Through the eyes of a descriptor: Constructing complete, invertible descriptions of atomic environments

In this work we apply methods for describing three-dimensional images to the problem of encoding atomic environments in a way that is invariant to rotations, translations, and permutations of the atoms and, crucially, can be decoded back into the original environment modulo global orientation without the need for training a model. From the point of view of decoding, the descriptor is optimally complete and can be extended to arbitrary order, allowing for a systematic convergence of the fidelity of the description. In experiments on molecules ranging from 3 to 29 atoms in size, we demonstrate that positions can be decoded with a 97% success rate and positions plus species with a 70% rate of success, rising to 95% if a second fingerprint is used. In all cases, consistent recovery is observed for molecules with 17 or fewer atoms. Additionally, we evaluate the descriptor's performance in predicting the energies and forces of bulk Ni, Cu, Li, Mo, Si, and Ge by means of a neural network model trained on DFT data. When comparing to six machine learning interaction potential methods that use various descriptors and regression schemes, our descriptor is found to be competitive, in several cases outperforming well established methods. The combined ability to both decode and make property predictions from a representation that does not need to be learned lays the foundations for a novel way of building generative models that are tasked with solving the inverse problem of predicting atomic arrangements that are statistically likely to have certain desired properties.

Figure 1

Difference in fingerprint vector components between atomic arrangements (labeled $B$ and $B^{\prime }$) that cannot be distinguished when using $\nu =2$ and $\nu =3$ invariants. All of the nonzero values correspond to $\nu =4$ invariants.

Figure 2

Reconstructions from Zernike moments with increasing maximum expansion order $N$. The contour data is taken as a slice through a 3D grid corresponding to the plane of the molecule, a reduced opacity version of which is shown over $N=15$. Atomic species have been mapped to the following weights of delta functions: $w_{\text{H}}=1.125,w_{\text{C}}=2.375,w_{\text{N}}=3.625,w_{\text{O}}=4.875$. The fractional color scale has been chosen to correspond to the standard CPK colors for each element.

Figure 3

Peak finding from an approximation of the original environment $\tilde{f}$ based on Zernike moments. Atoms are located one by one whereupon they are replaced by an orange marker and the signal is subtracted for the next iteration. The final panel shows the original locations of the atoms as green markers.

Figure 4

Iterative scheme that finds atomic positions, and optionally species, by alternating between an optimization of the moments and an optimization of the atomic positions, both with respect to the known fingerprints.

Figure 5

Results showing the RMSD of the original fingerprints versus those calculated from the reconstructed molecules. The leftmost panel shows results aggregated by number of atoms. The right panels shows results for each molecule individually where the marker is placed at the lowest RMSD. Error bars indicate the min/max for that data point over several reconstruction attempts. A value of $\approx {10}^{-6}$ (dashed line) represents a faithful reconstruction, the legend shows the proportion of markers falling below this threshold.

Figure 6

A plot showing recovery of atomic species using a second fingerprint vector including species information. Results from decoding atomic positions only (“no species”) are replotted from Fig. 5 for ease of comparison.

Figure 7

Breakdown by point group showing the proportion of molecules correctly inverted. Numbers above bars indicate the absolute number correctly inverted out of the total. (a) No species, (b) With species (single fingerprint), and (c) With species (two fingerprints).

Figure 8

The Behler-Parinello like ANN scheme used to fit total energies and local forces. A rotation invariant MILAD vector ${\mathit{\Phi }}_i$ is created for each atomic environment by including neighbors up to the cutoff radius. Each ${\mathit{\Phi }}_i$ is passed to a feed-forward neural network which outputs the corresponding contribution to the total energy ${\epsilon }_i$. Finally, the total energy $E$ is given by the sum of local contributions.