  In recent years, there have been a surge in applications of neural networks
(NNs) in physical sciences. Although various algorithmic advances have been
proposed, there are, thus far, limited number of studies that assess the
interpretability of neural networks. This has contributed to the hasty
characterization of most NN methods as "black boxes" and hindering wider
acceptance of more powerful machine learning algorithms for physics. In an
effort to address such issues in fluid flow modeling, we use a probabilistic
neural network (PNN) that provide confidence intervals for its predictions in a
computationally effective manner. The model is first assessed considering the
estimation of proper orthogonal decomposition (POD) coefficients from local
sensor measurements of solution of the shallow water equation. We find that the
present model outperforms a well-known linear method with regard to estimation.
This model is then applied to the estimation of the temporal evolution of POD
coefficients with considering the wake of a NACA0012 airfoil with a Gurney flap
and the NOAA sea surface temperature. The present model can accurately estimate
the POD coefficients over time in addition to providing confidence intervals
thereby quantifying the uncertainty in the output given a particular training
data set.
