Deep learning methods for partial differential equations and related parameter identification problems

• Published in: Inverse problems Vol. 39; no. 10; pp. 103001 - 103075
• Main Authors: Nganyu Tanyu, Derick, Ning, Jianfeng, Freudenberg, Tom, Heilenkötter, Nick, Rademacher, Andreas, Iben, Uwe, Maass, Peter
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.10.2023
• Subjects: deep learning; inverse problems; inverse problems for PDEs; neural networks; neural operators; parameter identification; partial differential equations
• ISSN: 0266-5611; 1361-6420
• DOI: 10.1088/1361-6420/ace9d4

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 (NN) 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 NNs, recurrent NNs, or convolutional neural networks. This has had a great impact in the area of mathematical modelling 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 also show their relevance in various industrial applications.