本文为西班牙加泰罗尼亚理工大学（作者：Albert Prat Baucells）的硕士论文，共86页。

在文献[Velasco et al., 2014]中，介绍了一种经典人工神经网络结构的新方法，称为“LTI ODE值神经网络”，其中LTI ODE代表线性时不变序微分方程。在这个新的系统中，人工神经网络中的节点特征是：以可微连续时间信号的形式输入；以线性时不变序微分方程（LTI ODE）作为连接权值；以及在频域中评估的激活函数。结果表明，这种新的结构可以同时使用一个共同的神经结构来解决多个问题。然而，文章最后提出了开发新的神经网络模型的学习算法的必要性。本硕士论文以Velasco等人2014年指出的缺点为出发点，主要目的是开发一种LTI ODE值神经网络的训练算法。作为第一种自然的方法，反向传播算法的改进被认为是一个通用的框架。此外，由于输入信号的性质是可微的连续时间信号，因此分析了如何获得一个可以以模拟电路的形式物理实现的模型。

In [Velasco et al., 2014], a new approachof the classical artificial neural network architecture is introduced, named’LTI ODE-valued neural networks’, where LTI ODE stands for Linear TimeInvariant Ordinal Differential Equation. In this novel system, nodes in theartificial neural network are characterized by: inputs in the form ofdifferentiable continuous-time signals; linear time-invariant ordinarydifferential equations (LTI ODE) as connection weights; and activationfunctions evaluated in the frequency domain. It was shown that this newconfiguration allows solving multiple problems at the same time using a commonneural structure. However, the article concludes with the need for developinglearning algorithms for the new model of neural network. Taking as startingpoint the drawback pointed out in [Velasco et al., 2014], the main objective ofthis master thesis is to develop a training algorithm for a LTI ODE-valuedneural network. As a first and natural approach, modifications of theBackPropagation algorithm is considered as a general framework. Moreover, sincethe nature of the inputs are differentiable continuous-time signals, it isanalyzed how to obtain a model that can be physically implemented in the formof an analogical circuit.

1. 引言

2. 人工神经网络

3. LTI-ODEVNN：LTI ODE算子

4. 静态LTI-ODE算子

5. 代价

附录A Matlab代码

附录B 静态LTI ODE算子

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