单位网络的概率划分：基于聚类的深度近似
Probabilistic partition of unity networks: clustering based deep approximation
摘要：
单位网络的划分（POU-Nets）已被证明能够实现回归和偏微分方程解的代数收敛率，但需要经验调整训练参数。我们丰富的POU网与高斯噪声模型，以获得一个概率泛化服从基于梯度的最大似然损失的最小化。由此产生的架构提供了无噪声和噪声数据的空间表示为高斯混合物与封闭形式的方差表达式，它提供了一个估计的局部误差。训练过程基于函数值的相关性产生对输入空间的非常清晰的划分。这种训练点的分类适用于分层细化策略，该策略显着改善了回归的局部化，允许使用高阶多项式近似。与高斯过程回归相比，该框架更有利于扩展到大数据集，并允许空间变化的不确定性，利用深度神经网络的表达能力，同时绕过与其他概率深度学习方法相关的昂贵训练。与标准的深度神经网络相比，该框架在不使用正则化器来调整分区局部化的情况下证明了hp收敛。我们提供的基准量化性能在高/低维，表明收敛速度只取决于高维空间内的数据的潜在维度。最后，我们引入了一个新的开放源数据集的半导体器件的基于偏微分方程的模拟和执行无监督提取的物理可解释的降阶基础。
英文摘要：
Partition of unity networks (POU-Nets) have been shown capable of realizing algebraic convergence rates for regression and solution of PDEs, but require empirical tuning of training parameters. We enrich POU-Nets with a Gaussian noise model to obtain a probabilistic generalization amenable to gradient-based minimization of a maximum likelihood loss. The resulting architecture provides spatial representations of both noiseless and noisy data as Gaussian mixtures with closed form expressions for variance which provides an estimator of local error. The training process yields remarkably sharp partitions of input space based upon correlation of function values. This classification of training points is amenable to a hierarchical refinement strategy that significantly improves the localization of the regression, allowing for higher-order polynomial approximation to be utilized. The framework scales more favorably to large data sets as compared to Gaussian process regression and allows for spatially varying uncertainty, leveraging the expressive power of deep neural networks while bypassing expensive training associated with other probabilistic deep learning methods. Compared to standard deep neural networks, the framework demonstrates hp-convergence without the use of regularizers to tune the localization of partitions. We provide benchmarks quantifying performance in high/low-dimensions, demonstrating that convergence rates depend only on the latent dimension of data within high-dimensional space. Finally, we introduce a new open-source data set of PDE-based simulations of a semiconductor device and perform unsupervised extraction of a physically interpretable reduced-order basis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
