SFFT-based homogenization: using tensor trains to enhance FFT-based homogenization

Homogenization is a fundamental technique for estimating the macroscopic properties of materials with microscale heterogeneity. Among homogenization methods, the fast Fourier transform (FFT)-based homogenization algorithm has become widely used due to its computational efficiency and ability to handle complex microstructures. Nevertheless, even with GPU acceleration, FFT-based homogenization for industrial applications remains excessively time-consuming, particularly when generating elastic training data for AI models. This is due to the curse of dimensionality, which arises from the algorithms reliance on the FFT, creating a fundamental bottleneck. In this paper, we propose a quantum-inspired superfast Fourier transform (SFFT)-based homogenization algorithm that leverages the improved time complexity of a tensor train variant of the Quantum Fourier Transform. By additionally exploiting structural properties of the underlying microstructure, our method achieves exponential improvements in time complexity and memory efficiency compared to the traditional FFT-based technique—all while remaining executable on classical hardware. We evaluate the performance of our algorithm across increasingly complex microstructures, demonstrating its potential advantages and limitations.