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Parallel-in-time and consistent training methods for solving hyperbolic PDEs via physics informed neural networks


   Department of Mathematics

   Friday, November 25, 2022  Competition Funded PhD Project (Students Worldwide)

About the Project

In recent years, neural networks (NNs) have been applied in various fields of numerical modelling to learn solutions of mathematical models from data rather than by approximating equations and solving them by standard numerical methods. Learning a model for a system of interest from data alone is particularly advantageous when there is no understanding of the underlying processes and/or when no relevant mathematical equations are available. However, for many physical systems, not only has there been decades of research on how to accurately describe them mathematically, but also there is very important information encoded in these formulations. Hence, it would be a pity not to use this information which is already available about the system and just train the NNs from data. Based on this philosophy, various approaches have been introduced in the literature [Ra19, Bi22] in which the training of NNs is able to take partial differential equations (PDEs) into account. This project aims at contributing to this field by increasing both accuracy and efficiency of NNs by enhancing their training.

More specifically, this project intends to develop AI algorithms based on conservative machine learning approaches that apply physics-informed neural networks (PINNs) [Ra19, Bi22]. Another aspect of the project is to address the efficiency of training of PINNs that describe hyperbolic PDEs. We will aim at significantly improve the training efficiency of PINNs that are used to describe the solutions of PDEs. Similarly to time slices in hyperbolic PDEs, we will train the layers of multiple PINNs parallel-in-time (PIT) as this is expected to deliver significant impact on the efficiency and scalability of training AI algorithms. Here, we intend to explore the ParaDIAG idea [Ga20] in combination with training PINNs that describe time-dependent hyperbolic PDEs.

Supervisor: Werner Bauer

This project is open to UK and international students starting in January 2023. Later start dates may be possible.

Entry requirements

Applicants should have a minimum of a first class honours degree in mathematics, the physical sciences or engineering. Preferably applicants will hold a MMath, MPhys or MSc degree, though exceptional BSc students will be considered.

English language requirements: IELTS minimum 6.5 overall with 6.0 in Writing, or equivalent.

How to apply

Applications should be submitted via the Mathematics PhD programme page on the "Apply" tab.

Please state clearly the studentship project that you would like to apply for and the name of your intended supervisor.


Funding Notes

The Department of Mathematics has a number of fully funded PhD studentships for UK nationals. These studentships will include the tuition fees and a tax-free stipend in line with the standard UKRI rate (£15,609 p.a. for 2021-22). The Department has also a few scholarships for partial and full funding for overseas fees for exceptional applicants. However, funding for overseas students is limited and applicants are encouraged to find suitable funding themselves.

References

[Bi22] Bihlo etal. [2022], Physics-informed neural networks for the shallow-water equations on the sphere, J. Comp. Physics Journal of Computational Physics, 456, 111024.
[Ga20] Gander etal. [2020], ParaDiag: parallel-in-time algorithms based on the diagonalization technique, arXiv:2005.09158.
[Ra19] Raissi etal. [2019], Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comp. Physics, 378, 686-707.

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