While it is assumed in many applications that components are characterized by a homogeneous microstructure, this is not always the case. In fact, materials often exhibit heterogeneities, which can affect the material behavior drastically. Pronounced examples for this are Metal-Matrix Composites (MMCs). To determine the material behavior in multi-scale simulations, homogenization problems have to be solved that are often discretized using the Finite Element Method (FEM). The resulting linear system can then be solved by a Conjugate Gradient (CG) scheme. However, it is a common problem that the system becomes ill-conditioned for finely resolved discretizations leading to slow convergence.

This is problematic, when solutions are required for many different microstructures and material parameters, e.g., in virtual materials design. Hence, preconditioners are used for CG, but these often depend on the matrix and have to be reassembled for any new microstructure. An exception are FFT-based preconditioners such as in Fourier-Accelerated Nodal Solvers (FANS) that are highly effective for homogenization problems with periodic boundary conditions. FANS act on a Fourier representation of the residual field and perform convolutions with a problem-specific fundamental solution.

Recently, Fourier Neural Operators (FNOs) are emerging as a machine learning method to predict the solution of parametric PDEs for given parameter fields. However, there is no guarantee that the predictions fulfill a given accuracy or are even physically consistent. A promising option is, thus, to use hybrid methods that combine the advantages of both data-driven methods and classical solvers.

With FNO-CG, we propose a novel hybrid solver with guaranteed convergence involving special variants of FNOs that are admissible as preconditioners for CG and can be interpreted as a machine-learned extension of FANS. In contrast to FANS, FNO-CG is not restricted to periodic boundary conditions. For various problem formulations, we demonstrate that FNO-CG is competitive with machine learning models in terms of performance but can achieve results with an accuracy that is multiple orders of magnitude higher.