arXiv 2407.050782024-07math.NAcs.NA

浅层神经网络的函数与导数逼近

Function and derivative approximation by shallow neural networks

Yuanyuan Li, Shuai Lu

AI总结：

该研究针对含噪测量下单位立方域的函数及导数逼近反问题，提出适配浅层网络的含三类相关（半）范数惩罚的Tikhonov正则化方案，分析范数关联与维数影响，给出误差界并指导网络设计。

摘要：

我们针对求解一类经典反问题的场景，研究了一种专门为浅层神经网络设计的Tikhonov正则化方案：该反问题为基于含噪测量值，在单位立方域内逼近未知函数及其导数。所提出的Tikhonov正则化方案包含一个惩罚项，该惩罚项采用三种不同但密切相关的网络（半）范数：扩展Barron范数、变分范数以及Radon-BV半范数。惩罚项的这些选择取决于所使用的神经网络的具体架构。我们建立了各类网络范数之间的联系，尤其追踪了维数指标的依赖关系，旨在加深对这些范数如何相互作用的理解。我们重新探讨了通过各类范数实现函数逼近的通用性，为该Tikhonov正则化方案建立了严格的误差界分析，并明确阐释了维数指标的依赖关系，从而更清晰地揭示了维数如何影响逼近性能，以及人们应如何设计适用于多样化逼近任务的神经网络。

英文摘要：

We investigate a Tikhonov regularization scheme specifically tailored for shallow neural networks within the context of solving a classic inverse problem: approximating an unknown function and its derivatives within a unit cubic domain based on noisy measurements. The proposed Tikhonov regularization scheme incorporates a penalty term that takes three distinct yet intricately related network (semi)norms: the extended Barron norm, the variation norm, and the Radon-BV seminorm. These choices of the penalty term are contingent upon the specific architecture of the neural network being utilized. We establish the connection between various network norms and particularly trace the dependence of the dimensionality index, aiming to deepen our understanding of how these norms interplay with each other. We revisit the universality of function approximation through various norms, establish rigorous error-bound analysis for the Tikhonov regularization scheme, and explicitly elucidate the dependency of the dimensionality index, providing a clearer understanding of how the dimensionality affects the approximation performance and how one designs a neural network with diverse approximating tasks.

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