Solving well known physical problems modelled by partial differential equations by Machine Learning (ML) techniques remains a difficult task, and a lot of questions of a methodological nature need to be answered. The purpose of this project is to develop new solution methods at the interface of ML and scientific computing for coupled complex models in advanced manufacturing. This is a collaborative project with the Advanced Forming Research Centre (AFRC).

Unlike in image processing, for many complex scientific and engineering problems there is usually a limited availability of experimental data and for this reason using data driven machine learning (ML) tools and algorithms (working under the assumption of the availability of a large amount of data) is not an option. The development of ML techniques where sparse data is supplemented with well-established physical and numerical models, has only recently gained attention giving rise to a new paradigm: Scientific Machine Learning (SciML). Computing observables of complex numerical simulations, solving high-dimensional parametrised PDEs, data assimilation and solving inverse problems, or reconstructing incomplete physics were all made possible thanks to the use of neural networks complemented in part with data and in part with mathematical models and physical laws.

However, solving well known physical problems modelled by partial differential equations by ML techniques remains a difficult task, and a lot of questions of a methodological nature need to be answered. The purpose of this project is to develop faster solution methods at the interface of ML and scientific computing for complex, non-linear and time-dependent models (and improve existing models with incomplete physics) and apply these to problems in advanced manufacturing. These are typically coupled multi-physics fluid dynamics problems for which state-of-the art numerical methods are not satisfactory.

This is a collaborative project with the Advanced Forming Research Centre (AFRC).

In the whole process, besides the practical aspect, methodological aspects will be tackled/improved, i.e. numerical modelling of non-linear PDEs, fast solution methods in the optimisation process, testing of different neural networks and their combination with reduced order modelling for non-linear problems (which remains a challenge).

The ideal candidate will have a strong background in some of the following areas:  linear algebra, numerical analysis, optimisation and/or machine learning.  Experience in programmng (eg MATLAB, Python or Julia) is highly desirable.

Applicants should have, or be expecting to obtain in the near future, a first class or good 2.1 honours degree (or equivalent) in mathematics or a mathematical science. An MMath, MPhys or MSc degree is desirable.

Interested candidates are strongly encouraged to contact the first supervisor, Prof Victorita Dolean, (victorita.dolean@strath.ac.uk) for informal discussions about the project.

The successful candidate will receive comprehensive research training related to all aspects of the research and opportunities to participate in conferences, workshops, and seminars to develop professional skills and their research network.