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This paper presents the first classical Convolutional Neural Network (CNN)that can be applied directly to data from unstructured finite element meshes orcontrol volume grids. CNNs have been hugely influential in the areas of imageclassification and image compression, both of which typically deal with data onstructured grids. Unstructured meshes are frequently used to solve partialdifferential equations and are particularly suitable for problems that requirethe mesh to conform to complex geometries or for problems that require variablemesh resolution. Central to the approach are space-filling curves, whichtraverse the nodes or cells of a mesh tracing out a path that is as short aspossible (in terms of numbers of edges) and that visits each node or cellexactly once. The space-filling curves (SFCs) are used to find an ordering ofthe nodes or cells that can transform multi-dimensional solutions onunstructured meshes into a one-dimensional (1D) representation, to which 1Dconvolutional 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 convolutionalautoencoder (CAE) although other types of CNN could be used. The approach istested 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 useof multiple SFCs is explored. We compare the accuracy of the SFC-based CAE withthat 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 usingthe finite-element method and an unstructured mesh.

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