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Crystal Structure Generation by Constrained Combinatorial Sampling

Funding
Funded
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Full-time
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Subject area
Computer Science
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Overview

The goal of this project is to rethink crystal structure prediction (CSP) using logic-based constrained sampling to generate chemically sensible, unbiased starting structures. We will work at the interface of chemistry, AI, formal reasoning and optimisation, building and testing samplers, integrating them into state-of-the-art material discovery workflows, and using them against and together with generative AI to explore structural space more efficiently.

About this opportunity

Designing new materials by predicting stable crystal structures from composition is a central goal in materials science and one of the key promises of AI for Science initiatives. Mathematically, Crystal Structure Prediction (CSP) can be seen as an optimisation problem asking to allocate atoms in space while minimising their interaction energy.It is an extremely challenging class of problems: the underlying quantum-mechanical interactions give rise to a vast, high-dimensional energy landscape with an enormous number of possible structures and many competing local minima.

Practical CSP workflows evolve a population of structures toward energy minima [1]. Initial structures are typically generated by sampling random structures and keeping only those that are “sensible” (realistic densities, interatomic distances, coordination) [2], but this is costly and becomes impractical as more constraints are enforced. Alternatively, structures are assembled from chemically plausible modules or obtained by generative AI [3], but these methods can introduce uncontrolled bias and provide no guarantee of even coverage across structural space.

This project treats structure generation as a combinatorial problem [4] and builds on advances in constrained sampling [5] to obtain chemically sensible structures. We will provide a flexible, uniform way to express chemistry constraints in formal logic; efficiently and (approximately) uniformly sample structures satisfying these constraints; integrate these samplers into CSP pipelines to accelerate convergence and broaden exploration; and quantitatively evaluate coverage and starting energies, including comparisons with generative AI using distribution testing. The student will work at the chemistry/computer science interface, gaining experience in AI, machine learning, automated reasoning, and structural chemistry, while contributing to the next-generation CSP methods. The project is multi-disciplinary, and we specifically welcome students with backgrounds in mathematics, physics, chemistry, engineering, and computer science. We do not expect the student to be familiar with all the topics outlined above. We will facilitate additional training and evolve the project with the student skills in mind as we have successfully done in the past.

This project will be supervised by Dr Friedrich Slivovsky (School of Computer Science & Informatics), Dr Vladimir Gusev (School of Computer Science & Informatics) and Dr Matthew Dyer (Department of Chemistry).

Dr Slivovsky is an expert in satisfiability (SAT) solving, the core technology behind our constrained sampling approach. He has advanced both the theory and practice of SAT solving and co-chaired the leading SAT conference, ensuring the student has experienced mentorship on algorithms, encodings, and solver engineering.

Dr Dyer has a strong record of applying computer-science methods within inorganic materials chemistry and of developing new approaches to crystal structure prediction. He is a co-investigator in major UK initiatives in digital materials discovery and a supervisor in the CDT in Digital and Automated Materials Chemistry, with active industrial collaborations and patents in materials discovery.

Dr Gusev is an interdisciplinary scientist at the CS-chemistry interface, with high-impact publications on optimisation for CSP and AI-driven formulation in robotic workflows, and a broad track record across theoretical computer science, chemistry, and AI venues. He has played a leading role in building collaborations between computer science and chemistry, exemplified by joint projects, community initiatives, and co-supervised PhDs.

This project is expected to start in October 2026 and is offered under the EPSRC Centre for Doctoral Training in Digital and Automated Materials Chemistry based in the Materials Innovation Factory at the University of Liverpool, the largest industry-academia colocation in UK physical science. The successful candidate will benefit from training in robotic, digital, chemical and physical thinking, which they will apply in their domain-specific research in materials design, discovery and processing. PhD training has been developed with 35 industrial partners and is designed to generate flexible, employable, enterprising researchers who can communicate across domains.

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Who is this for?

Candidates will have, or be due to obtain, a Master’s Degree or equivalent related to Applied Mathematics, Computer Science, or Computational Chemistry. Exceptional candidates with a First Class undergraduate degree or equivalent in an appropriate field will also be considered.

The minimum English Language requirements for international candidates is IELTS 6.5 overall (with no band below 5.5) or equivalent. Find out more about English language requirements.

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How to apply

  1. 1. Contact supervisors

    We strongly encourage candidates to get in touch with the supervisory team to get a better idea of the project before making a formal application online. Any informal enquiries about the project can be directed to F.Slivovsky@liverpool.ac.uk.

    Supervisors:

    Dr Friedrich Slivovsky F.Slivovsky@liverpool.ac.uk https://www.liverpool.ac.uk/people/friedrich-slivovsky
    Dr Vladimir Gusev Vladimir.Gusev@liverpool.ac.uk https://www.liverpool.ac.uk/people/vladimir-gusev
    Dr Matthew Dyer msd30@liverpool.ac.uk https://www.liverpool.ac.uk/people/matthew-dyer
  2. 2. Prepare your application documents

    Review our CDT guide on “How to Apply carefully as it may differ from a standard application process.

  3. 3. Apply

    Finally, register and apply online. You'll receive an email acknowledgment once you've submitted your application. We'll be in touch with further details about what happens next.

    You’ll receive an email acknowledgment once you’ve submitted your application. We’ll be in touch with further details about what happens next.

    Candidates are strongly encouraged to apply early before the deadline and indicate the subject area as Computer Science in your application. This position will remain open until a suitable candidate has been found.

    We want all our Staff and Students to feel that Liverpool is an inclusive and welcoming environment that actively celebrates and encourages diversity. We are committed to working with students to make all reasonable project adaptations including supporting those with caring responsibilities, disabilities or other personal circumstances.

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Funding your PhD

The EPSRC DAMC CDT Studentship will cover full home tuition fees and a maintenance grant for 4 years starting at the UKRI minimum (for the 2025-26 academic year this was £5,006 pa tuition fees and £20,780 pa maintenance grant; rates for 2026-27 academic year TBC). The Studentship also comes with a Research Training Support Grant to fund consumables, conference attendance, etc.

Studentships are available to any prospective student wishing to apply including both home and international students. While EPSRC funding will not cover international fees, a limited number of scholarships to meet the fee difference will be available to support outstanding international students.

If you have a disability you may be entitled to a Disabled Students’ Allowance on top of your studentship to help cover the costs of any additional support that a person studying for a doctorate might need as a result

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Contact us

Have a question about this research opportunity or studying a PhD with us? Please get in touch with us, using the contact details below, and we’ll be happy to assist you.

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