MAC-MIGS 2024+: Projects 2025

The familiar waves at the surface of the sea are crucial to all forms of air-sea interactions and hence have a strong impact on our climate. They are also important for human activities, commercial (shipping, off-shore turbines) and recreational (sailing, surfing). Open questions about sea surface waves concern the factors which control the spatial distribution of wave energy (proportional to the root-mean-square of the sea-surface height). Recent advances, combining observations, numerical and theoretical modelling, have shown that surface currents play a major role in shaping the spatial distribution of wave energy, through advection and refraction. This motivates the present project.

The project will examine aspects of wave-current interactions, based on a phase-averaged description of the wave fields. This description relies on a partial differential equation governing the dynamics of wave energy in a four-dimensional phase space capturing position, wave frequency and direction. The high dimensionality poses exciting challenges for both analytical and numerical modelling. The project can take several directions, depending on the student’s interests including: the development of a theory that relates wave energy to currents (extending recent work to capture wave groups and/or applying ideas of Hamiltonian averaging and homogenisation), the study of feedback of the waves on the currents that modulate them, and the design of efficient numerical methods for the solution of the wave energy equation. The project will benefit from an ongoing collaboration with oceanographers at the University of Hamburg, University of California at San Diego, and Colorado School of Mines.

Related references

Wang et al 2024, Scattering of ocean surface waves by currents: the U2H map, https://arxiv.org/abs/2402.05652

Wang et al 2023, Scattering of swell by currents, https://arxiv.org/abs/2305.12163

This project will take advantage of recent technological advances in differentiable programming, some of which have been developed for applications in machine learning, and apply these to tackle problems arising in physical systems with non-local interactions.

Such non-local interaction problems include the flow of particle-laden fluids in complex geometries, the evaporation of droplets, flocking and swarming of animals, pedestrian and vehicle dynamics, and opinion formation. Crucially the partial differential equations contain integral convolution terms, and these terms may pose a challenge for differentiable programming tools.

This project will apply a range of cutting edge technologies, such as the Firedrake finite element code generation library which implements high-level differentiable programming, and will also apply more general purpose differentiable programming tools such as JAX.

Successful codes could be applied to study problems from a range of application areas, which could include: complex fluids in biology, engineering, and physics; industrial processes such as mixing and demixing of fluids and sedimentation of particles; and formation of opinions, flow of cars and pedestrians, and disease control in social science.

Rathgeber et al, Firedrake: Automating the finite element method by composing abstractions, ACM Transactions on Mathematical Software 43(3), 2016

Frostig et al, Compiling machine learning programs via high-level tracing, SYSML’18, 2018

The ocean is an enormously complicated turbulent fluid. A crucial problem in the ocean is that important large scale properties, such as the transports of climate relevant quantities such as heat and carbon, depend upon the details of the smaller scale turbulence — but it is very challenging to simulate ocean turbulence directly using numerical models.

Significant effort has been invested in studying ocean turbulence on just the largest scales — the ocean ‘mesoscale’. This project will investigate theories and closures for the ocean mesoscale, and see how and if these can extended to higher resolution models, or models which represent smaller scale processes (the ‘submesoscale’). The key approach will be to consider idealized numerical experiments, which are cheap enough to run at higher resolution, but complicated enough that key processes can be introduced and studied in detail.

Bachman et al, Ocean Modelling 109, pp. 44-54, 2017,
https://doi.org/10.1016/j.ocemod.2016.12.004

Taylor and Thompson, Annual review of fluid mechanics 55, 2023,
https://doi.org/10.1146/annurev-fluid-031422-095147

The rise of machine learning has had a tremendous impact on computational science, although one under-exploited area is the scope for automatic differentiation (AD) to revolutionise “traditional” numerical solvers [e.g. see 1]. Unlike standard machine learning models, AD has no impact on interpretability but allows us to approach complex nonlinear problems in a new way, with a clear methodology to search for trajectories of interest via gradient-based optimisation. This project will explore the use of AD in building robust low-order (N-body) models to explain observations in both large scale turbulence and stellar dynamics.

The student will first adapt an N-body point-vortex code, originally used to find vortex crystals in superfluids [2], to explore the role of unstable vortex crystals in upscale energy transfer in full two-dimensional Navier-Stokes turbulence, both in the plane and on the surface of a sphere. The hope would be to build a simple symbolic dynamical picture with a small handful of relevant point-vortex states.

Similarly, this approach can be exploited also to search for new equilibria in low-order systems of bodies interacting by gravitational forces. Elegant periodic solutions have been occasionally discovered [3], but further progress has been hampered by the prohibitive computational cost of traditional numerical methods. This project, therefore, carries significant potential to extend to a new regime the insightful analogy [4] between the properties of two-dimensional point vortices and three-dimensional self-gravitating systems, as initiated by Onsager [5] and Chandrasekhar [6].

Requirements: Strong undergraduate background in mathematics, physics or engineering and with programming experience, preferably in an object-oriented language. Previous exposure to with machine learning libraries (e.g. JAX/TensorFlow) is a plus.

