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Deep learning based numerical methods for quantum systems


Department of Mathematical Sciences

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Dr Pranav Singh No more applications being accepted Competition Funded PhD Project (European/UK Students Only)
Bath United Kingdom Applied Mathematics Computational Mathematics Computational Physics Machine Learning Operational Research Quantum Mechanics

About the Project

The University of Bath is inviting applications for the following PhD project commencing in October 2021.

Funding is available to candidates who qualify for ‘Home’ fee status. Following the UK’s departure from the European Union, the rules governing fee status have changed and, therefore, candidates from the EU/EEA are advised to check their eligibility before applying. Please see the Funding Eligibility section below for more information.

This project will focus on developing and analysing novel numerical methods for equations of quantum mechanics such as the Schrödinger, Dirac and Liouville-von Neumman equations. Numerical solutions of these equations are required for predicting the dynamics of atoms, molecules, and spins, and have applications in atomic physics, optical fibres, quantum biology and quantum computing, among others.

The numerical solution of these equations is extremely challenging due to the highly oscillatory nature of solutions, long temporal windows of integration, a need for high accuracy to resolve subtle quantum effects, and a requirement for conserving physical quantities such as mass, energy, and angular momentum. These requirements are met to various degrees by specialised methods such as exponential splittings and Krylov subspace methods, which are designed carefully using algebraic and analytic means. Nevertheless, some of the most challenging applications remain intractable.

In stark contrast to these carefully crafted and analysed numerical schemes are deep neural networks, which have recently emerged as promising contenders for efficiently solving partial differential equations. However, the current approaches are black box approaches that have proven difficult to analyse, are found to lack stability, the solutions typically do not respect conservation laws, and come with little to no guarantee of accuracy.

This project will develop a new class of numerical methods by combining the flexibility of deep learning approaches with the more traditional analytic and algebraic techniques for developing specialised numerical methods for quantum systems. The aim will be to design, implement and analyse methods that are efficient but come with provable conservation properties and convergence guarantees. The student will be expected to develop a good understanding of techniques for designing and analysing existing numerical methods, explore new deep learning architectures motivated by these, utilise state-of-art optimisation and training strategies, and extend existing analysis techniques for providing convergence guarantees for the resulting numerical solutions.

Candidate Requirements:

Applicants should hold, or expect to receive, a First Class or high Upper Second Class UK Honours degree (or the equivalent qualification gained outside the UK) in Mathematics or closely related disciplines with strong mathematical component and background in Numerical Analysis. Background in deep learning, optimisation, or quantum mechanics will be advantageous. A master’s level qualification would also be advantageous. The candidate should have good programming skills. 

Non-UK applicants must meet our English language entry requirement.

Enquiries and Applications:

Informal enquiries are welcomed and should be directed to Dr Pranav Singh ([Email Address Removed]).

Formal applications should be made via the University of Bath’s online application form for a PhD in Mathematics (full-time).

More information about applying for a PhD at Bath may be found on our website.

Funding Eligibility:

In order to be considered for a studentship, you must qualify as a ‘Home’ student. In determining ‘Home’ student status, we follow the UK government’s fee regulations and guidance which, when available, will be set out by the UK Council for International Student Affairs (UKCISA) on their website.  At the time of advertising this project, the fee regulations for 2021/22 have not yet been published, but we expect (subject to confirmation) that the main categories of students generally eligible for ‘Home’ fee status will be:

  • UK nationals (who have lived in the UK, EU, EEA or Switzerland continuously since September 2018)
  • Irish nationals (who have lived in the UK or Ireland continuously since September 2018)
  • EU/EEA applicants with settled status in the UK under the EU Settlement Scheme (who have lived in the UK continuously since September 2018)
  • EU/EEA applicants with pre-settled status in the UK under the EU Settlement Scheme (who have lived in the UK, EU, EEA, Switzerland or Gibraltar continuously since September 2018)
  • Applicants with indefinite leave to enter/remain in the UK (who have been resident in the UK continuously since September 2018)

EU/EEA citizens who live outside the UK are unlikely to be eligible for ‘Home’ fees and funding.

Additional information may be found on our fee status guidance webpage, on the GOV.UK website and on the UKCISA website


Funding Notes

A studentship includes ‘Home’ tuition fees, a stipend (£15,285 per annum, 2020/21 rate) and research/training expenses (£1,000 per annum) for up to 3.5 years.


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