  The study of neural operators has paved the way for the development of
efficient approaches for solving partial differential equations (PDEs) compared
with traditional methods. However, most of the existing neural operators lack
the capability to provide uncertainty measures for their predictions, a crucial
aspect, especially in data-driven scenarios with limited available data. In
this work, we propose a novel Neural Operator-induced Gaussian Process (NOGaP),
which exploits the probabilistic characteristics of Gaussian Processes (GPs)
while leveraging the learning prowess of operator learning. The proposed
framework leads to improved prediction accuracy and offers a quantifiable
measure of uncertainty. The proposed framework is extensively evaluated through
experiments on various PDE examples, including Burger's equation, Darcy flow,
non-homogeneous Poisson, and wave-advection equations. Furthermore, a
comparative study with state-of-the-art operator learning algorithms is
presented to highlight the advantages of NOGaP. The results demonstrate
superior accuracy and expected uncertainty characteristics, suggesting the
promising potential of the proposed framework.
