Learning HTL-ORAS Algorithmic Components and Neural Solvers for ADR and Linear Elasticity Problems

In recent years, employing deep learning to improve the overall performance of iterative algorithms for solving large-scale PDE discrete systems has emerged as a new research hotspot in the field of scientific computing and engineering. Based on the optimized restricted additive Schwarz (ORAS) method and the corresponding hybrid two-level (HTL-ORAS) algorithm and theoretical framework, this work first develops a neural solver, ORAS(η), which is capable of optimizing the transmission parameter η in the ORAS method. For the finite element discretizations of the advection-diffusion-reaction (ADR) and linear elasticity equations, the proposed neural network is employed to learn the optimized parameter ηDL. Numerical experiments comparing with the default value of parameter η demonstrate that both the ORAS(ηDL) iterative method and its preconditioned FGMRES method achieve significantly improved convergence rates. Moreover, they exhibit strong generalization capabilities with respect to the reaction coefficient in the ADR equation and the Poisson ratio parameter in the linear elasticity equation. Furthermore, based on the HTL-ORAS method, lightweight neural networks capable of learning the coarse space and restriction operator are proposed. For the finite element discrete systems of the ADR and linear elasticity equations, the corresponding preconditioned FGMRES solvers, denoted as HTL-ORAS(R0ξ,p, ηDL)-FGMRES (ξ=I,II corresponding to two different learning strategies; Strategy I learns basis functions with prescribed subdomain supports, whereas Strategy II learns a small collection of dense global basis functions), are developed. Compared with ORAS(ηDL)-FGMRES, the proposed HTL-ORAS(R0ξ,p, ηDL)-FGMRES achieves a significant improvement in convergence speed. Compared with the HTL-ORAS-FGMRES method using the ideal coarse space, HTL-ORAS(R0ξ,p, ηDL)-FGMRES substantially reduces the size of the coarse space, thereby significantly improving computational efficiency.