  We propose a nonlinear manifold learning technique based on deep
convolutional autoencoders that is appropriate for model order reduction of
physical systems in complex geometries. Convolutional neural networks have
proven to be highly advantageous for compressing data arising from systems
demonstrating a slow-decaying Kolmogorov n-width. However, these networks are
restricted to data on structured meshes. Unstructured meshes are often required
for performing analyses of real systems with complex geometry. Our custom graph
convolution operators based on the available differential operators for a given
spatial discretization effectively extend the application space of deep
convolutional autoencoders to systems with arbitrarily complex geometry that
are typically discretized using unstructured meshes. We propose sets of
convolution operators based on the spatial derivative operators for the
underlying spatial discretization, making the method particularly well suited
to data arising from the solution of partial differential equations. We
demonstrate the method using examples from heat transfer and fluid mechanics
and show better than an order of magnitude improvement in accuracy over linear
methods.
