  The accessibility of spatially distributed data, enabled by affordable
sensors, field, and numerical experiments, has facilitated the development of
data-driven solutions for scientific problems, including climate change,
weather prediction, and urban planning. Neural Partial Differential Equations
(Neural PDEs), which combine deep learning (DL) techniques with domain
expertise (e.g., governing equations) for parameterization, have proven to be
effective in capturing valuable correlations within spatiotemporal datasets.
However, sparse and noisy measurements coupled with modeling approximation
introduce aleatoric and epistemic uncertainties. Therefore, quantifying
uncertainties propagated from model inputs to outputs remains a challenge and
an essential goal for establishing the trustworthiness of Neural PDEs. This
work evaluates various Uncertainty Quantification (UQ) approaches for both
Forward and Inverse Problems in scientific applications. Specifically, we
investigate the effectiveness of Bayesian methods, such as Hamiltonian Monte
Carlo (HMC) and Monte-Carlo Dropout (MCD), and a more conventional approach,
Deep Ensembles (DE). To illustrate their performance, we take two canonical
PDEs: Burger's equation and the Navier-Stokes equation. Our results indicate
that Neural PDEs can effectively reconstruct flow systems and predict the
associated unknown parameters. However, it is noteworthy that the results
derived from Bayesian methods, based on our observations, tend to display a
higher degree of certainty in their predictions as compared to those obtained
using the DE. This elevated certainty in predictions suggests that Bayesian
techniques might underestimate the true underlying uncertainty, thereby
appearing more confident in their predictions than the DE approach.
