Deep learning has shown successful application in visual recognition and certain artificial intelligence tasks. Deep learning is also considered as a powerful tool with high flexibility to approximate functions. In the present work, functions with desired properties are devised to approximate the solutions of PDEs. Our approach is based on a posteriori error estimation in which the adjoint problem is solved for the error localization to formulate an error estimator within the framework of neural network. An efficient and easy to implement algorithm is developed to obtain a posteriori error estimate for multiple goal functionals by employing the dual-weighted residual approach, which is followed by the computation of both primal and adjoint solutions using the neural network. The present study shows that such a data-driven model based learning has superior approximation of quantities of interest even with relatively less training data. The novel algorithmic developments are substantiated with numerical test examples. The advantages of using deep neural network over the shallow neural network are demonstrated and the convergence enhancing techniques are also presented
翻译:深层学习显示在视觉识别和某些人工智能任务中成功应用了视觉识别和某些人工智能任务。深层学习也被视为一种强大的工具,具有与近似功能的高度灵活性。在目前的工作中,设计了具有理想特性的功能,以近似PDEs的解决方案。我们的方法是基于事后误差估计,解决了误差的附带问题,从而在神经网络框架内设定了误差估计器。开发了一种高效和易于实施的算法,以便通过采用双重加权剩余法,从而获得多个目标功能的事后误差估计,然后利用神经网络计算原始和联合解决方案。本项研究显示,这种以数据驱动的模型的学习,即使以相对较少的培训数据为根据,其兴趣程度也较高。新的算法发展得到了数字测试示例的证实。在浅线网络上使用深神经网络的优势得到了证明,并展示了增强趋同的技术。