In this talk, we present a data-driven framework for constructing efficient and accurate approximate inverse preconditioners for solving elliptic partial differential equations (PDEs). The core idea is to train a neural network to approximate the Green’s function of the underlying operator. The architecture exploits the intrinsic multiscale structure of Green’s functions and integrates exact inverse information from a coarse-grid discretization, thereby improving both training efficiency and accuracy. To enable a scalable application, we introduce a data-driven hierarchical matrix construction algorithm that circumvents the prohibitive cost of evaluating all Green’s function entries. This yields an almost linear-cost preconditioner application cost that is naturally suited for parallel architectures. We demonstrate the effectiveness of the approach on a range of challenging elliptic PDE and saddle point problems. Numerical experiments show significant reductions in iteration counts and total solution time, highlighting the potential of learning-based preconditioning techniques for large-scale scientific computing.