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Machine Learning and Domain Decomposition methods for Fluid Dynamics


   Faculty of Engineering, Computing and the Environment

  Dr Michal Bosy, Dr G Hunter  Applications accepted all year round  Self-Funded PhD Students Only

About the Project

Modelling of many modern applications leads to linear systems whose size is too large to allow the use of direct solvers. Thus, parallel solvers are becoming increasingly important in scientific computing. A natural paradigm to take advantage of modern parallel architectures is the Domain Decomposition method that uses iterative solvers based on a decomposition of a global domain into subdomains. At each iteration, one (or two) boundary value problem(s) are solved in each subdomain and the continuity of the solution at the interfaces between subdomains is only satisfied at convergence of the iterative procedure.

Another powerful and quickly developing tool is Machine Learning. The combination of Machine Learning and Domain Decomposition methods can be used in various way. On the one hand, Machine Learning techniques are used to improve convergence or the computational efficiency [1]. On the other hand, “deep” neural networks are used as discretization methods [2, 3]. Considering potential advantages and fitting the most suitable approach for Fluid Dynamic systems, we will propose an efficient solver based on combination of appropriate Machine learning technique and Domain Decomposition method.

This project would suit graduates in the areas of computing, mathematics, physics or engineering, and would provide an excellent training in mathematical and computational modelling. If you are interested in being involved in developing a leading-edge system that has multiple practical applications, then you should apply. Please feel free to contact either of the supervisors via email for an informal discussion.

Candidates must have a First or Upper Second Class Honours degree;, and should have experience of programming in at least one high-level computer language, such as C/C++/C# or Python, and have some knowledge of Finite Element methods. Some background in Numerical Analysis, Fluid Dynamics or Machine Learning would be an advantage. All enquiries would be welcome, and we are keen to ensure that equality, diversity and inclusion are manifested in our selection of candidates.


References

[1] Heinlein, A., Klawonn, A., Lanser, M., & Weber, J., "Machine Learning in Adaptive Domain Decomposition Methods---Predicting the Geometric Location of Constraints." SIAM Journal on Scientific Computing 41.6 (2019): A3887-A3912.
[2] Li, K., Tang, K., Wu, T., & Liao, Q., "D3M: A deep domain decomposition method for partial differential equations." IEEE Access 8 (2019): 5283-5294.
[3] Li, W., Xiang, X., & Xu, Y., "Deep domain decomposition method: Elliptic problems." Mathematical and Scientific Machine Learning. PMLR, (2020): pp. 269-286.
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