论文标题
通过隐藏的串联极端学习机对部分微分方程的数值计算
Numerical Computation of Partial Differential Equations by Hidden-Layer Concatenated Extreme Learning Machine
论文作者
论文摘要
极端学习机（ELM）方法可以为线性/非线性偏微分方程（PDE）产生高度准确的解决方案，但要求神经网络的最后一个隐藏层要宽，以实现高精度。如果最后一个隐藏层狭窄，则现有ELM方法的准确性将很差，而与网络配置的其余部分无关。在本文中，我们提出了一种修改后的ELM方法，称为HLCONCELM（隐藏层串联ELM），以克服常规ELM方法的上述缺点。当网络的最后一个隐藏层狭窄且宽时，HLCONCELM方法可以为线性/非线性PDE产生高度精确的解决方案。新方法基于一种修改后的前馈神经网络（FNN），称为HLCONCFNN（隐藏层串联FNN），该网络融合了网络中隐藏层的逻辑串联，并将所有隐藏的节点暴露于输出层节点。 HLCONCFNN具有一个有趣的属性，即在网络体系结构，将其他隐藏层附加到网络上或将额外的节点添加到现有隐藏层中时，与新体系结构相关的HLCONCFNN的表示能力保证不小于原始网络体系结构的HLCONCFNN。在这里，表示能力是指所有函数的集合，这些功能可以由给定体系结构的神经网络准确表示。我们使用线性/非线性PDE提出了足够的基准测试，以证明HLCONCELM方法的计算准确性和性能以及该方法比以前工作的常规ELM的优越性。
The extreme learning machine (ELM) method can yield highly accurate solutions to linear/nonlinear partial differential equations (PDEs), but requires the last hidden layer of the neural network to be wide to achieve a high accuracy. If the last hidden layer is narrow, the accuracy of the existing ELM method will be poor, irrespective of the rest of the network configuration. In this paper we present a modified ELM method, termed HLConcELM (hidden-layer concatenated ELM), to overcome the above drawback of the conventional ELM method. The HLConcELM method can produce highly accurate solutions to linear/nonlinear PDEs when the last hidden layer of the network is narrow and when it is wide. The new method is based on a type of modified feedforward neural networks (FNN), termed HLConcFNN (hidden-layer concatenated FNN), which incorporates a logical concatenation of the hidden layers in the network and exposes all the hidden nodes to the output-layer nodes. HLConcFNNs have the interesting property that, given a network architecture, when additional hidden layers are appended to the network or when extra nodes are added to the existing hidden layers the representation capacity of the HLConcFNN associated with the new architecture is guaranteed to be not smaller than that of the original network architecture. Here representation capacity refers to the set of all functions that can be exactly represented by the neural network of a given architecture. We present ample benchmark tests with linear/nonlinear PDEs to demonstrate the computational accuracy and performance of the HLConcELM method and the superiority of this method to the conventional ELM from previous works.
