  Non-intrusive reduced-order models (ROMs) have recently generated
considerable interest for constructing computationally efficient counterparts
of nonlinear dynamical systems emerging from various domain sciences. They
provide a low-dimensional emulation framework for systems that may be
intrinsically high-dimensional. This is accomplished by utilizing a
construction algorithm that is purely data-driven. It is no surprise,
therefore, that the algorithmic advances of machine learning have led to
non-intrusive ROMs with greater accuracy and computational gains. However, in
bypassing the utilization of an equation-based evolution, it is often seen that
the interpretability of the ROM framework suffers. This becomes more
problematic when black-box deep learning methods are used which are notorious
for lacking robustness outside the physical regime of the observed data. In
this article, we propose the use of a novel latent-space interpolation
algorithm based on Gaussian process regression. Notably, this reduced-order
evolution of the system is parameterized by control parameters to allow for
interpolation in space. The use of this procedure also allows for a continuous
interpretation of time which allows for temporal interpolation. The latter
aspect provides information, with quantified uncertainty, about full-state
evolution at a finer resolution than that utilized for training the ROMs. We
assess the viability of this algorithm for an advection-dominated system given
by the inviscid shallow water equations.

Computational Physics;Machine Learning;Machine Learning