The complexity of real-world geophysical systems is often compounded by the fact that the observed measurements depend on hidden variables. These latent variables include unresolved small scales and/or rapidly evolving processes, partially observed couplings, or forcings in coupled systems. This is the case in ocean-atmosphere dynamics, for which unknown interior dynamics can affect surface observations. The identification of computationally-relevant representations of such partially-observed and highly nonlinear systems is thus challenging and often limited to short-term forecast applications. Here, we investigate the physics-constrained learning of implicit dynamical embeddings, leveraging neural ordinary differential equation (NODE) representations. In particular, we restrict the NODE representation to linear-quadratic dynamics and enforce a global boundedness constraint, which promotes the generalization of the learned dynamics to arbitrary initial conditions. The proposed architecture is implemented within a deep learning framework, and its relevance is demonstrated with respect to state-of-the-art schemes for different case studies representative of geophysical dynamics.