitation and shock compression can bring matter into highly nonequilibrium states with complex structural transformation, metallization, and dissociation dynamics. To understand and model the dramatic change of both electronic structures and ion dynamics during such dynamic processes, the traditional method faces difficulties. Here, we demonstrate the ability of a deep neural network (DNN) to capture the atomic local-environment dependence of the electronic density of states (DOS) for both multicomponent systems under exoplanet thermodynamic conditions and nonequilibrium systems during superheated melting processes. Large-scale and time-resolved diagnosis of DOS can be efficiently achieved within the accuracy of the ab initio method. Moreover, the atomic contribution to DOS given by the DNN model accurately reveals the information of the local neighborhood for the selected atom, which thus can serve as robust order parameters to identify different phases and intermediate local structures, strongly highlighting the efficacy of this DNN model in studying dynamic processes.

Figure 1

Workflow of DeepDOS scheme. (a) Generating the local-environmental matrices ${\tilde{\mathcal{R}}}^i$ from atomic configurations $\left\{{\mathbf{r}}_i\right\}$ to preserve the translation invariance. (b) Mapping smoothed relative distances $s\left(r_{ji}\right)$ into $M$-dimension feature space to obtain local embedding matrices ${\mathcal{G}}^i$ via an embedding net, where $s(...)$ is the smooth function [17]. (c) Generating descriptor ${\mathcal{D}}^i$ by multiplication of local embedding matrices and local-environmental matrices to preserve the permutation and rotation invariance. (d) Mapping the descriptor of atom $i$ into the atomic DOS $g^i\left(\epsilon \right)$ via a fitting net.

Figure 2

(a)–(c) Comparison of DOS/ADOS between the KS-DFT calculation and DNN prediction at $p=298\mathrm{GPa}, T=5000\mathrm{K}$. (d), (e) Parity plot of ADOS/width of the band gap between DFT and DNN prediction in the validation set, where five thermodynamic conditions are included. In (d), the marker color indicates the frequency. In (e), the blue and gray dots represent the predictions of the band gap by the well-trained DNN model (training steps of $2\times {10}^6$) and undertrained DNN model ($5\times {10}^4$), respectively.

Figure 3

(a) Radar plot of the test statistics chosen to measure the band gap prediction accuracy for the validation set and test set. (b) DNN-predicted band gaps vs KS-DFT band gaps on the test set. The black dashed line is for visual reference, showing what the exact band gap prediction corresponds to. (c) Configuration-averaged band gaps at different pressures along the isotherm of 5000 K, KS-DFT calculations (gray), and DNN predictions on the validation set (blue) and on the test set (red) are presented.

Figure 4

(a)–(c) Temporal evolution of temperature, pressure, and density during isobaric heating of cubic diamond silicon at 5 GPa (containing 64 atoms), with the orange line indicating the onset of melting. (d) Electronic density of states prediction for the unseen snapshots during the isobaric melting process, where the solid (dashed) line indicates the result from the KS-DFT calculation (DNN prediction). The inset shows the snapshots of the atomic configurations colored by the ext-CNA method [26].

Figure 5

(a)–(c) Temporal evolution of temperature, pressure, and density during isobaric heating of cubic diamond silicon at 5 GPa (containing 4096 atoms). (d) DOS prediction for the unseen snapshots of the solid-liquid coexistence phase during the process of nucleation and growth of the liquid phase. The black dotted lines mark the three local minima to give the characteristic parameter $Q$.

Figure 6

(a), (b) PCA scatter plot of 81 920 ADOS from 20 snapshots of a 4096-atom system during the melting process, where the marker color indicates the frequency and ADOS-derived order parameter $Q$, respectively. The cyan line in (a) indicates the trajectories of averaged ADOS in PCA space during the melting process. In (b), a trajectory of Si-3182 atoms in ADOS-PCA space is highlighted with a black line.

Figure 7

(a) Snapshots of atomic configurations, colored by $Q$. (b), (c) Temporal evolution of ADOS in PCA subspace colored by the ADOS-derived order parameter $Q$ and ext-CNA method, respectively.