Partition of Unity Networks: Deep HP-Approximation
Partition of Unity Networks: Deep HP-Approximation
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DOI：
10.2172/1856303
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发表时间：
2021-01
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通讯作者：
Kookjin Lee;N. Trask;Ravi G. Patel;Mamikon A. Gulian;E. Cyr
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作者：
Kookjin Lee;N. Trask;Ravi G. Patel;Mamikon A. Gulian;E. Cyr
近似理论家通过利用深度神经网络同时模拟单位和单项式分区的能力，建立了深度神经网络的同类最佳近似率。出于这一动机，我们提出了统一网络（POUnets）的分区，将这些元素直接纳入架构。用于学习概率度量的类型的分类架构用于构建空间的无网格分区，而具有可学习系数的多项式空间与每个分区相关联。所得到的类似hp元素的近似允许使用快速最小二乘优化器，并且所得到的架构大小不需要随空间维度呈指数级扩展，从而打破了维度灾难。一个抽象的近似结果建立理想的属性，以指导网络设计。两种选择的架构的数值结果表明，POUnets产生Hp收敛的光滑函数和一贯优于MLP的分段多项式函数与大量的不连续性。
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.