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

TL;DR

The paper tackles the bottleneck of solving Poisson equations with mixed Dirichlet/Neumann boundary conditions on evolving domains. It introduces a neural preconditioner that approximates the inverse of the discrete Laplacian, integrated into a neural-preconditioned PSDO iterative solver, with a lightweight, geometry-aware architecture using spatially varying convolutions. Training relies on a residual-based loss across diverse domain geometries and boundary conditions, enabling generalization beyond the training set. Empirical results show significant speedups over AMG, IC, CHOLMOD, and prior neural preconditioners across fluid-physics benchmarks, highlighting practical impact for real-time or large-scale incompressible flow simulations.

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.

Figures (15)

• Figure 1: Sketches of our network architecture.
• Figure 2: We demonstrate our solver with incompressible flow simulations requiring the solution of mixed Neumann/Dirichlet boundary conditions for the pressure Poisson equation.
• Figure 3: Histograms of solution speedup vs. a baseline of unpreconditioned CG (a) for all solves; and (b-f) for certain subsets of the systems to help tease apart different modes of the distribution.
• Figure 4: Solver scaling for mixed BC system matrices originating from a fixed-resolution domain $(n_c = 256^3)$; matrix row/col size $n_f$ is determined by the proportion of cells occupied by fluid. The vast majority of total solve time is contributed by the high-occupancy systems clustered to the right, where our method outperforms the rest.
• Figure 5: Comparisons among AMG, IC, CG and NSPDO (Ours) on a single frame at $256^3$ with Neumann only BC (top two) and mixed BC (bottom two).