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：针对偏微分方程的高效数值求解器推动了科学与工程的发展。其中常用的数值求解器之一是预条件共轭梯度（PCG）算法，该算法能以给定精度水平求解大型系统。PCG求解器面临的一个挑战是预条件子的选择，因为不同问题依赖的系统可从不同的预条件子中受益。我们提出了一种新方法，在共轭梯度算法中引入归纳偏置进行预条件。给定一个系统矩阵和一组来源于某潜在分布的求解向量，我们训练一个图神经网络来获取系统矩阵的近似分解，并将其作为PCG求解器中的预条件子。我们进行了大量实验，证明了所提方法在求解各种二维和三维线性二阶PDE时的有效性和泛化能力。