Dear Colleagues,

Partial differential equations (PDEs) are the main tool of modeling in a wide range of applications from physics via economics and epidemiology to computer vision and graphics. Deep learning (DL) emerged in the last decade as the best tool for many scientific and technology tasks. While the success of DL is huge, our understanding, control, and design of the neural networks (NN) involved is very shallow.

In this Special Issue, we invite two types of research papers. One type is the use of PDEs to enhance understanding of NN and DL. In particular, we are interested in viewing NN as a discrete approximation of a PDE in a high dimension, but any other mathematical modeling of DL and NN that improves our understanding and control over these technologies is most welcome. The second type of research we are seeking is numerical solution methods of PDEs and inverse problems (IP) via the use of NN and DL. In particular, we are interested in the ways that this new emerging method which solves PDEs and IP can be analyzed to bound the approximation error as a function of the NN architecture. Inverse problems are the most important types of problems of applied value, and it is ubiquitous in medical imaging and environmental studies among other domains of interest with a huge impact on our daily life. The possibility to design a good numerical machinery that can solve these types of problems on different domains and high dimensions with proven control of the error can revolutionize many domains in science and technology. We aim to have a Special Issue that advances the state of the art in this important domain of research and be a good basis for further rapid advancement in this field.

Prof. Dr. Nir Sochen
Dr. Leah Bar
Guest Editors

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• partial differential equations
• inverse problems
• deep learning
• neural networks
• numerical analysis