The numerical modelling of Partial Differential Equations (PDEs) is at the core of modern science and engineering. Traditional high-fidelity methods are robust and accurate, but often computationally expensive, particularly for many-query scenarios. Hence, Scientific Machine Learning (SciML) has emerged as a promising paradigm to accelerate simulations, combining data-driven deep-learning techniques with physical and mathematical knowledge. The Latent Dynamics Network (LDNet) is a recent SciML architecture that has demonstrated remarkable performance in predicting the response of spatio-temporal systems, combining Neural ODEs with machine-learning-based reduced-order modelling. In this work, we build upon the LDNet architecture, extending it to handle variable initial conditions. This significantly broadens the applicability of LDNet, enabling it to tackle real-world problems, such as predicting the evolution of a system from a measured initial state. We propose novel auto-decoding strategies to effectively represent the initial conditions within the lower-dimensional latent space, while retaining the encoder-free design of the original LDNet architecture, thus preserving data-type and resolution independence. Moreover, we explore how meta-learning techniques can be integrated into our framework, treating the initial condition as task-specific information, in order to further improve performance and generalization. Finally, we demonstrate the capabilities of the proposed approach on several benchmark problems, from the advection-diffusion-reaction equation to computational fluid dynamics.