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We present DE-VAE, a variational autoencoder (VAE) architecture to search fora compressed representation of dynamical dark energy (DE) models inobservational studies of the cosmic large-scale structure. DE-VAE is trained onmatter 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 mosttypical dynamical DE parametrization with two extra parameters describing anevolving DE equation of state. The boosts are compressed to a lower-dimensionalrepresentation, which is concatenated with standard cold dark matter (CDM)parameters and then mapped back to reconstructed boosts; both the compressionand 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 cosmologicalparameters within $1\sigma$ ($2\sigma$) of a Gaussian error which includescosmic variance, shot noise and systematic effects for a Stage IV-like survey.This single parameter shows a high mutual information with the two DEparameters, and these three variables can be linked together with an explicitequation through symbolic regression. Considering a model with two latentvariables only marginally improves the accuracy of the predictions, and addinga 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 conceptto a general framework to be employed in the search for a commonlower-dimensional parametrization of a wide range of beyond-$\Lambda$CDM modelsand for different cosmological datasets. Such a framework could then bothinform the development of cosmological surveys by targeting optimal probes, andprovide theoretical insight into the common phenomenological aspects ofbeyond-$\Lambda$CDM models.

A representation learning approach to probe for dynamical dark energy in matter power spectra