Latent space design of interatomic potentials

TL;DR

It is described how latent space patterns and associated quantum embeddings can be constructed using first-principles methods based on theorems of density functional theory (DFT) and known, analytic constraints to enable a parsimonious, physics-based representation of energies and densities.

Abstract

The advent of neural-network-based deep learning techniques has led to the emergence of increasingly sophisticated numerical interatomic potentials, including graph neural networks and large language-motivated foundation models. Parameterized to reproduce large, precomputed quantum mechanical training datasets for molecules and materials, models can be fine-tuned for greater accuracy on specific problems. Despite notable successes, machine learning (ML) models of potentials still face intrinsic challenges due to the combinatoric complexity of the underlying quantum chemical interactions, the existence of as-yet-undiscovered but potentially relevant bonding motifs absent from training datasets, and the need for post-prediction interpretability analysis. Drawing inspiration from autoencoder methods, we propose a constructive approach to interatomic potential design. In standard autoencoder architectures, a ML model self-organizes numerical training data in an unsupervised manner to discover underlying patterns and construct a compressed representation or latent space embedding model of the data, which is then used for prediction and inference. In the present work, we describe how latent space patterns and associated quantum embeddings can be constructed using first-principles methods based on theorems of density functional theory (DFT) and known, analytic constraints. This enables a parsimonious, physics-based representation of energies and densities, formally coupling the electronic and atomic length scales through the electron density, and linking ground, excited, and charge-transfer states of the interacting atoms. We describe the complete set of latent space components providing the foundation for a recently-proposed ensemble charge-transfer potential, and discuss opportunities for synergy in the design and explainability of contemporary machine-learned interatomic potentials.

Figures (3)

• Figure 1: The "curse of dimensionality." Atomic configuration vectors $S_i$ normalized to a common magnitude (sphere radius), are only sparsely distributed on the surface of the sphere in $\mathbb{R}^{3N_A}$, where $N_A$ = number of atoms. The sparse coverage of configuration space makes it difficult to discern and cluster new or rare bonding patterns.
• Figure 2: (a) Schematic representation of Behler-Parrinellobehler2007 and ANI-type MLIPs.smith2017 Geometric information about the local environment of an atom is encoded in a descriptor comprised of local symmetry functions. A clear separation exists between geometry (the atomic descriptor, red circles) and the physico-chemical interactions (the neural network, in green). (b) Generic graph neural network (GNN) architecture. Atoms (fuzzy red disks) can encode both geometry and chemical information (atomic number, electronegativity, etc.) within a descriptor, termed an embedding. The atoms corespond to nodes of a neural network, and interactions between the atoms are encoded as "messages" (mathematical operations on the atom embeddings), indicated by edges in the GNN. In this architecture, geometry and chemistry are effectively convolved. (c) Constructive latent space representation (present work). In the ensemble charge-transfer embedded atom (ECT-EAM) interatomic potential based on this representation,muralidharan2007EDAtlas2021 each atom $i$ contributes an embedding energy consisting of the weighted sum of state embedding energies $F_{i,M_i}$, where $M_i$ is an index ranging over all contributing isolated atom states: ground state (purple), positive (green) and negative (red) ions, and excited states (aqua). The positive ion is indicated with slightly contracted disk, and the negative ion with a more diffuse representation. The excited state is indicated with a distorted density representation, as a visual suggestion of charge polarization. The embedding energy function for each atomic state is computed as described in the text. The net embedding energy contribution from atom $i$ is the weighted sum of the state-specific embedding functions. The statistical weights $\omega_{i,M_i}$ appearing in the ensemble are required to sum to 1, and provide a measure of the importance of the $M_i$th state to the overall embedding energy contribution from atom $i$. The sum over all atoms in the system gives the total embedding energy $E_{\rm emb}.$ The additional electronstatic contribution to the total cohesive energy (interatomic potential) is computed as described in the text; see Eq. (\ref{['eq:ensFF']}).
• Figure 3: Schematic illustration of an encoder/decoder architecture implementing a physics-based, constructive latent space representation of a molecular interatomic potential. The box at left corresponds to a database of energies. densities, and forces computed for a set of exemplar molecular configurations. The constructive latent space representation is specified a priori, in terms of density functional constructs---ensembles of atomic radial basis functions (RBFs) and embedded atom functions $F_i$---satisfying known physical constraints. The fuzzified box at right corresponds to the same input configurational dataset, but with molecular energies, densities, and forces computed from the lower-dimensional latent space representation. Since the latent space structures compress input molecular information onto a lower-dimensional representation (manifold), the decoded version of the data will contain errors (indicated by fuzzy edges). A high-quality latent space representation will nevertheless preserve essential molecular characteristics such as bond orders, effective charges, and dissociation behavior (indicated here by the retention of the original cube geometry.) By contrast, in a conventional autoencoder architecture,hinton2006 the latent space representation is learned through iterative optimization of neural network parameters to numerically reproduce digitized data, and the identification and interpretation of any emergent structure contained within the learned latent space must be deduced via downstream analysis techniques. After [bank2023autoencoders].