This paper presents a learnable solver tailored to solve discretized linear partial differential equations (PDEs). This solver requires only problem-specific training data, without using specialized expertise. Its development is anchored by three core principles: (1) a multilevel hierarchy to promote rapid convergence, (2) adherence to linearity concerning the right-hand side of equations, and (3) weights sharing across different levels to facilitate adaptability to various problem sizes. Built on these foundational principles, we introduce a network adept at solving PDEs discretized on structured grids, even when faced with heterogeneous coefficients. The cornerstone of our proposed solver is the convolutional neural network (CNN), chosen for its capacity to learn from structured data and its similar computation pattern as multigrid components. To evaluate its effectiveness, the solver was trained to solve convection-diffusion equations featuring heterogeneous diffusion coefficients. The solver exhibited swift convergence to high accuracy over a range of grid sizes, extending from $31 \times 31$ to $4095 \times 4095$. Remarkably, our method outperformed the classical Geometric Multigrid (GMG) solver, demonstrating a speedup of approximately 3 to 8 times. Furthermore, we explored the solver's generalizability to untrained coefficient distributions. The findings showed consistent reliability across various other coefficient distributions, revealing that when trained on a mixed coefficient distribution, the solver is nearly as effective in generalizing to all types of coefficient distributions.

翻译：本文提出了一种可学习的求解器，专门用于求解离散化的线性偏微分方程（PDE）。该求解器仅需问题特定的训练数据，无需依赖专业领域知识。其开发基于三个核心原则：（1）采用多层层次结构以促进快速收敛；（2）保持对等式右端项的线性约束；（3）跨层级共享权重以适配不同问题规模。基于这些基本原则，我们引入了一种能够求解结构化网格上离散化PDE的网络，即使面临非均匀系数的情况。该求解器的核心是卷积神经网络（CNN），因其具备从结构化数据中学习的能力以及与多重网格组件相似的计算模式。为评估有效性，我们对求解器进行了训练以求解具有非均匀扩散系数的对流-扩散方程。该求解器在从$31 \times 31$到$4095 \times 4095$的网格规模范围内均展现出快速收敛至高精度。值得注意的是，我们的方法优于经典几何多重网格（GMG）求解器，实现了约3至8倍的加速比。此外，我们探究了求解器对未训练系数分布的泛化能力。结果表明，在多种系数分布下均保持稳定可靠性，且当基于混合系数分布训练时，求解器几乎能同等有效地泛化至所有类型的系数分布。