Symbolic regression for defect interactions in 2D materials

TL;DR

This work tackles the challenge of interpretability in modeling defect interactions in 2D materials by applying SEGVAE, a deep symbolic regression method, to learn closed-form, physically constrained pairwise interaction kernels from DFT data. It demonstrates that symbolic expressions can achieve comparable or superior predictive accuracy to state-of-the-art graph neural networks on sparse representations, while yielding human-readable formulas that reveal underlying physics such as oscillatory interactions with decaying envelopes reminiscent of RKKY and Friedel oscillations. The approach constructs a general energy framework by combining pairwise kernels into $E_{ ext{formation}} = frac{1}{2}\sum_{i,j} V_{i,j}(r) + \sum_i E_i$ and normalizes to $E_{ ext{per site}} = E_{ ext{formation}}/N$, and defines the HOMO-LUMO gap via $E_{ ext{HOMO-LUMO gap}} = \min_{i,j} E_{i,j}(r)$, enabling a Pareto optimization between accuracy and complexity. By enforcing physical priors, pre-training, and a Bank of Best Formulas, the method yields interpretable, deployable expressions with potential for rapid inverse-defect design, while highlighting limitations in extending to new defect types or materials without new formulas.

Abstract

Machine learning models have become firmly established across all scientific fields. Extracting features from data and making inferences based on them with neural network models often yields high accuracy; however, this approach has several drawbacks. Symbolic regression is a powerful technique for discovering analytical equations that describe data, providing interpretable and generalizable models capable of predicting unseen data. Symbolic regression methods have gained new momentum with the advancement of neural network technologies and offer several advantages, the main one being the interpretability of results. In this work, we examined the application of the deep symbolic regression algorithm SEGVAE to determine the properties of two-dimensional materials with defects. Comparing the results with state-of-the-art graph neural network-based methods shows comparable or, in some cases, even identical outcomes. We also discuss the applicability of this class of methods in natural sciences.

Figures (12)

• Figure 1: Upper part: Science paradigm over time. Taken from schleder2019dft. Lower part: Simulation methods based on physics theories. Taken from lin2023multiscale.
• Figure 2: One crystal can be represented in various ways using different representations. These representations include: a graph containing atom and bond weights, Coulomb matrix, diffraction fingerprint, using topological descriptors and etc. Taken from li2022encoding with permission.
• Figure 3: Convolutional GNN (a) Example of graph structure (b) For each node, a computational graph is constructed, where each layer contains aggregation functions with shared weights across that layer (c) Upscaled vision of сonvolutional GNN.
• Figure 4: An overview of a MEGNet module. The initial graph is represented by the set of atomic attributes $V=v_i$, bond attributes $E =\{(e_k, r_k, s_k)\}$, and global state attributes $u$. In the first update step, the bond attributes are updated. Information flows from atoms that form the bond, the state attributes, and the previous bond attribute to the new bond attributes. Similarly, the second and third steps update the atomic and global state attributes, respectively, by information flow among all three attributes. The final result is a new graph representation. Taken from chen2019graph.
• Figure 5: Left side: pristine MoS2 crystal. Right side: it's a sparse representation (defects only).