Combining machine learning and adaptive coarse spaces

Abstract

The hybrid ML-FETI-DP algorithm combines the advantages of adaptive coarse spaces in domain decomposition methods and certain supervised machine learning techniques. Adaptive coarse spaces ensure robustness of highly scalable domain decomposition solvers, even for highly heterogeneous coefficient distributions with arbitrary coefficient jumps. However, their construction requires the setup and solution of local generalized eigenvalue problems, which is typically computationally expensive. The idea of ML-FETI-DP is to interpret the coefficient distribution as image data and predict whether an eigenvalue problem has to be solved or can be neglected while still maintaining robustness of the adaptive FETI-DP method. For this purpose, neural networks are used as image classifiers. In the present work, the ML-FETI-DP algorithm is extended to three dimensions, which requires both a complex data preprocessing procedure to construct consistent input data for the neural network as well as a representative training and validation data set to ensure generalization properties of the machine learning model. Numerical experiments for stationary diffusion and linear elasticity problems with realistic coefficient distributions show that a large number of eigenvalue problems can be saved; in the best case of the numerical results presented here, 97% of the eigenvalue problems can be avoided being set up and solved.