Learning Adaptive Coarse Spaces Using Transferable Neural Network Models for Linear and Nonlinear Overlapping Domain Decomposition Methods

Domain decomposition methods have been established as efficient and parallel scalable iterative solvers and preconditioners for the solution of large-scale systems arising from the discretization of partial differential equations. In particular, overlapping Schwarz methods have been successfully applied to a wide range of linear and nonlinear problems. However, for problems with highly heterogeneous coefficients, standard domain decomposition methods typically suffer from deteriorating convergence rates. Robustness with respect to the coefficient contrast can be achieved by enriching the coarse space with adaptively selected constraints obtained from local generalized eigenvalue problems. The construction of these adaptive coarse spaces, however, can account for a significant part of the overall computing time.
In the present work, machine learning techniques are employed to reduce this part of the computing time in the context of the adaptive Generalized Dryja-Smith-Widlund (AGDSW) coarse space. A two-stage approach is proposed in which regression neural networks are used to predict the adaptive coarse basis functions, while a classification neural network is employed to predict the number of basis functions required to ensure robustness. As a consequence, adaptive coarse spaces can be set up in the online phase without solving any eigenvalue problem. Particular attention is paid to problem-specific aspects, including sign-invariant loss functions and post-processing strategies to significantly improve the predicted constraints. The proposed approach is first investigated for scalar diffusion problems with high coefficient contrasts and is subsequently transferred, without retraining, to problems of linear elasticity and to nonlinear $p$-Laplace problems, also within a nonlinear Schwarz framework.