Symbolic Regression for Learning Scale Transition Equations in Synthetic Fractal Surface Roughness

Abstract

Modeling surface roughness in materials science is a challenging multiscale problem, as surface textures often exhibit hierarchical (fractal-like) structure across multiple scales. In this work, we present a synthetic data-driven approach to studying scale transitions in surface roughness using fractal data generation and symbolic regression. We construct coarse-grained representations of synthetic fractal surfaces and apply symbolic regression to derive interpretable mathematical expressions that map fine-scale features to coarse-scale behavior. On controlled synthetic data, our approach achieves high predictive accuracy (R² near 1, low MSE), serving as a baseline validation. While the data is idealized, these results suggest that symbolic regression can capture scale-transition relationships in hierarchical surface structures and may also be able to support future efforts in data-driven multiscale modeling. This work highlights the potential of symbolic learning in accelerating modeling workflows for complex physical systems.