Title: Utilizing Machine Learning Surrogate Models to Homogenize Hyperelastic Materials with Boolean Microstructures

Abstract: Data-driven surrogate models offer a viable alternative to traditional numerical homogenization techniques for heterogeneous materials. This study introduces a supervised learning framework designed to predict the effective Lamé parameters of hyperelastic composites using low-dimensional microstructural descriptors. The underlying dataset comprises numerical homogenization results from prior research on two-phase stochastic microstructures, which were generated via planar Boolean models and encompass variations in inclusion shape, phase contrast, and area fraction; these findings were originally detailed by Brändel, Brands, Maike, Rheinbach, Schröder, Schwarz, and Stoyan (2022).

We trained a neural network utilizing a mix of scalar and curve-valued statistical descriptors, specifically the area fraction, a derived scalar shape descriptor denoted as $\tau$, the two-point correlation function $S_2(r)$, and the lineal-path function $\ell(z)$. To enhance training stability and improve extrapolation performance, we also integrated data representing limiting cases within the parameter space. The surrogate model’s ability to generalize to unseen grain geometries was assessed through leave-one-grain-type-out cross-validation.

Our numerical findings indicate that incorporating additional descriptors helps lower relative errors. A predictor utilizing both $\tau$ and $S_2(r)$ achieves a compact representation with strong quantitative accuracy and consistent, regular dense response behavior. While adding the lineal-path function $\ell(z)$ further decreases error at existing data points—suggesting its potential as a valuable descriptor—dense evaluations of the post-training response reveal that higher pointwise accuracy does not necessarily ensure physically admissible behavior between sampled parameter values. These results highlight the need for future research into physically constrained surrogate models, refined loss formulations, bounded output parametrizations, and more systematic approaches to representing curve-valued geometric descriptors.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC