前馈神经网络(Feedforward Neural Network)是设计的第一种也是最简单的人工神经网络。在此网络中,信息仅在一个方向上移动,即从输入节点向前经过隐藏节点(如果有)并到达输出节点。 网络中没有周期或循环。

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深度学习(DL)在我们的生活中扮演着越来越重要的角色。它已经在癌症诊断、精准医疗、自动驾驶汽车、预测预测和语音识别等领域产生了巨大的影响。在传统的学习、分类和模式识别系统中使用的人工制作的特征提取器对于大型数据集是不可扩展的。在许多情况下,根据问题的复杂性,DL还可以克服早期浅层网络的限制,这些限制阻碍了有效的训练和多维培训数据分层表示的抽象。深度神经网络(DNN)使用多个(深度)单元层,具有高度优化的算法和体系结构。来自美国AJAY SHRESTHA等学者撰写了深度学习算法与架构回顾综述论文,包括深度学习算法类型与训练方法,深入研究了最近深度网络中使用的训练算法背后的数学原理。本文还介绍了深度卷积网络、深度残差网络、递归神经网络、增强学习、变分自编码器等不同类型的深度结构。

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In this work, we are concerned with neural network guided goal-oriented a posteriori error estimation and adaptivity using the dual weighted residual method. The primal problem is solved using classical Galerkin finite elements. The adjoint problem is solved in strong form with a feedforward neural network using two or three hidden layers. The main objective of our approach is to explore alternatives for solving the adjoint problem with greater potential of a numerical cost reduction. The proposed algorithm is based on the general goal-oriented error estimation theorem including both linear and nonlinear stationary partial differential equations and goal functionals. Our developments are substantiated with some numerical experiments that include comparisons of neural network computed adjoints and classical finite element solutions of the adjoints. In the programming software, the open-source library deal.II is successfully coupled with LibTorch, the PyTorch C++ application programming interface.

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In this work, we are concerned with neural network guided goal-oriented a posteriori error estimation and adaptivity using the dual weighted residual method. The primal problem is solved using classical Galerkin finite elements. The adjoint problem is solved in strong form with a feedforward neural network using two or three hidden layers. The main objective of our approach is to explore alternatives for solving the adjoint problem with greater potential of a numerical cost reduction. The proposed algorithm is based on the general goal-oriented error estimation theorem including both linear and nonlinear stationary partial differential equations and goal functionals. Our developments are substantiated with some numerical experiments that include comparisons of neural network computed adjoints and classical finite element solutions of the adjoints. In the programming software, the open-source library deal.II is successfully coupled with LibTorch, the PyTorch C++ application programming interface.

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