  Research in machine learning has polarized into two general approaches for
regression tasks: Transductive methods construct estimates directly from
available data but are usually problem unspecific. Inductive methods can be
much more specific but generally require compute-intensive solution searches.
In this work, we propose a hybrid approach and show that transductive
regression principles can be meta-learned through gradient descent to form
efficient in-context neural approximators by leveraging the theory of
vector-valued Reproducing Kernel Banach Spaces (RKBS). We apply this approach
to function spaces defined over finite and infinite-dimensional spaces
(function-valued operators) and show that once trained, the Transducer can
almost instantaneously capture an infinity of functional relationships given a
few pairs of input and output examples and return new image estimates. We
demonstrate the benefit of our meta-learned transductive approach to model
complex physical systems influenced by varying external factors with little
data at a fraction of the usual deep learning training computational cost for
partial differential equations and climate modeling applications.
