  Modeling dynamical systems, e.g. in climate and engineering sciences, often
necessitates solving partial differential equations. Neural operators are deep
neural networks designed to learn nontrivial solution operators of such
differential equations from data. As for all statistical models, the
predictions of these models are imperfect and exhibit errors. Such errors are
particularly difficult to spot in the complex nonlinear behaviour of dynamical
systems. We introduce a new framework for approximate Bayesian uncertainty
quantification in neural operators using function-valued Gaussian processes.
Our approach can be interpreted as a probabilistic analogue of the concept of
currying from functional programming and provides a practical yet theoretically
sound way to apply the linearized Laplace approximation to neural operators. In
a case study on Fourier neural operators, we show that, even for a discretized
input, our method yields a Gaussian closure--a structured Gaussian process
posterior capturing the uncertainty in the output function of the neural
operator, which can be evaluated at an arbitrary set of points. The method adds
minimal prediction overhead, can be applied post-hoc without retraining the
neural operator, and scales to large models and datasets. We showcase the
efficacy of our approach through applications to different types of partial
differential equations.
