Universal Fourier Neural Operators for periodic homogenization problems in linear elasticity

TL;DR

This paper introduces Fourier Neural Operators (FNOs) tailored for periodic homogenization in linear elasticity, achieving fast, accurate, and general solutions for complex micromechanical cell problems without restrictions on material symmetry or geometry.

Contribution

It develops a Fourier Neural Operator framework inspired by FFT-based methods, providing a universal, training-free surrogate capable of handling large-scale, complex homogenization problems efficiently.

Findings

FNO surrogate predicts solutions with high accuracy across diverse material configurations.

The method handles large-scale problems with over 100 million voxels efficiently.

Explicit guarantees and physical insights enhance the reliability of the FNO approach.

Abstract

Solving cell problems in homogenization is hard, and available deep-learning frameworks fail to match the speed and generality of traditional computational frameworks. More to the point, it is generally unclear what to expect of machine-learning approaches, let alone single out which approaches are promising. In the work at hand, we advocate Fourier Neural Operators (FNOs) for micromechanics, empowering them by insights from computational micromechanics methods based on the fast Fourier transform (FFT). We construct an FNO surrogate mimicking the basic scheme foundational for FFT-based methods and show that the resulting operator predicts solutions to cell problems with arbitrary stiffness distribution only subject to a material-contrast constraint up to a desired accuracy. In particular, there are no restrictions on the material symmetry like isotropy, on the number of phases and on the geometry of the interfaces between materials. Also, the provided fidelity is sharp and uniform, providing explicit guarantees leveraging our physical empowerment of FNOs. To show the desired universal approximation property, we construct an FNO explicitly that requires no training to begin with. Still, the obtained neural operator complies with the same memory requirements as the basic scheme and comes with runtimes proportional to classical FFT solvers. In particular, large-scale problems with more than 100 million voxels are readily handled. The goal of this work is to underline the potential of FNOs for solving micromechanical problems, linking FFT-based methods to FNOs. This connection is expected to provide a fruitful exchange between both worlds.