[1] Kochkov et al, “Machine learning-accelerated computational fluid dynamics”, Proc. Nat. Acad. Sci. 118 (2021)

[2] Cleary & Page, “Exploring the free-energy landscape of a rotating superfluid” (arXiv:2306.10870)

[3] Chenciner & Montgomery, “A remarkable periodic solution of the three-body problem in the case of equal masses”, Ann. of Math. 2, 152 (2000)

[4] Chavanis, Sommeria, Robert, “Statistical Mechanics of Two-Dimensional Vortices and Collisionless Stellar Systems”, Astrophys. J. 471, 1, (1996)

[5] Onsager, “Statistical hydrodynamics”, Nuovo Cimento Suppl. 6, 279 (1949)

[6] Chandrasekhar, “Principles of stellar dynamics”, Dover, (1942)

Efficient mixing of fluids is vital in a variety of industrial settings, for example in drug development/discovery or in building ventilation. At low Reynolds numbers one must rely on chaotic advection to mix [1], while mixing in high Reynolds number flows is aided by the presence of turbulence. This project will seek optimal mixing strategies in parallel shear flows using a time-dependent forcing (either pressure gradient or boundary oscillation). The mixing strategies will be determined via a nonlinear optimisation algorithm the student will implement within the Dedalus codebase [2]. There is a strong Dedalus community at the University of Edinburgh and scope for a range of collaborations over the course of this project.

Nonlinear optimisation in fluids has been effective in determining “optimal” initial conditions to trigger transition to turbulence (so-called minimal seeds, [3]). There are many subtleties in extending this approach to mixing, including the definition of an appropriate norm to quantify the mixing [4]. These will be explored by the student over the course of the project. Time permitting, we will also consider the application of these ideas to viscoelastic flows, where turbulent-like states can be triggered in the absence of inertia [5], and where the existence of chaos presents an exciting opportunity for new mixing strategies in micro-flow devices.  Moreover the methods being explored can be used to calculate the optimal flows for triggering dynamo action — the mechanism for generating the magnetic fields of the Earth and Sun.

[1] Aref, “Stirring by chaotic advection”, J. Fluid Mech. 143 (1984)

[2] Burns et al, “Dedalus: A flexible framework for numerical simulations with spectral methods”, Phys. Rev. Research 2 (2020)

[3] Kerswell, “Nonlinear nonmodal stability theory”, Ann. Rev. Fluid Mech. 50 (2018)

[4] Foures et al, “Optimal mixing in two-dimensional plane Poiseuille flow at finite Peclet number”, J. Fluid Mech. 748 (2014)

[5] Groisman & Steinberg, “Elastic turbulence ina polymer solution flow”, Nature 405 (2000)

Cerebrospinal fluid (CSF) is a clear, water-like fluid that surrounds the brain and spinal cord, providing protection and nourishment. It’s produced in the brain’s choroid plexus, flows through the brain’s ventricular system, and circulates around the brain and spinal canal. Eventually, it’s drained into the superior sagittal sinus and the venous system. The balance between CSF production and drainage controls the brain’s pressure. When this pressure becomes too high, it’s associated with a condition called hydrocephalus, which involves an excessive buildup of CSF.

The goal of this project is to create mathematical models that can help us better understand how CSF is produced, how it flows and drains; and relate it to pathological conditions. In this project, we’ll tackle simplified systems of second-order partial differential equations, ordinary differential equations, and algebraic equations. We may also use techniques like Bayesian inference and sensitivity analysis for parameter estimation and uncertainty quantification. Knowledge of continuum or fluid mechanics and partial differential equations is a desirable prerequisite for this project.

Kelley, D. H., & Thomas, J. H. (2023). Cerebrospinal fluid flow. Annual Review of Fluid Mechanics, 55(1), 237-264.

Vallet, A., Del Campo, N., Hoogendijk, E. O., Lokossou, A., Balédent, O., Czosnyka, Z., … & Schmidt, E. (2020). Biomechanical response of the CNS is associated with frailty in NPH-suspected patients. Journal of Neurology, 267, 1389-1400.

Linninger, A. A., Tangen, K., Hsu, C. Y., & Frim, D. (2016). Cerebrospinal fluid mechanics and its coupling to cerebrovascular dynamics. Annual Review of Fluid Mechanics, 48, 219-257.

For this project only, the candidate will not be based at the University of Edinburgh. The candidate will be based at the Culham Centre for Fusion Energy in Oxfordshire, with regular visits to the University of Edinburgh.

Characterisation of the transition from low (L) to high (H) confinement mode regimes in tokamak plasmas is a key step towards improving predictions for fusion power plants such as STEP. In the world leading JET tokamak, two density branches have been observed for the L-H transition as the input heating power is ramped up: (a) the low density branch for which the threshold power decreases with increasing density, and (b) the high density branch in which the converse is true. This observation suggests that the nature of the microinstabilities and turbulent transport changes in the L-mode edge of the tokamak depending on the local conditions.

