本文为西班牙加泰罗尼亚技术大学（作者：Roberto L′opez Gonz′alez）的博士论文，共237页。

科学和工程中许多问题的目的是寻找一个函数，它是某种特定功能的最佳表达。比如最优控制、反向分析和最优形状设计等。只有其中的一些变分问题能够被解析地求解，因此，唯一通用的技术是使用直接方法近似逼近解。然而，变分问题很难求解，为了克服这些困难，有必要在数值方法领域进行创新。

这篇博士论文的目的是从泛函分析和变分演算的角度发展神经网络的概念理论。在这种公式中，学习意味着通过最小化与神经网络相关联的目标函数来解决变分问题。目标函数的选择取决于特定的应用。另一方面，性能评估可能需要函数、常微分方程或偏微分方程的积分。

神经网络能够处理广泛的数学和物理应用问题。更具体地说，多层感知器的变分公式提供了解决变分问题的直接方法，包括诸如函数回归、模式识别或时间序列预测等典型应用，还包括诸如最优控制、反向问题和最优形状设计等新应用。这种新应用的增加导致标准神经网络不能处理某些特定问题，需要对其进行扩展。

本文开发了一类扩展的多层感知器，除了传统的神经元模型和网络结构外，还包括独立的参数、边界条件和上下界。本文通过求解不同验证问题的解析解，研究了该数值方法的计算性能。此外，对于最优控制、反向问题和最优形状设计中的几个工程实例，应用了扩展型多层感知器的变分公式。最后，结合开源神经网络C++库Flood，在泛函分析和变分理论的基础上实现了这一工作。

Many problems arising in science andengineering aim to find a function which is the optimal value of a specifiedfunctional. Some examples include optimal control, inverse analysis and optimalshape design. Only some of these, regarded as variational problems, can besolved analytically, and the only general technique is to approximate thesolution using direct methods. Unfortunately, variational problems are verydifficult to solve, and it becomes necessary to innovate in the field ofnumerical methods in order to overcome the difficulties. The objective of thisPhD Thesis is to develop a conceptual theory of neural networks from theperspective of functional analysis and variational calculus. Within thisformulation, learning means to solve a variational problem by minimizing anobjective functional associated to the neural network. The choice of theobjective functional depends on the particular application. On the other side,its evaluation might need the integration of functions, ordinary differentialequations or partial differential equations. As it will be shown, neuralnetworks are able to deal with a wide range of applications in mathematics andphysics. More specifically, a variational formulation for the multilayerperceptron provides a direct method for solving variational problems. Thisincludes typical applications such as function regression, pattern recognitionor time series prediction, but also new ones such as optimal control, inverseproblems and optimal shape design. This addition of applications causes that astandard neural network is not able to deal with some particular problems, andit needs to be augmented. In this work an extended class of multilayerperceptron is developed which, besides the traditional neuron models andnetwork architectures, includes independent parameters, boundary conditions andlower and upper bounds. The computational performance of this numerical methodis investigated here through the solution of different validation problems withanalytical solution. Moreover, a variational formulation for an extended classof multilayer perceptron is applied to several engineering cases within optimalcontrol, inverse problems or optimal shape design. Finally, this work comeswith the open source neural networks C++ library Flood, which has beenimplemented following the functional analysis and calculus of variationstheories.

1 引言
2 基础知识
3 多层感知器的变分公式
4 数据建模
5 变分演算的传统问题
6 最优控制问题
7 反向问题
8 最优形状设计
9 结论与未来工作展望
附录A Flood软件模型
附录B 数值积分
附录C 相关论文
附录D 相关工程
附录E 相关软件
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