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This paper presents a learnable solver tailored to iteratively solve sparselinear systems from discretized partial differential equations (PDEs). Unliketraditional approaches relying on specialized expertise, our solver streamlinesthe algorithm design process for a class of PDEs through training, whichrequires only training data of coefficient distributions. The proposed methodis anchored by three core principles: (1) a multilevel hierarchy to promoterapid convergence, (2) adherence to linearity concerning the right-hand-side ofequations, and (3) weights sharing across different levels to facilitateadaptability to various problem sizes. Built on these foundational principlesand considering the similar computation pattern of the convolutional neuralnetwork (CNN) as multigrid components, we introduce a network adept at solvinglinear systems from PDEs with heterogeneous coefficients, discretized onstructured grids. Notably, our proposed solver possesses the ability togeneralize over right-hand-side terms, PDE coefficients, and grid sizes,thereby ensuring its training is purely offline. To evaluate its effectiveness,we train the solver on convection-diffusion equations featuring heterogeneousdiffusion coefficients. The solver exhibits swift convergence to high accuracyover a range of grid sizes, extending from $31 \times 31$ to $4095 \times4095$. Remarkably, our method outperforms the classical Geometric Multigrid(GMG) solver, demonstrating a speedup of approximately 3 to 8 times.Furthermore, our numerical investigation into the solver's capacity togeneralize to untrained coefficient distributions reveals promising outcomes.

MGCNN: a learnable multigrid solver for sparse linear systems from PDEs on structured grids