Learning interpretable surface elasticity properties from bulk properties

TL;DR

The paper tackles the problem of deriving interpretable, closed-form relationships between bulk material properties and surface elastic properties. It introduces a neural network equation learner (nn-EQL) that uses customized activations and pruning to yield compact, symbolic expressions mapping bulk descriptors to surface invariants across seven FCC metals, yielding expressions like $\Gamma^\alpha = \Gamma_0^\alpha g_\Gamma(\theta,\phi)$ and $T^\alpha = T_0^\alpha g_T(\theta,\phi)$. The results show that surface tension exhibits a nearly universal orientation dependence, while residual surface stress and stiffness display material-specific anisotropy, and a linear bulk-to-surface map highlights stacking-fault energy as a dominant predictor. This interpretable neurosymbolic approach bridges atomistic simulations and continuum mechanics, enabling generalizable structure–property insights and providing a pathway to incorporate higher-fidelity data and electronic descriptors in the future.

Abstract

Surface elasticity is central to understanding the mechanics and stability of surfaces and interfaces. It is characterized by quantities such as surface tension, residual surface stress, and surface stiffness, however their analytical expressions are typically difficult to derive from atomistic data, and depend strongly on modeling choices. This work presents a neural network-based equation learner which combines customized activation functions and connection-based pruning to discover parsimonious, closed-form equations for surface elasticity from atomistic simulations. Applying the method to seven face centered cubic (FCC) metals, our equation learner uncovers interpretable equations that describe both low-Miller index and high-Miller index surface properties, capturing long-tail property distributions accurately. The discovered expressions are decoupled into two components: a universal, geometry-driven orientation function, and material-specific baseline coefficients. We find that lower-order properties such as surface tension are fundamentally geometry dependent, while higher-order properties such as surface stress and elasticity show more complex geometry and material dependence. We also relate material dependent coefficients to bulk properties, forming a clear map from bulk material properties to surface elasticity. Overall, this approach demonstrates that interpretable neurosymbolic machine learning can bridge the gap between atomistic simulations and physical laws, enabling the discovery of generalizable structure-property relationships for materials science phenomena such as surface elasticity.

Figures (5)

• Figure 1: The dataset of surface properties: (a) Distributions of surface tension ($\Gamma$) for seven FCC metals. (b) Range of surface orientations covered and corresponding surface tension values for Au, displayed in terms of spherical coordinates, $\theta$ and $\phi$.
• Figure 2: Equation learner (nn-EQL) workflow: We learn interpretable relationships between bulk properties of materials (e.g., bulk modulus, shear modulus, lattice parameter) and surface properties of materials (invariants) using equation learner networks. nn-EQLs are trained in three stages: Stage 1 is the training of an initial network with customized activations. Stage 2 uses connection-based pruning to reduce network size. Stage 3 (post training) is a readout of the network weights and activations as an equation.
• Figure 3: Visualizing the predictions of the nn-EQL: (a), (c), and (e) show distributions of the surface tension $\Gamma$, residual surface stress invariant $T$, and the isotropic surface elasticity stiffness invariant $2 T_1+T_0$, respectively, comparing results from the semi-analytical method (blue) and the equations learned by the nn-EQL (orange). Surface plots in (b), (d), and (f) illustrate the orientation dependence of $\Gamma$, $T$, and $2 T_1+T_0$ in Au predicted by the nn-EQL. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
• Figure 4: Comparing nn-EQL predictions to current literature: (a) Comparing nn-EQL predicted surface energy/stress/elastic constants to literature shenoy2005atomistic (molecular dynamics, open symbols), jain2013commentary (density functional theory, filled symbols) (b) Trends in nn-EQL predicted surface properties with surface orientation and material.
• Figure 5: Parity plots of the material-specific coefficients in the equations for the surface invariants. 'Ground Truth' values are from the nn-EQL discovery and 'Predictions' values are from the linear model based on normalized bulk properties from Eq. \ref{['eq:linmodel']}. Plots correspond to coefficients for (a) $\Gamma$ and (b) $T$.