Abstract: Approximation theorists have established best-in-class optimal approximation
rates of deep neural networks by utilizing their ability to simultaneously
emulate partitions of unity and monomials. Motivated by this, we propose
partition of unity networks (POUnets) which incorporate these elements directly
into the architecture. Classification architectures of the type used to learn
probability measures are used to build a meshfree partition of space, while
polynomial spaces with learnable coefficients are associated to each partition.
The resulting hp-element-like approximation allows use of a fast least-squares
optimizer, and the resulting architecture size need not scale exponentially
with spatial dimension, breaking the curse of dimensionality. An abstract
approximation result establishes desirable properties to guide network design.
Numerical results for two choices of architecture demonstrate that POUnets
yield hp-convergence for smooth functions and consistently outperform MLPs for
piecewise polynomial functions with large numbers of discontinuities.