Machine learning for preconditioning elliptic equations in porous microstructures: A path to error control

Elliptic equations on complex porous microstructures govern the flow of fluids inside subsurface rocks in underground CO₂ and H₂ storage, and the transport of heat and solute within electrochemical devices like batteries and fuel cells. The algebraic systems arising from the discretization of these equations are often prohibitively large and must be solved via iterative (e.g., Krylov) methods, for which effective preconditioning is key to ensure rapid convergence. In recent work, the authors proposed a scalable two-level preconditioner whose performance was superior to existing algebraic multigrid variants for pore-scale problems. The preconditioner was based on the pore-level multiscale method (PLMM) and consisted of a coarse preconditioner, M_G, and a fine smoother, M_L. Similar two-level preconditioners based on the multiscale finite element/volume and variational multiscale methods also exist for solving continuum-scale PDEs in porous media. The most expensive step in building such two-level preconditioners is computing M_G, for which many numerical bases on a set of subdomains must be calculated to yield a prolongation matrix. Here, we show that machine learning (ML) can dramatically reduce this cost. Moreover, by embedding ML within a preconditioning framework, we enable the rarity of estimating and controlling ML errors to any desired level. We systematically probe the ML-built preconditioner in solving the Poisson and linear-elasticity equations over complex 2D/3D geometries and show that it performs comparably to its solver-built counterpart. Implications and future extensions are discussed.