Optimal prediction of dynamical systems with incomplete data

About the Project

Dynamical systems describe a variety of real-life problems related to evolution over time. It is often the case that the problems are so complicated that mathematical models for them either do not exist or are not accurate enough. Nowadays, there are data-driven techniques that allow us to obtain a mathematical model from measurement data over time. One of such techniques is the method of dynamic mode decomposition (DMD). It was originally proposed by Schmid (2010) as a data-processing algorithm. Then, it was realized that DMD can be immediately derived from the Koopman analysis.

The key idea of the Koopman theory is that any nonlinear dynamical system can be made linear if we extend the space of dependent (or state) variables via their nonlinear functions, called the observables. In that space, the problem becomes linear but the dimension of space is infinite. The Koopman operator provides the dynamics (trajectory) of any observable. This operator is unknown and can be approximated from the measurement data. DMD provides a linear approximation of the Koopman operator. As a result, this approach allows us to identify key modes. As such, one can obtain a low-rank approximate solution that can be used for both the analysis of the system and its prediction. Thus, DMD is a factorization and low dimensionality algorithm. There are numerous variants of DMD. One of them, called the higher-order DMD (HODMD) is based on the generalization of the Koopman operator with delayed snapshots. It has been demonstrated that for a variety of problems, HODMD is capable of significantly increasing the accuracy of prediction.

A key problem with the application of DMD to real-life problems is related to incomplete data. The measurement data is practically always limited. In addition, as a rule, this data is noisy and inaccurate. A recently developed modification of DMD based on the optimal prediction (Katrutsa et al, 2023) is capable of taking into account the effect of uncertainties in the input data in the most optimal way. In the project, this approach is supposed to be extended to HODMD with a higher accuracy of approximation in the algorithm to make it more accurate and efficient. It is supposed to be applied to the analysis of electric power grids. One of the advantages of the algorithm is that it can be potentially applied to a variety of problems since it is entirely based on the input data.

Eligibility

Applicants should have, or expect to achieve, at least a 2.1 honours degree or a master’s (or international equivalent) in a relevant science or engineering related discipline.

Funding

This 3.5-year PhD is for self-funded students. Exceptional candidates will be considered for Faculty funding (this will include an annual tax-free stipend of £20,780 and tuition fees will be paid. We expect the stipend to increase each year).

At The University of Manchester, we offer a range of scholarships, studentships and awards at university, faculty and department level, to support both UK and overseas postgraduate researchers.

For more information, visit our funding page or search our funding database for specific scholarships, studentships and awards you may be eligible for.

Before you apply

We strongly recommend that you contact the supervisor for this project before you apply. Please include details of your current level of study, academic background and any relevant experience and include a paragraph about your motivation to study this PhD project.

How to apply

Apply online through our website: https://uom.link/pgr-apply-2425

When applying, you’ll need to specify the full name of this project, the name of your supervisor, if you already having funding or if you wish to be considered for available funding through the university, details of your previous study, and names and contact details of two referees.

Your application will not be processed without all of the required documents submitted at the time of application, and we cannot accept responsibility for late or missed deadlines. Incomplete applications will not be considered.

After you have applied you will be asked to upload the following supporting documents:

• Final Transcript and certificates of all awarded university level qualifications
• Interim Transcript of any university level qualifications in progress
• CV
• Supporting statement: A one or two page statement outlining your motivation to pursue postgraduate research and why you want to undertake postgraduate research at Manchester, any relevant research or work experience, the key findings of your previous research experience, and techniques and skills you’ve developed. (This is mandatory for all applicants and the application will be put on hold without it).
• Contact details for two referees (please make sure that the contact email you provide is an official university/work email address as we may need to verify the reference)
• English Language certificate (if applicable)

If you have any questions about making an application, please contact our admissions team by emailing FSE.doctoralacademy.admissions@manchester.ac.uk.