The performance of iterative solvers such as the conjugate gradient methods (CG), GMRES and others depends heavily on the conditioning of the underlying system. In the context of microstructure simulations, the condition number depends on the phase contrast and the discretization. To improve convergence, the use of preconditioners is basically mandatory in order to be competitive.
We propose a novel class of machine-learned preconditioners that employ a unitary transform (e.g., Fourier Transform) on regular grid data. We unveil previously unknown correspondence of the resulting neural operator to a well-known class of spectral solvers used in microstructure solid mechanical simulations with unparalleled success. We demonstrate that our learned precondition is almost reaching the performance of the domain specific method while generalizing easily to other problem settings and being purely data-based.
We think that the connection between the increasingly popular method belonging to the field of neural operators and established theoretical methods will pave the way to new insights in the future.
The article will appear soon (DOI link given below but not active yet as of Feb 3, 2026).