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We introduce a neural-preconditioned iterative solver for Poisson equationswith mixed boundary conditions. The Poisson equation is ubiquitous inscientific computing: it governs a wide array of physical phenomena, arises asa subproblem in many numerical algorithms, and serves as a model problem forthe broader class of elliptic PDEs. The most popular Poisson discretizationsyield large sparse linear systems. At high resolution, and forperformance-critical applications, iterative solvers can be advantageous forthese -- but only when paired with powerful preconditioners. The core of oursolver is a neural network trained to approximate the inverse of a discretestructured-grid Laplace operator for a domain of arbitrary shape and with mixedboundary conditions. The structure of this problem motivates a novel networkarchitecture that we demonstrate is highly effective as a preconditioner evenfor boundary conditions outside the training set. We show that on challengingtest cases arising from an incompressible fluid simulation, our methodoutperforms state-of-the-art solvers like algebraic multigrid as well as somerecent neural preconditioners.

A Neural-preconditioned Poisson Solver for Mixed Dirichlet and Neumann Boundary Conditions