Deep Learning Methods for Partial Differential Equations and Related
Parameter Identification Problems
- URL: http://arxiv.org/abs/2212.03130v2
- Date: Tue, 16 May 2023 16:53:53 GMT
- Title: Deep Learning Methods for Partial Differential Equations and Related
Parameter Identification Problems
- Authors: Derick Nganyu Tanyu, Jianfeng Ning, Tom Freudenberg, Nick
Heilenk\"otter, Andreas Rademacher, Uwe Iben, and Peter Maass
- Abstract summary: 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.
- Score: 1.7150329136228712
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: 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.
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