Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than standard feed-forward neural networks, recurrent neural networks, or convolutional neural networks. This has had a great impact in the area of mathematical modeling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering. In this work, we review such methods as well as their extensions for parametric studies and for solving the related inverse problems. We equally proceed to show their relevance in some industrial applications.

翻译：近年来，深度学习的数学理论——旨在用数学深入理解深度学习概念并探索其鲁棒性——以及基于深度学习的数学研究——即利用深度学习算法解决数学问题——均取得了显著发展。后者推动了科学机器学习领域的普及，其中深度学习被应用于科学计算问题。具体而言，越来越多专门针对特定类别偏微分方程的神经网络架构被提出。这类方法利用偏微分方程的固有特性，在求解方程时优于标准前馈神经网络、循环神经网络或卷积神经网络。这对数学建模领域产生了重大影响，其中参数化偏微分方程被广泛用于模拟科学与工程中大多数自然及物理过程。本文综述了此类方法及其在参数化研究和相关反问题求解中的扩展应用，同时展示了它们在部分工业应用中的相关性。