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：我们提出一种适用于混合边界条件泊松方程的神经预处理迭代求解器。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，作为子问题出现在众多数值算法中，并且作为更广泛的椭圆型偏微分方程类的模型问题而存在。最常用的泊松离散化方法会产生大型稀疏线性系统。在高分辨率下以及对性能关键的应用中，迭代求解器对此类系统具有优势——但仅当与强大的预处理子结合时。我们求解器的核心是一个神经网络，该网络被训练为近似模拟任意形状域中具有混合边界条件的离散结构化网格拉普拉斯算子的逆算子。该问题的结构启发了一种新颖的网络架构，我们证明即使对于训练集之外的边界条件，该架构作为预处理子也非常有效。我们展示，在来自不可压缩流体模拟的具有挑战性的测试案例中，我们的方法优于代数多重网格等最先进的求解器以及一些近期提出的神经预处理子。