  Efficient downscaling of large ensembles of coarse-scale information is
crucial in several applications, such as oceanic and atmospheric modeling. The
determining form map is a theoretical lifting function from the low-resolution
solution trajectories of a dissipative dynamical system to their corresponding
fine-scale counterparts. Recently, a physics-informed deep neural network
("CDAnet") was introduced, providing a surrogate of the determining form map
for efficient downscaling. CDAnet was demonstrated to efficiently downscale
noise-free coarse-scale data in a deterministic setting. Herein, the
performance of well-trained CDAnet models is analyzed in a stochastic setting
involving (i) observational noise, (ii) model noise, and (iii) a combination of
observational and model noises. The analysis is performed employing the
Rayleigh-Benard convection paradigm, under three training conditions, namely,
training with perfect, noisy, or downscaled data. Furthermore, the effects of
noises, Rayleigh number, and spatial and temporal resolutions of the input
coarse-scale information on the downscaled fields are examined. The results
suggest that the expected l2-error of CDAnet behaves quadratically in terms of
the standard deviations of the observational and model noises. The results also
suggest that CDAnet responds to uncertainties similar to the theorized and
numerically-validated CDA behavior with an additional error overhead due to
CDAnet being a surrogate model of the determining form map.
