Learned Adaptive Mesh Generation

TL;DR

This work trains a neural network to mimic a computationally expensive method that requires multiple (iterative) FEM solves, and achieves the solution via one FEM computation on the adaptive mesh through a one-shot method.

Abstract

The distribution and evolution of several real-world quantities, such as temperature, pressure, light, and heat, are modelled mathematically using Partial Differential Equations (PDEs). Solving PDEs defined on arbitrary 3D domains, 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. We use sparse and approximate MC estimates to infer adaptive discretization. We achieve this by training a neural network that is lightweight and that generalizes across shapes and boundary conditions. Our algorithm, Learned Adaptive Mesh Generation (LAMG), maps a set of sparse MC estimates of the solution to a sizing field that defines a local (adaptive) spatial resolution. We then use standard methods to generate tetrahedral meshes that respect the sizing field, and obtain the solution via one FEM computation on the adaptive mesh. We train the network to mimic a computationally expensive method that requires multiple (iterative) FEM solves. Thus, our one-shot method is $2\times$ to $4\times$ faster than adaptive methods for FEM or MC while achieving similar error. Our learning framework is lightweight and versatile. We demonstrate its effectiveness across a large dataset of meshes.

Figures (9)

• Figure 1: An overview of our approach to solve Poisson equations in 3D meshed domains. (a) Monte Carlo methods estimate the solution at sampled domain locations given a boundary mesh. (b) Finite Element Methods operate on a tetrahedral mesh and yield a dense solution via interpolation. (c) Adaptive Mesh Refinement applies FEM iteratively while subdividing portions of the mesh where error is high. (d) Our method, Learned Adaptive Mesh Generation (LAMG), uses sparse MC estimates as inputs to a lightweight Neural Network that predicts the sizing field (without iterative solves). We use standard tools for mesh followed by an FEM step. (e) We train our method using reference sizing fields estimated on adaptive meshes produced by AMR.
• Figure 2: Relative error (top row) and execution time (bottom row) on three of our datasets-- single mesh (a), small dataset (b) and large dataset (c) using three differently trained models $h_{\theta 1}$ , $h_{\theta 2}$ and $h_{\theta 4}$ respectively. We compare our (lamg) results with AMR (amr), a Monte Carlo solver (wos) and AMG (amg). Regardless of the dataset, the result remains the same: for similar levels of error, LAMG is consistently faster than the other methods.
• Figure 3: The figure shows that training on many shapes improves runtime (better sizing fields) for similar error. Increasing the number of parameters does not impact error or runtime. Since $h_{\theta2}$ is only trained on 6 shapes, it conservatively predicts fine meshes on some shapes, increasing runtime.
• Figure 4: Our method is robust to a wide range of $m$$[50,2000]$ (top) and $n$$[50,3000]$ (bottom). We plot relative $L_2$ error (first column), number of vertices in the resulting mesh (second column) and total computation time (third column) for 4 different meshes. Parameter settings that are 'out of training distribution' are shaded grey. Further details in Section \ref{['sec:robust']}.
• Figure 5: Although large uniform meshes can achieve lower error than LAMG (with $\eta=1$), the computation time is significantly larger. The plots show RE ($L_2$) vs time (left) and number of vertices (right) for uniform meshing compared to LAMG.