Machine learning, especially deep learning is gaining much attention due to the breakthrough performance in various cognitive applications. Recently, neural networks (NN) have been intensively explored to model partial differential equations as NN can be viewed as universal approximators for nonlinear functions. A deep network operator (DeepONet) architecture was proposed to model the general non-linear continuous operators for partial differential equations (PDE) due to its better generalization capabilities than existing mainstream deep neural network architectures. However, existing DeepONet can only accept one input function, which limits its application. In this work, we explore the DeepONet architecture to extend it to accept two or more input functions. We propose new Enhanced DeepONet or EDeepONet high-level neural network structure, in which two input functions are represented by two branch DNN sub-networks, which are then connected with output truck network via inner product to generate the output of the whole neural network. The proposed EDeepONet structure can be easily extended to deal with multiple input functions. Our numerical results on modeling two partial differential equation examples shows that the proposed enhanced DeepONet is about 7X-17X or about one order of magnitude more accurate than the fully connected neural network and is about 2X-3X more accurate than a simple extended DeepONet for both training and test.
翻译:由于各种认知应用的突破性工作,人们越来越重视机器学习,特别是深层次学习。最近,对神经网络(NN)进行了深入的探索,以模拟部分差异方程,因为NN可以被视为非线性功能的通用近似方程。一个深网络操作员(DeepONet)架构建议对非线性方程(PDE)的一般非线性连续操作员(PDE)进行模拟,因为其通用能力优于现有主流深神经网络结构。然而,现有的DeepONet只能接受一个限制其应用的输入功能。在这项工作中,我们探索DeepONet结构以扩大部分差异方程以接受两个或更多输入功能。我们提出了一个新的增强型DeepONet或EdeepONet高级神经网络结构,其中两个输入功能由两个分支DNNNE子网络代表,然后通过内部产品与输出卡车网络连接,以产生整个神经网络的输出。提议的EdeepOnet结构可以很容易扩展至处理多个输入功能。我们关于两个部分差异方程式示例的数值结果显示,对于一个更精确的深X级或更精确的深X级网络来说,一个比一个级的高级测试级的高级网络比一个级,一个级比一个级更高级的高级的深度和一个级比一个级。