"I have recently jumped on a control project that involves predicting
output of a nonlinear system given some input."
good luck...no one's found a way of doing this in decades of work.
"The team has used $N$ training input/output relationships to build a
'dictionary' of $N$ linear models. They would like to match query
inputs to the most similar model (cross-correlate new input against
training inputs etc.) and use the corresponding linear model for
This sounds a lot like gain scheduling to me, which is not really a way of predicting a nonlinear output from an input a priori, but rather of systematically representing a nonlinear system as a linear system about a chosen series of points (or continuous trajectory). This technique is standard and ubiquitous.
"Is there established theory around this piecewise/dictionary approach
to nonlinear systems? I can't find anything in literature, but perhaps
my terminology is incorrect."
Yes. It goes under a variety of names: gain scheduling, perturbation theory, Linear time-dependent systems, etc. This is basically the standard approach to controlling a nonlinear plant which is used in practical/industrial controllers.