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

Weixian Lan, Elias Gueidon, Ayano Kaneda, Julian Panetta, Joseph Teran

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

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Keywords: Computational Linear Algebra, Neural Network, Conjugate Gradients, Partial Differential Equations, Fluid Simulation

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Abstract:

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.

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Submission Number: 6390