An operator learning perspective on parameter-to-observable maps
Daniel Zhengyu Huang Nicholas H. Nelsen Margaret Trautner
Computationally efficient surrogates for parametrized physical models play a
crucial role in science and engineering. Operator learning provides data-driven
surrogates that map between function spaces. However, instead of full-field
measurements, often the available data are only finite-dimensional
parametrizations of model inputs or finite observables of model outputs.
Building off of Fourier Neural Operators, this paper introduces the Fourier
Neural Mappings (FNMs) framework that is able to accommodate such
finite-dimensional inputs and outputs. The paper develops universal
approximation theorems for the method. Moreover, in many applications the
underlying parameter-to-observable (PtO) map is defined implicitly through an
infinite-dimensional operator, such as the solution operator of a partial
differential equation. A natural question is whether it is more data-efficient
to learn the PtO map end-to-end or first learn the solution operator and
subsequently compute the observable from the full-field solution. A theoretical
analysis of Bayesian nonparametric regression of linear functionals, which is
of independent interest, suggests that the end-to-end approach can actually
have worse sample complexity. Extending beyond the theory, numerical results
for the FNM approximation of three nonlinear PtO maps demonstrate the benefits
of the operator learning perspective that this paper adopts.