  Uncertainty quantification (UQ) in scientific machine learning (SciML)
becomes increasingly critical as neural networks (NNs) are being widely adopted
in addressing complex problems across various scientific disciplines.
Representative SciML models are physics-informed neural networks (PINNs) and
neural operators (NOs). While UQ in SciML has been increasingly investigated in
recent years, very few works have focused on addressing the uncertainty caused
by the noisy inputs, such as spatial-temporal coordinates in PINNs and input
functions in NOs. The presence of noise in the inputs of the models can pose
significantly more challenges compared to noise in the outputs of the models,
primarily due to the inherent nonlinearity of most SciML algorithms. As a
result, UQ for noisy inputs becomes a crucial factor for reliable and
trustworthy deployment of these models in applications involving physical
knowledge. To this end, we introduce a Bayesian approach to quantify
uncertainty arising from noisy inputs-outputs in PINNs and NOs. We show that
this approach can be seamlessly integrated into PINNs and NOs, when they are
employed to encode the physical information. PINNs incorporate physics by
including physics-informed terms via automatic differentiation, either in the
loss function or the likelihood, and often take as input the spatial-temporal
coordinate. Therefore, the present method equips PINNs with the capability to
address problems where the observed coordinate is subject to noise. On the
other hand, pretrained NOs are also commonly employed as equation-free
surrogates in solving differential equations and Bayesian inverse problems, in
which they take functions as inputs. The proposed approach enables them to
handle noisy measurements for both input and output functions with UQ.
