  We present DE-VAE, a variational autoencoder (VAE) architecture to search for
a compressed representation of dynamical dark energy (DE) models in
observational studies of the cosmic large-scale structure. DE-VAE is trained on
matter power spectra boosts generated at wavenumbers $k\in(0.01-2.5) \
h/\rm{Mpc}$ and at four redshift values $z\in(0.1,0.48,0.78,1.5)$ for the most
typical dynamical DE parametrization with two extra parameters describing an
evolving DE equation of state. The boosts are compressed to a lower-dimensional
representation, which is concatenated with standard cold dark matter (CDM)
parameters and then mapped back to reconstructed boosts; both the compression
and the reconstruction components are parametrized as neural networks.
Remarkably, we find that a single latent parameter is sufficient to predict 95%
(99%) of DE power spectra generated over a broad range of cosmological
parameters within $1\sigma$ ($2\sigma$) of a Gaussian error which includes
cosmic variance, shot noise and systematic effects for a Stage IV-like survey.
This single parameter shows a high mutual information with the two DE
parameters, and these three variables can be linked together with an explicit
equation through symbolic regression. Considering a model with two latent
variables only marginally improves the accuracy of the predictions, and adding
a third latent variable has no significant impact on the model's performance.
We discuss how the DE-VAE architecture can be extended from a proof of concept
to a general framework to be employed in the search for a common
lower-dimensional parametrization of a wide range of beyond-$\Lambda$CDM models
and for different cosmological datasets. Such a framework could then both
inform the development of cosmological surveys by targeting optimal probes, and
provide theoretical insight into the common phenomenological aspects of
beyond-$\Lambda$CDM models.
