  Neural operators are a type of deep architecture that learns to solve (i.e.
learns the nonlinear solution operator of) partial differential equations
(PDEs). The current state of the art for these models does not provide explicit
uncertainty quantification. This is arguably even more of a problem for this
kind of tasks than elsewhere in machine learning, because the dynamical systems
typically described by PDEs often exhibit subtle, multiscale structure that
makes errors hard to spot by humans. In this work, we first provide a
mathematically detailed Bayesian formulation of the ''shallow'' (linear)
version of neural operators in the formalism of Gaussian processes. We then
extend this analytic treatment to general deep neural operators using
approximate methods from Bayesian deep learning. We extend previous results on
neural operators by providing them with uncertainty quantification. As a
result, our approach is able to identify cases, and provide structured
uncertainty estimates, where the neural operator fails to predict well.
