论文标题
组成函数的神经网络近似，并应用于动态系统
Neural Network Approximations of Compositional Functions With Applications to Dynamical Systems
论文作者
论文摘要
正如在现实生活应用的许多领域所证明的那样，神经网络具有处理高维数据的能力。在最佳控制和动力学系统的领域中，近年来在许多发布的结果中研究和验证了相同的功能。为了揭示神经网络能够解决一些高维问题的根本原因，我们为组成函数及其神经网络近似而开发了代数框架和近似理论。理论基础是以某种方式开发的，因此它不仅支持函数作为投入输出关系，还支持数值算法。此功能至关重要，因为它可以分析无法获得分析解决方案的问题的近似错误，例如微分方程和最佳控制。我们确定组成函数的一组关键特征以及神经网络的特征与复杂性之间的关系。除功能近似外，我们还证明了神经网络的多个误差上限公式，这些公式近似于微分方程，优化和最佳控制的解决方案。
As demonstrated in many areas of real-life applications, neural networks have the capability of dealing with high dimensional data. In the fields of optimal control and dynamical systems, the same capability was studied and verified in many published results in recent years. Towards the goal of revealing the underlying reason why neural networks are capable of solving some high dimensional problems, we develop an algebraic framework and an approximation theory for compositional functions and their neural network approximations. The theoretical foundation is developed in a way so that it supports the error analysis for not only functions as input-output relations, but also numerical algorithms. This capability is critical because it enables the analysis of approximation errors for problems for which analytic solutions are not available, such as differential equations and optimal control. We identify a set of key features of compositional functions and the relationship between the features and the complexity of neural networks. In addition to function approximations, we prove several formulae of error upper bounds for neural networks that approximate the solutions to differential equations, optimization, and optimal control.
