Efficient numerical solvers for partial differential equations empower science and engineering. One of the commonly employed numerical solvers is the preconditioned conjugate gradient (PCG) algorithm which can solve large systems to a given precision level. One challenge in PCG solvers is the selection of preconditioners, as different problem-dependent systems can benefit from different preconditioners. We present a new method to introduce \emph{inductive bias} in preconditioning conjugate gradient algorithm. Given a system matrix and a set of solution vectors arise from an underlying distribution, we train a graph neural network to obtain an approximate decomposition to the system matrix to be used as a preconditioner in the context of PCG solvers. We conduct extensive experiments to demonstrate the efficacy and generalizability of our proposed approach in solving various 2D and 3D linear second-order PDEs.

翻译：高效的偏微分方程数值求解器是科学与工程领域的重要支撑。其中，预处理共轭梯度（PCG）算法是常用的数值求解器之一，能对大规模系统在指定精度下完成求解。PCG求解器面临的关键挑战在于预处理器的选择——不同问题依赖的系统需匹配合适的预处理器。本文提出一种在预处理共轭梯度算法中引入**归纳偏置**的新方法。给定系统矩阵及基于底层分布生成的解向量集合，我们训练图神经网络获取系统矩阵的近似分解，并将其作为PCG求解器中的预处理器。通过大量实验验证，该方法在求解各类二维及三维线性二阶偏微分方程时表现出优异的有效性和泛化能力。