Deep-learning approach to the structure of amorphous silicon

We present a deep-learning approach for modeling the atomic structure of amorphous silicon ($a$-Si). While accurate models of disordered systems require an ab initio description of the energy landscape which severely limits the attainable system size, large-scale models rely on empirical potentials, at the price of reduced reliability and a computational load that is still restricting for many purposes. In this paper, we explore an approach based on deep learning, particularly generative modeling that could reconcile both requirements of accuracy and efficiency by learning structural features from data. When trained on a set of observations, such models can generate new structures very efficiently with the desired level of accuracy, as determined by the data set. We first validate our approach by training a convolutional neural network to approximate the potential-energy surface of $a$-Si, as given by the Stillinger-Weber potential, which results in a root-mean-square error of 5.05 meV per atom—about $0.16\%$ of the atomic energy. We then train a deep generative model, the Wasserstein autoencoder, for the generation of $a$-Si configurations. Our approach leads to models which exhibit some of the essential features of $a$-Si while possessing too much structural disorder, thus suggesting that the method is viable; we indicate avenues for improving it towards the generation of state-of-the-art structures.

Figure 1

Architecture of the MLP (top) and the CNN (bottom) used for learning the PES. Each block represents a neuron layer; the color indicates its type, and the dimensions of the output tensor are indicated. The first dimension corresponds to that of the minibatches: the corresponding elements are treated in parallel by the NN as different observations. The last dimension is the one where the operation is applied; the second dimension in the convolutional layers therefore corresponds to channels. The last layer performs a sum over the atomic energies and yields the total potential energy.

Figure 2

Mean-square error (in ${\text{eV}}^2$) over 128 configurations between predicted and target energies during training for the two architectures considered, as indicated. The full lines are obtained by smoothing the data via a moving average.

Figure 3

Architecture of the WAE used to model the structure of $a$-Si. The color code and symbols are the same as in Fig. 1. The dimension of the output tensor of each layer is indicated. Inputs $\mathbf{x}$ and $\mathbf{y}$ stand for a minibatch of 32 data-set configurations and the corresponding total potential energies. Layers $\mathcal{N}(0,1)$ and $\mathcal{N}(\mu ,\mathrm{\Sigma })$ perform a draw from the normal distribution, where $\mu$ and $\mathrm{\Sigma }$ are the encoder's output vectors.

Figure 4

Separate contributions to the training error of the WAE with respect to gradient iterations (unitless). The discriminator cost (blue) is computed from ${\mathcal{L}}_{\text{D}}$ in Eq. (6). The reconstruction cost corresponds to the first term of ${\mathcal{L}}_{\text{AE}}$ in Eq. (6) while the autoencoder cost corresponds to the second term, i.e., the adversarial part. The reconstruction cost has been evaluated on the validation set. The curves are obtained by smoothing the raw data via a moving average.

Figure 5

Atomic configuration (of energy $-302.38\mathrm{eV}$) generated using the WAE algorithm. The color code indicates the distribution of distances in the range 1.7–$2.9\AA$. The image was produced using the ovito software [42].

Figure 6

Average radial distribution functions of generated samples (blue) and data-set samples (red), normalized by the number density ($5.00\times {10}^{-2}{\AA }^3$). The averages have been taken over 1600 samples (WAE) and $20000$ samples (data set), and the curves have been smoothed via a moving average.

Figure 7

Average nearest-neighbor bond-angle distribution over the 1600 generated configurations. The distribution is approximately Gaussian, while presenting a small peak at its low tail.