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# Hierarchical partition of unity networks: fast multilevel training

*Proceedings of Mathematical and Scientific Machine Learning*, PMLR 190:271-286, 2022.

#### Abstract

We present a probabilistic mixture of experts framework to perform nonparametric piecewise polynomial approximation without the need for an underlying mesh partitioning space. Deep neural networks traditionally used for classification provide a means of localizing polynomial approximation, and the probabilistic formulation admits a trivially parallelizable expectation maximization (EM) strategy. We then introduce a hierarchical architecture whose EM loss naturally decomposes into coarse and fine scale terms and small decoupled least squares problems. We exploit this hierarchical structure to formulate a V-cycle multigrid-inspired training algorithm. A suite of benchmarks demonstrate the ability of the scheme to: realize for smooth data algebraic convergence with respect to number of partitions, exponential convergence with respect to polynomial order; exactly reproduce piecewise polynomial functions; and demonstrate through an application to data-driven semiconductor modeling the ability to accurately treat data spanning several orders of magnitude.