Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems
This work addresses the problem of solving PDEs and related inverse problems for researchers and practitioners in scientific computing and mathematical modeling, but it is incremental as it primarily reviews and extends existing methods.
The paper reviews neural network architectures developed to solve specific classes of partial differential equations (PDEs), exploiting inherent PDE properties to outperform standard neural networks, and extends these methods for parametric studies and inverse problems with applications in industrial settings.
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.