Symbolic Regression in Materials Science: Discovering Interatomic Potentials from Data

Abstract

Particle-based modeling of materials at atomic scale plays an important role in the development of new materials and the understanding of their properties. The accuracy of particle simulations is determined by interatomic potentials, which allow calculating the potential energy of an atomic system as a function of atomic coordinates and potentially other properties. First-principles-based ab initio potentials can reach arbitrary levels of accuracy, however, their applicability is limited by their high computational cost. Machine learning (ML) has recently emerged as an effective way to offset the high computational costs of ab initio atomic potentials by replacing expensive models with highly efficient surrogates trained on electronic structure data. Among a plethora of current methods, symbolic regression (SR) is gaining traction as a powerful “white-box” approach for discovering functional forms of interatomic potentials. This contribution discusses the role of symbolic regression in Materials Science (MS) and offers a comprehensive overview of current methodological challenges and state-of-the-art results. A genetic programming-based approach for modeling atomic potentials from raw data (consisting of snapshots of atomic positions and associated potential energy) is presented and empirically validated on ab initio electronic structure data.

5 Appendix

Empirical potentials

For a comprehensive overview of empirical potentials, we recommend the work of Araújo and Ballester [2]. Below, we give a casual overview of the most important empirical potentials mentioned in this contribution.

Morse potential

This is an empirical potential used to model diatomic molecules:

$$\begin{aligned} V_\text {M}(r) = D\Big ( 1 - \exp \big ( -a(r - r_0) \big ) ^2\Big ) \end{aligned}$$

(17)

where D is the dissociation energy, r is the distance between atoms, a is a set of parameters and $r_0$ is the equilibrium bond distance.

Lennard-Jones potential

The Lennard-Jones potential models soft repulsive and attractive interactions and can describe electronically neutral atoms or molecules. Interacting particles repel each other at very close distances, attract each other at moderate distances, and do not interact at infinite distances:

$$\begin{aligned} V_\text {LJ}(r) = 4 \varepsilon \bigg [ \Big ( \frac{\sigma }{r} \Big )^{12} - \Big ( \frac{\sigma }{r} \Big )^6 \bigg ] \end{aligned}$$

(18)

where r is the distance between atoms, $\varepsilon$ is the dispersion energy and $\sigma$ is the distance at which the particle-particle potential energy V is zero.

Lippincott potential

Lippincott [49] potential involves an exponential of interatomic distances

$$\begin{aligned} V_\text {LIP}(r) = D\Bigg (1 - \exp \Big (\frac{-n(r - r_0)^2}{2r} \Big ) \Bigg )\Big (1 + aF(r) \Big ) \end{aligned}$$

(19)

where D is the dissociation energy, r is the distance between atoms, $r_0$ is the equilibrium bond distance and a and n are parameters. F(r) is a function of internuclear distance such that $F(r) = 0$ when $r=\infty$ and $F(r) = \infty$ when $r=0$.

Stillinger-Weber potential

The Stillinger-Weber potential [50] models two- and three-body interactions by taking into account not only the distances between atoms but also the bond angles:

$$\begin{aligned} V_\text {SW}(r) = \sum _{\langle i, j \rangle } \phi _2(r_{ij}) + \sum _{\langle i, j, k \rangle } \phi _3(r_{ij}, r_{ik}, \theta _{ijk}) \end{aligned}$$

(20)

where

$$\begin{aligned} \phi _2(r_{ij})&= A \varepsilon \left[ B\left( \frac{\sigma }{r_{ij}} \right) ^p - \left( \frac{\sigma }{r_{ij}} \right) ^q \right] \exp \left( \frac{\sigma }{r_{ij} - a\sigma } \right) \text { and}\end{aligned}$$

(21)

$$\begin{aligned} \phi _3(r_{ij}, r_{ik}, \theta _{ijk})&= \lambda \varepsilon \left[ \cos \theta _{ijk} - \cos \theta _0 \right] ^2 \times \exp \left( \frac{\gamma \sigma }{r_{ij} - a\sigma } \right) \exp \left( \frac{\gamma \sigma }{r_{ik} - a \sigma } \right) \end{aligned}$$

(22)

Sutton-Chen potential

The Sutton-Chen potential [51] has been used in molecular dynamics and Monte Carlo simulations of metallic systems. It offers a reasonable description of various bulk properties, with an approximate many-body representation of the delocalized metallic bonding:

$$\begin{aligned} V_\text {SC} = \sum _{\langle i,j \rangle } U(r_{ij}) - \sum _i u\sqrt{\rho _i} \end{aligned}$$

(23)

Here, the first term represents the repulsion between atomic cores, and the second term models the bonding energy due to the electrons. Both terms are further defined in terms of reciprocal power so that the complete expression is

$$\begin{aligned} V_\text {SC} = \epsilon \left[ \sum _{\langle i,j \rangle } \left( \frac{a}{r_{ij}} \right) ^n - C \sum _i \sqrt{\sum _j \left( \frac{a}{r_{ij}} \right) ^m} \right] \end{aligned}$$

(24)

where C is a dimensionless parameter, $\epsilon$ is a parameter with dimensions of energy, a is the lattice constant, m, n are positive integers with $n > m$ and $r_{ij}$ is the distance between the ith and jth atoms.