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

TL;DR

This paper presents a neural-preconditioned iterative solver for Poisson equations with mixed boundary conditions, offering efficient and adaptable solutions that outperform traditional methods in fluid simulation scenarios.

Contribution

The authors introduce a neural preconditioner that generalizes across domain shapes and boundary conditions, reducing setup costs and improving solver efficiency.

Findings

Outperforms algebraic multigrid in test cases

Generalizes to unseen domain shapes and boundary conditions

Supports fast inference with a lightweight neural network

Abstract

We introduce a neural-preconditioned iterative solver for Poisson equations with mixed boundary conditions. Typical Poisson discretizations yield large, ill-conditioned linear systems. Iterative solvers can be effective for these problems, but only when equipped with powerful preconditioners. Unfortunately, effective preconditioners like multigrid require costly setup phases that must be re-executed every time domain shapes or boundary conditions change, forming a severe bottleneck for problems with evolving boundaries. In contrast, we present a neural preconditioner trained to efficiently approximate the inverse of the discrete Laplacian in the presence of such changes. Our approach generalizes to domain shapes, boundary conditions, and grid sizes outside the training set. The key to our preconditioner's success is a novel, lightweight neural network architecture featuring spatially varying convolution kernels and supporting fast inference. We demonstrate that our solver outperforms state-of-the-art methods like algebraic multigrid as well as recently proposed neural preconditioners on challenging test cases arising from incompressible fluid simulations.