  Operator regression provides a powerful means of constructing
discretization-invariant emulators for partial-differential equations (PDEs)
describing physical systems. Neural operators specifically employ deep neural
networks to approximate mappings between infinite-dimensional Banach spaces. As
data-driven models, neural operators require the generation of labeled
observations, which in cases of complex high-fidelity models result in
high-dimensional datasets containing redundant and noisy features, which can
hinder gradient-based optimization. Mapping these high-dimensional datasets to
a low-dimensional latent space of salient features can make it easier to work
with the data and also enhance learning. In this work, we investigate the
latent deep operator network (L-DeepONet), an extension of standard DeepONet,
which leverages latent representations of high-dimensional PDE input and output
functions identified with suitable autoencoders. We illustrate that L-DeepONet
outperforms the standard approach in terms of both accuracy and computational
efficiency across diverse time-dependent PDEs, e.g., modeling the growth of
fracture in brittle materials, convective fluid flows, and large-scale
atmospheric flows exhibiting multiscale dynamical features.
