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.

我们介绍了一种用于具有混合边界条件的 Poisson 方程的神经预处理迭代求解器。我们的求解器的核心是一个神经网络，用于逼近具有任意形状和混合边界条件的离散结构网格 Laplace 算子的逆运算。我们通过在不可压缩流体模拟中产生的具有挑战性的测试案例上展示，我们的方法优于最先进的求解器如代数多重网格和一些最近的神经预处理器。