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

Modeling and predicting surface roughness in materials science is a challenging multiscale problem, as surface textures exhibit hierarchical (fractal) structures across multiple scales. In this work, we propose a synthetic data-driven approach to understanding scale transitions in surface roughness using fractal data generation and symbolic regression. We introduce a method for constructing coarse-grained representations of synthetic fractal surfaces and employ symbolic regression to derive interpretable mathematical expressions that map fine-scale features to coarse-scale behavior. Our approach demonstrates high predictive accuracy, achieving an R² value close to 1 with a low mean squared error. These findings suggest that symbolic regression can effectively capture scale transitions in hierarchical surface structures, offering a promising direction for data-driven multiscale modeling in materials science. While our work is preliminary, it highlights the potential for AI-driven approaches to improve the efficiency of multiscale modeling in computationally expensive scientific domains.