  We consider the use of probabilistic neural networks for fluid flow
{surrogate modeling} and data recovery. This framework is constructed by
assuming that the target variables are sampled from a Gaussian distribution
conditioned on the inputs. Consequently, the overall formulation sets up a
procedure to predict the hyperparameters of this distribution which are then
used to compute an objective function given training data. We demonstrate that
this framework has the ability to provide for prediction confidence intervals
based on the assumption of a probabilistic posterior, given an appropriate
model architecture and adequate training data. The applicability of the present
framework to cases with noisy measurements and limited observations is also
assessed. To demonstrate the capabilities of this framework, we consider
canonical regression problems of fluid dynamics from the viewpoint of
reduced-order modeling and spatial data recovery for four canonical data sets.
The examples considered in this study arise from (1) the shallow water
equations, (2) a two-dimensional cylinder flow, (3) the wake of NACA0012
airfoil with a Gurney flap, and (4) the NOAA sea surface temperature data set.
The present results indicate that the probabilistic neural network not only
produces a machine-learning-based fluid flow {surrogate} model but also
systematically quantifies the uncertainty therein to assist with model
interpretability.
