Applying Convolutional Neural Networks to Data on Unstructured Meshes with Space-Filling Curves

TL;DR

This paper introduces a novel method for applying convolutional neural networks directly to unstructured mesh data using space-filling curves to linearize the data, enabling CNN techniques on complex geometries.

Contribution

The paper develops a CNN approach that leverages space-filling curves to handle unstructured mesh data, extending CNN applicability beyond structured grids.

Findings

Effective transformation of unstructured mesh data into 1D for CNN processing

Comparable accuracy of SFC-based CAE with classical CAE on structured data

Successful application to flow simulation data on unstructured meshes

Abstract

This paper presents the first classical Convolutional Neural Network (CNN) that can be applied directly to data from unstructured finite element meshes or control volume grids. CNNs have been hugely influential in the areas of image classification and image compression, both of which typically deal with data on structured grids. Unstructured meshes are frequently used to solve partial differential equations and are particularly suitable for problems that require the mesh to conform to complex geometries or for problems that require variable mesh resolution. Central to the approach are space-filling curves, which traverse the nodes or cells of a mesh tracing out a path that is as short as possible (in terms of numbers of edges) and that visits each node or cell exactly once. The space-filling curves (SFCs) are used to find an ordering of the nodes or cells that can transform multi-dimensional solutions on unstructured meshes into a one-dimensional (1D) representation, to which 1D convolutional layers can then be applied. Although developed in two dimensions, the approach is applicable to higher dimensional problems. To demonstrate the approach, the network we choose is a convolutional autoencoder (CAE) although other types of CNN could be used. The approach is tested by applying CAEs to data sets that have been reordered with an SFC. Sparse layers are used at the input and output of the autoencoder, and the use of multiple SFCs is explored. We compare the accuracy of the SFC-based CAE with that of a classical CAE applied to two idealised problems on structured meshes, and then apply the approach to solutions of flow past a cylinder obtained using the finite-element method and an unstructured mesh.