The recent adoption of data-driven deep learning (DL) –based approaches in the context of reduced order modeling has allowed to overcome many of the limitations affecting traditional reduced basis methods. These techniques aim at addressing the two main tasks involved in the construction of ROMs within a time-dependent parameterized context: nonlinear dimensionality reduction by means of autoencoders (AEs), and modeling of the latent dynamics of the resulting reduced representation, either via regressive, autoregressive or recurrent approaches. However, both traditional and DL-based ROMs are intrinsically tied to the temporal discretization employed during the offline training phase. This aspect represents a critical constraint, restricting ROMs' ability to adapt to different temporal resolutions during the online phase. Therefore, it is often necessary to rebuild the reduced basis or undergo additional fine-tuning stages when a novel temporal discretization is encountered, thereby entailing increased offline computational costs. To this end, a central aspect in reduced order modeling resides in whether the ROM solution is a meaningful time-continuous approximation of the FOM, in which case it could be possible to query the learned ROM at any given time by retaining a prescribed level of accuracy.
To address this problem, we introduce the latent dynamics model (LDM) mathematical framework, which enables us to: (1) devise a ROM architecture in a continuous setting, (2) consecutively analyze the impact of employing a numerical integration scheme for the solution of the latent dynamics, and (3) address its approximation capabilities in a learnable setting, rigorously demonstrating that it is able to provide an accurate and time-continuous approximation of the FOM solution.
The resulting reduced order modeling framework is characterized in a DL-context by relying on the AE-Neural ODE architecture, for which different architectural choices are proposed, aimed at enhancing interpretability and efficiency. The numerical experiments, performed on high-dimensional dynamical systems arising from the semi-discretization of parameterized nonlinear time-dependent PDEs, demonstrate that the framework exhibits a time-continuous approximation property, enabling zero-shot generalization to finer temporal discretizations than the one employed at training time.