Evolve Smoothly, Fit Consistently: Learning Smooth Latent Dynamics For Advection-Dominated Systems

We present a data-driven, space-time continuous framework to learn surrogate models for complex physical systems described by advection-dominated partial differential equations. Those systems have slow-decaying Kolmogorov n-width that hinders standard methods, including reduced order modeling, from producing high-fidelity simulations at low cost. In this work, we construct hypernetworkbased latent dynamical models directly on the parameter space of a compact representation network. We leverage the expressive power of the network and a specially designed consistency-inducing regularization to obtain latent trajectories that are both low-dimensional and smooth. These properties render our surrogate models highly efficient at inference time. We show the efficacy of our framework by learning models that generate accurate multi-step rollout predictions at much faster inference speed compared to competitors, for several challenging examples. High-fidelity numerical simulation of physical systems modeled by time-dependent partial differential equations (PDEs) has been at the center of many technological advances in the last century. However, for engineering applications such as design, control, optimization, data assimilation, and uncertainty quantification, which require repeated model evaluation over a potentially large number of parameters, or initial conditions, high-fidelity simulations remain prohibitively expensive, even with state-of-art PDE solvers. The necessity of reducing the overall cost for such downstream applications has led to the development of surrogate models, which captures the core behavior of the target system but at a fraction of the cost. One of the most popular frameworks in the last decades (Aubry et al., 1988) to build such surrogates has been reduced order models (ROMs). In a nutshell, they construct lower-dimensional representations and their corresponding reduced dynamics that capture the system's behavior of interest. The computational gains then stem from the evolution of a lower-dimensional latent representation (see Benner et al. (2015) for a comprehensive review).