  Autoencoders have found widespread application, in both their original
deterministic form and in their variational formulation (VAEs). In scientific
applications it is often of interest to consider data that are comprised of
functions; the same perspective is useful in image processing. In practice,
discretisation (of differential equations arising in the sciences) or
pixellation (of images) renders problems finite dimensional, but conceiving
first of algorithms that operate on functions, and only then discretising or
pixellating, leads to better algorithms that smoothly operate between different
levels of discretisation or pixellation. In this paper function-space versions
of the autoencoder (FAE) and variational autoencoder (FVAE) are introduced,
analysed, and deployed. Well-definedness of the objective function governing
VAEs is a subtle issue, even in finite dimension, and more so on function
space. The FVAE objective is well defined whenever the data distribution is
compatible with the chosen generative model; this happens, for example, when
the data arise from a stochastic differential equation. The FAE objective is
valid much more broadly, and can be straightforwardly applied to data governed
by differential equations. Pairing these objectives with neural operator
architectures, which can thus be evaluated on any mesh, enables new
applications of autoencoders to inpainting, superresolution, and generative
modelling of scientific data.
