  Representing the parton distribution functions (PDFs) of the proton and other
hadrons through flexible, high-fidelity parametrizations has been a
long-standing goal of particle physics phenomenology. This is particularly true
since the chosen parametrization methodology can play an influential role in
the ultimate PDF uncertainties as extracted in QCD global analyses; these, in
turn, are often determinative of the reach of experiments at the LHC and other
facilities to non-standard physics, including at large $x$, where
parametrization effects can be significant. In this study, we explore a series
of encoder-decoder machine-learning (ML) models with various neural-network
topologies as efficient means of reconstructing PDFs from meaningful
information stored in an interpretable latent space. Given recent effort to
pioneer synergies between QCD analyses and lattice-gauge calculations, we
formulate a latent representation based on the behavior of PDFs in Mellin
space, i.e., their integrated moments, and test the ability of various models
to decode PDFs from this information faithfully. We introduce a numerical
package, $\texttt{PDFdecoder}$, which implements several encoder-decoder models
to reconstruct PDFs with high fidelity and use this tool to explore strengths
and pitfalls of neural-network approaches to PDF parametrization. We
additionally dissect patterns of learned correlations between encoded Mellin
moments and reconstructed PDFs which suggest opportunities for further
improvements to ML-based approaches to PDF parametrizations and uncertainty
quantification.
