Partition of unity networks: deep hp-approximation
This work addresses the curse of dimensionality in neural network approximation for computational mathematics, offering a novel architecture with potential broad impact, though it appears incremental as it builds on existing approximation theory.
The authors tackled the problem of achieving optimal approximation rates in deep neural networks by proposing partition of unity networks (POUnets), which incorporate partitions of unity and monomials directly into the architecture, resulting in hp-convergence for smooth functions and outperforming MLPs for piecewise polynomial functions with many discontinuities.
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.