Homogenization of Composites using the Derivative-Free Loss Method for Neural Networks

Simulating the behavior of a composite material requires resolving fine-scale heterogeneities, which leads to extremely large systems of linear equations that remain computationally infeasible. In this work, we investigate a neural-network–based, derivative-free loss approach for estimating the effective homogeneous coefficient of heterogeneous composite materials. This method inherently yields an averaged representation of the solution, corresponding to macroscopic material behavior. First, we apply the derivative-free framework to compute the macroscopic response of a prototypical composite problem, and then we construct a physics-informed neural network to extract the effective, homogenized coefficient. Finally, we compare the learned coefficient against values obtained by standard numerical homogenization techniques. Our results demonstrate that the proposed approach accurately predicts effective material parameters in the periodic material setting, which suggests a promising direction for integrating data-driven procedures in the composite material analysis.