arXiv:2510.13034v3 Announce Type: replace Abstract: In this paper, we propose a learning-based framework for constructing approximate inverse preconditioners for partial differential equations (PDEs) by approximating the Green's function of the underlying differential operator with neural networks (NNs). The training framework combines four components: an adaptive multiscale neural architecture ($\alpha$MSNN) that resolves hierarchical features across near-, middle-, and far-field regimes; optional coarse-grid solution anchors that improve identifiability; a multistage training strategy that progressively refines the Green's function representation across spatial scales; and an overlapping domain decomposition that enables local adaptation while preserving global consistency. Once trained, the learned Green's function provides a continuous approximation to the inverse operator that can be discretized on a given mesh to construct approximate inverse preconditioners. Within the domain decomposition framework, selected blocks of the learned inverse can also be used to approximate the inverse Schur complement, avoiding the need to assemble a full dense inverse. Numerical experiments on challenging PDE problems demonstrate that the resulting preconditioners provide robust and scalable convergence and are competitive with classical two-level preconditioning methods.