  Unsupervised learning with functional data is an emerging paradigm of machine
learning research with applications to computer vision, climate modeling and
physical systems. A natural way of modeling functional data is by learning
operators between infinite dimensional spaces, leading to discretization
invariant representations that scale independently of the sample grid
resolution. Here we present Variational Autoencoding Neural Operators (VANO), a
general strategy for making a large class of operator learning architectures
act as variational autoencoders. For this purpose, we provide a novel rigorous
mathematical formulation of the variational objective in function spaces for
training. VANO first maps an input function to a distribution over a latent
space using a parametric encoder and then decodes a sample from the latent
distribution to reconstruct the input, as in classic variational autoencoders.
We test VANO with different model set-ups and architecture choices for a
variety of benchmarks. We start from a simple Gaussian random field where we
can analytically track what the model learns and progressively transition to
more challenging benchmarks including modeling phase separation in
Cahn-Hilliard systems and real world satellite data for measuring Earth surface
deformation.
