Size and temperature transferability of direct and local deep neural networks for atomic forces

A direct and local deep learning (DL) model for atomic forces is presented. We demonstrate the model performance in bulk aluminum, sodium, and silicon and show that its errors are comparable to those found in state-of-the-art machine learning and DL models. We then analyze the model's performance as a function of the number of neighbors included and show that one can ascertain physical attributes of the system from the analysis of the deep learning model's behavior. Finally, we test the size scaling performance of the model and the transferability between different temperatures and show that our model performs well in both scaling to larger systems and high- to low-temperature predictability.

Figure 1

Schematic illustration of the NN model. The input layer consists of the distance vectors of the atom from its nearest neighbors. The output layer has three nodes with the values of the forces in each direction.

Figure 2

Comparison of the estimated forces to DFT forces at (a)–(c) 300 K and (d)–(f) 2000 K. Results are shown for (a) and (d) Na (27 atoms), (b) and (e) Al (27 atoms), and (c) and (f) Si (16 atoms). The training set results are shown with black dots, while the test set results are shown with red diamonds. The solid line is a result of linear fit between the model and DFT results.

Figure 3

Average radial distribution function (RDF) of the different structures: (a) Al, (b) Na, and (c) Si. The RDF of the ground-state crystal is shown with a solid black line, the RDF at $T=300\mathrm{K}$ is shown with a dash-dotted red line, and the $T=2000$ RDF is shown with a solid blue line.

Figure 4

Mean absolute error (MAE) as a function of the number of input neighbor atoms that are used for the model [data produced (a) at 300 K and (b) at 2000 K]. Al with red circles, Na is shown with blue triangles, and Si is shown with black squares.

Figure 5

Comparison of the estimated forces to DFT forces at (a)–(c) 300 K and (d)–(f) 2000 K. Results are shown for (a) and (d) Na (27 atoms), (b) and (e) Al (27 atoms), and (c) and (f) Si (16 atoms). The training set results are shown with black dots, while the test set results are shown with red diamonds. The solid line is a result of linear fit between the model and DFT results.

Figure 6

Comparison of estimated forces and DFT forces when the sizes of the training system and validation system are not the same. The following sets are shown: (a) Al, train with 27 atoms and validation with 125 atoms, (b) Na, train with 26 atoms and test with 125 atoms, and (c) Si, train with 16 atoms and test with 128 atoms. The training set results are shown with black dots, while the test set results are shown with red diamonds. The solid line is a result of linear fit between the model and DFT results.

Figure 7

MAE dependence on the number of nodes $L$ of the first hidden layer, (a) showing $T=300\mathrm{K}$ and (b) showing $T=2000\mathrm{K}$. Al is shown with red circles, Na is shown with blue triangles, and Si is shown with black squares.

Figure 8

MAE dependence on the number of hidden layers. The $x$ axis is the number of hidden layers, while the $y$ axis is the model's MAE: (a) $T=300\mathrm{K}$ and (b) 2000 K. Al is shown with red circles, Na is shown with blue triangles, and Si is shown with black squares.