We introduce a neural-preconditioned iterative solver for Poisson equations with mixed boundary conditions. The Poisson equation is ubiquitous in scientific computing: it governs a wide array of physical phenomena, arises as a subproblem in many numerical algorithms, and serves as a model problem for the broader class of elliptic PDEs. The most popular Poisson discretizations yield large sparse linear systems. At high resolution, and for performance-critical applications, iterative solvers can be advantageous for these -- but only when paired with powerful preconditioners. The core of our solver is a neural network trained to approximate the inverse of a discrete structured-grid Laplace operator for a domain of arbitrary shape and with mixed boundary conditions. The structure of this problem motivates a novel network architecture that we demonstrate is highly effective as a preconditioner even for boundary conditions outside the training set. We show that on challenging test cases arising from an incompressible fluid simulation, our method outperforms state-of-the-art solvers like algebraic multigrid as well as some recent neural preconditioners.

翻译：我们提出了一种基于神经预条件的迭代求解器，用于求解混合边界条件下的泊松方程。泊松方程在科学计算中无处不在：它支配着广泛的物理现象，常作为许多数值算法的子问题出现，同时是更广泛椭圆型偏微分方程类中的典范问题。最主流的泊松离散化方法会生成大型稀疏线性系统。在高分辨率及性能关键型应用中，迭代求解器需配合强大的预条件器才能发挥优势——这正是本求解器的核心。我们训练了一个神经网络，用于逼近任意形状域内、含混合边界条件的离散结构化网格拉普拉斯算子的逆算子。该问题的结构特性驱动我们设计了一种新型网络架构，实验证明即使面对训练集之外的边界条件，该架构作为预条件器依然高效。基于不可压缩流体模拟中的挑战性测试案例，我们证明了该方法在性能上超越了代数多重网格等顶尖求解器及近期提出的若干神经预条件器。