Integrating Neural Networks for Enhanced Solving of PDEs Full-time

CEDoc – UM6P – College of Computing: Integrating Neural Networks for Enhanced Solving of PDEs

Mohammed VI Polytechnic University is an institution dedicated to research and innovation in Africa and aims to position itself among world-renowned universities in its fields The University is engaged in economic and human development and puts research and innovation at the forefront of African development. A mechanism that enables it to consolidate Morocco’s frontline position in these fields, in a unique partnership-based approach and boosting skills training relevant for the future of Africa. Located in the municipality of Benguerir, in the very heart of the Green City, Mohammed VI Polytechnic University aspires to leave its mark nationally, continentally, and globally.

Context:

The search for more efficient and accurate computational methods of solving partial differential equations is of paramount importance to the development of very many scientific and engineering disciplines. Conventional numerical methods like finite difference, finite volume and finite element techniques are the cornerstone of computational research but are inherently expensive, with computational costs typically increasing dramatically as problems transition from one dimension to many and parameter spaces become highly complex. The emergence of neural network technologies heralds a sea change in scientific computing, wherein not only the current state-of-the-art methods will be augmented but also the very approach to solving complex mathematical problems.

Research objectives:

The aim of the thesis is to go deeper in recent advances in neural network methodologies, specifically Physics-Informed Neural Networks and neural operators (Fourier neural operators, DeepONets, …). Already, these methods have achieved significant success, going all the way from fluid dynamics to biomedical engineering. Research will further investigate, refine, and adapt these neural network approaches for robust, scalable solutions with excellence in speed, accuracy, and generality compared to traditional methods. The focus will be on establishing a comprehensive understanding of how diverse neural network models can be effectively utilized and expanded across various challenging scientific and technical tasks. With the latest machine learning and deep learning techniques, this project is committed to creating models that simulate as much as they explain complex systems, shifting the paradigm toward the integrated data-driven approach in computational science.

Selection criteria:

• A master’s degree in applied maths or equivalent.
• Aptitude for teamwork, problem solving, and collaborative relationships.
• Strong technical background in at least one of the following areas: Mathematical Modeling, High performance Computing, Numerical Methods

References:

[1] Wu, Guozheng, Fajie Wang, and Lin Qiu. “Physics-informed neural network for solving Hausdorff derivative Poisson equations.” Fractals 31.06 (2023): 2340103.

[2] Li, Zongyi, et al. “Neural operator: Graph kernel network for partial differential equations.” arXiv preprint arXiv:2003.03485 (2020).

[3] Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., & Anandkumar, A. (2020). Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895.

[4] Wang, Sifan, Hanwen Wang, and Paris Perdikaris. “Learning the solution operator of parametric partial differential equations with physics-informed DeepONets.” Science advances 7.40 (2021): eabi8605

UM6P.