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：摘要：我们提出了一种用于混合边界条件下泊松方程的神经预条件迭代求解器。泊松方程在科学计算中无处不在：它控制着广泛的物理现象，作为子问题出现在许多数值算法中，并且是更广泛的椭圆偏微分方程类别的模型问题。最流行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于性能关键型应用，迭代求解器对这些系统具有优势——但前提是配备了强大的预条件器。我们求解器的核心是一个神经网络，该网络经过训练，能够近似任意形状域上且具有混合边界条件的离散结构化网格拉普拉斯算子的逆。该问题的结构激发了一种新颖的网络架构，我们证明该架构即使对于训练集之外的边界条件，作为预条件器也极为有效。我们表明，在由不可压缩流体模拟产生的具有挑战性的测试案例上，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最新的神经预条件器。