Physically Motivated Recursively Embedded Atom Neural Networks: Incorporating Local Completeness and Nonlocality

Recent advances in machine-learned interatomic potentials largely benefit from the atomistic representation and locally invariant many-body descriptors. It was, however, recently argued that including three-body (or even four-body) features is incomplete to distinguish specific local structures. Utilizing an embedded density descriptor made by linear combinations of neighboring atomic orbitals and realizing that each orbital coefficient physically depends on its own local environment, we propose a recursively embedded atom neural network model. We formally prove that this model can efficiently incorporate complete many-body correlations without explicitly computing high-order terms. This model not only successfully addresses challenges regarding local completeness and nonlocality in representative systems, but also provides an easy and general way to update local many-body descriptors to have a message-passing form without changing their basic structures.

Figure 1

Two representative pairs of ${\mathrm{CH}}_4$ molecules that have the same set of distances and angles between central C atom (silver) and neighboring H atoms (white for identical ones and light red for the different one), for which the EANN (REANN) atomic energies (in electron volts) for C are identical (distinct).

Figure 2

(a) Schematic diagram of the REANN model showing how the density descriptor is recursively embedded. (b) An example of ${\mathrm{CH}}_4$ showing how the C-centered complete five-body correlation is achieved by twice iteration, where the path going through all atoms corresponds to the product of $F$ functions and arrows point from the central atom to neighbor atoms. (c) An illustration that how the number of three-body terms ($F$) increases in each iteration ($2^T$). Different colors correspond to different iteration times, namely $T=0$ (red), $T=1$ (blue), and $T=2$ (green).

Figure 3

(a) Comparison of the RMSEs for energies of random ${\mathrm{CH}}_4$ configurations of linear fits in Ref. [28] (with $3B+4B$ correlations) and [29] (with $5B$ correlation), and that with the recursively embedded density descriptor ($T=0$, 1, 2). (b) Similar to (a) but all results are now based on nonlinear NN fits.