For linear elliptic boundary value problems, the solution operator can be represented by a Green's function (GF)--an integral operator mapping the source term to the solution--but for variable-coefficient operators, this kernel is rarely available in closed form. In addition, the GF is singular at the source point and must satisfy boundary conditions; these non-smooth features and hard constraints are difficult for a plain neural network to learn accurately from training pairs of solutions and corresponding source terms. We propose a hybrid neural surrogate that builds in the analytic structure of Green's kernels and lets the network learn only a smooth correction, yielding accurate one-dimensional GFs. The main contributions are: (i) a hybrid Green's function kernel model for variable-coefficient elliptic operators, and (ii) an axial operator-learning framework enabling multi-dimensional reconstruction. For one-dimensional variable-coefficient operators, we introduce \emph{GreenNet}, a hybrid kernel model that starts from the closed-form Poisson Green's function and learns only a correction term needed to match the target operator. The analytic component enforces the delta-induced singular structure (including the correct flux jump) and homogeneous boundary conditions, so the network is tasked only with a smooth residual. GreenNet is trained using solution--source pairs via a kernel-based reconstruction loss. In two dimensions, an axial approximation~\cite{kim2008axial} rewrites the PDE as coupled 1D problems on horizontal and vertical lines, enabling reuse of 1D GreenNet kernels. We introduce \emph{CouplingNet}, an operator-learning model~\cite{lu2021deeponet} that predicts horizontal and vertical line-wise flux-divergence components and enforces axial balance and consistency constraints. As the coefficient changes from line to line, we use a single operator-learning model that adapts locally and is reused across all lines. We validate GreenNet on benchmark one-dimensional variable-coefficient elliptic problems, extend it with operator learning to obtain line-wise Green kernels in 2D, and demonstrate CouplingNet reconstructions using these kernels.