Learned Adaptive Mesh Generation

TL;DR

This paper introduces LAMG, a neural network-based method that generates adaptive meshes for solving elliptic PDEs efficiently, reducing computational costs by inferring local resolution from coarse solutions.

Contribution

The paper presents a novel neural network approach for one-shot adaptive mesh generation that generalizes across shapes and boundary conditions, improving PDE solution efficiency.

Findings

LAMG produces high-quality adaptive meshes across diverse shapes.

It reduces computational costs compared to traditional methods.

The approach is robust and versatile across different PDE problems.

Abstract

Elliptic Partial Differential Equations (PDEs) play a central role in computing the equilibrium conditions of physical problems (heat, gravitation, electrostatics, etc.). Efficient solutions to elliptic PDEs are also relevant to computer graphics since they encode global smoothness with local control leading to stable, well-behaved solutions. The Poisson equation is a linear elliptic PDE that serves as a prototypical candidate to assess newly-proposed solvers. Solving the Poisson equation on an arbitrary 3D domain, say a 3D scan of a turbine's blade, is computationally expensive and scales quadratically with discretization. Traditional workflows in research and industry exploit variants of the finite element method (FEM), but some key benefits of using Monte Carlo (MC) methods have been identified. Our key idea is to exploit a sparse and approximate solution (via FEM or MC) to the Poisson equation towards inferring an adaptive discretization in one shot. We achieve this by training a lightweight neural network that generalizes across shapes and boundary conditions. Our algorithm, Learned Adaptive Mesh Generation (LAMG), maps from a coarse solution to a sizing field that defines a local (adaptive) spatial resolution. This output space, rather than directly predicting a high-resolution solution, is a unique aspect of our approach. We use standard methods to generate tetrahedral meshes that respect the sizing field, and obtain the solution via one FEM computation on the adaptive mesh. That is, our neural network serves as a surrogate model of a computationally expensive method that requires multiple (iterative) FEM solves. We demonstrate the versatility, controllability, robustness and efficiency of LAMG via systematic experimentation.