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While Deep Learning (DL) -based surrogate modeling approaches for Partial Differential Equations (PDEs) -- ranging from DL reduced-order models (ROMs) to operator learning -- have demonstrated significant potential in addressing many limitations of traditional reduced basis techniques, their development and implementation still face several theoretical and empirical challenges. ; To enable the effective deployment of these approaches in real-world scenarios, several key macro-aspects require further investigation: (I) the ability of surrogates to generalize to unseen parameter configurations, extrapolate over longer time horizons, and operate across varying spatio-temporal discretizations; (II) the need for a robust mathematical framework to enhance the interpretability of the learned model components, promoting transparency of DL-based surrogates and explainability of the learned representations; (III) the interactions between numerical and computational learning aspects, in order to bridge numerical PDEs modeling and DL-based techniques, and to justify their employment through the characterization of their approximation capabilities. ; These challenges are addressed from the perspective of Latent Dynamics Learning, focusing on the task of approximating parameterized time-evolution maps associated with the solutions of time-dependent parameterized PDEs, by modeling the dynamics of a learned latent representation. The core of the work lies in the design of DL architectures based on the latent dynamics paradigm, leveraging tools at the intersection of differential equations and DL, such as Neural Ordinary/Stochastic Differential Equations, accompanied by a comprehensive theoretical analysis, including the derivation of approximation error bounds, generalization guarantees, and convergence and stability results.