  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.
