  Supervised operator learning is an emerging machine learning paradigm with
applications to modeling the evolution of spatio-temporal dynamical systems and
approximating general black-box relationships between functional data. We
propose a novel operator learning method, LOCA (Learning Operators with Coupled
Attention), motivated from the recent success of the attention mechanism. In
our architecture, the input functions are mapped to a finite set of features
which are then averaged with attention weights that depend on the output query
locations. By coupling these attention weights together with an integral
transform, LOCA is able to explicitly learn correlations in the target output
functions, enabling us to approximate nonlinear operators even when the number
of output function in the training set measurements is very small. Our
formulation is accompanied by rigorous approximation theoretic guarantees on
the universal expressiveness of the proposed model. Empirically, we evaluate
the performance of LOCA on several operator learning scenarios involving
systems governed by ordinary and partial differential equations, as well as a
black-box climate prediction problem. Through these scenarios we demonstrate
state of the art accuracy, robustness with respect to noisy input data, and a
consistently small spread of errors over testing data sets, even for
out-of-distribution prediction tasks.