There is no theory-based model that can reproduce the L-H transition. This project aims to develop a simulation informed model by using high fidelity nonlinear gyrokinetic simulations with the GENE code. To analyse the turbulent simulation data and place it in the context of JET Deuterium – Tritium experimental results, the candidate will use data decomposition techniques designed to either decompose signals into high and low-intensity components (proper orthogonal decomposition – POD), or, into spatio-temporally coherent structures (dynamic mode decomposition – DMD). Such techniques were recently applied to JET pedestals [B. Chapman-Oplopoiou et. al. Accepted in Physical Review Research] and are frequently used for model reduction in fluid dynamics and to understand the spatio-temporal structure of turbulent flows.

In summary, this project will exploit a dataset that is unique thanks to JET’s tritium handling capabilities, elucidating fundamental turbulence phenomena whilst contributing to the development of sustainable fusion energy.

Biological cells dynamically create, maintain, and disassemble cell surface structures that determine their shape and behaviour, such as directed migration. Mathematically, the study of the underlying biophysical mechanisms of intracellular morphogenesis — which includes biochemical reactions, molecular transport, and membrane dynamics, among others — frequently relies on the well-developed machinery of reaction-diffusion equations [1]. These equations can exhibit multiple-scale dynamics, giving rise to rich coherent structures that include travelling waves, excitable pulses, and spiral waves that exhibit complex bifurcations; the latter are frequently not well-understood mathematically [2,3].

In this project, the student will contribute to a rigorous mathematical understanding of bifurcations in systems of reaction-diffusion equations, building on recent advances in the study of pattern-forming models for intracellular morphogenesis. A first step could be the study of a “cartoon” ordinary differential equation model for activator, inhibitor, and depleted substrate that is based on the classic FitzhHugh-Nagumo equations. Ultimately, the aim of the project is a mathematically rigorous bifurcation analysis of pattern-forming reaction-diffusion models for intracellular morphogenesis in select, physiologically relevant, parameter regimes.

The project will give the student a solid foundation in the mathematical modelling of intracellular morphogenesis, with a particular focus on dynamical systems techniques, such as normal forms, invariant manifolds, geometric singular perturbation theory, and the desingularisation technique known as “blow-up”, that have proven fruitful in the study of multiple-scale dynamics [4].

No previous background on these topics is assumed, though experience in the analytical and numerical solution of ordinary and partial differential equations and some experience in coding is preferable.

[1] A. Michaud et al., A versatile cortical pattern-forming circuit based on Rho, F-actin, Ect2, and RGA-3/4, J. Cell Biol. 221(8), e202203017, 2022.

[2] L. Yang et al., Pattern formation arising from interactions between Turing and wave instabilities, J. Chem. Phys. 117(5), 7259-7265, 2002.

[3] A. Yochelis, C. Beta, and N.S. Gov, Excitable solitons: annihilation, crossover, and nucleation of pulses in mass-conserving activator-inhibitor media, Phys. Rev. E 101, 022213, 2020.

[4] C. Kuehn, Multiple Time Scale Dynamics, Applied Mathematical Sciences 191, 2015.

This project focuses on modelling and simulation for mixtures of liquids. Such mixtures not only allow us to probe fundamental questions in physics, but they also have a wide range of industrial applications. Particular mixtures of interest here are those of water, alcohol, and oil, which exhibit a range of behaviours depending on the precise proportions of the constituents. If the student is interested then there is an opportunity to perform simple experiments to inform and validate the modelling approaches, as has been done for other systems [1].

An everyday example of such mixtures can be seen while drinking common Mediterranean spirits, including ouzo and sambuca. The spirit sold in bottles is a clear, single-phase liquid consisting of around 60% water, 40% ethanol (alcohol), and a small amount of anise oil, which gives the drinks their distinctive taste. Water and oil are completely immiscible, but both are fully soluble in alcohol; alcohol has a strong preference for water. In pure Ouzo, there is sufficient alcohol to solubilise the oil, but when even a small amount of water is added, the alcohol partitions with the water, reducing the solubility of the oil and causing the drink to become cloudy. Recent work (combining modelling and experiment) has elucidated the mechanisms behind this behaviour [2], but there are many open questions around such mixtures.

The mathematical modelling is based on non-equilibrium statistical mechanics, which has its foundations in statistics and probability, and often uses stochastic and/or partial differential equations (SDEs/PDEs) to describe the dynamics of particles, liquids, and other systems. Here we focus on a well-established approach called Dynamic Density Functional Theory (DDFT), which has a wide range of applications ranging from colloid particles, to liquids (as studied here), cancer modelling, and even how people form and change their opinions [2].

This project will focus on two DDFT approaches, which are very closely related: one comes from lattice models and describes the dynamics of mixtures on liquids over a set of discrete sites. The second is a continuum description through (integro-)PDEs. The former is numerically more tractable, but has limitations such as a fixed length-scale, and poor scaling with the size of the domain. The second has challenges when trying to describe steep interfaces, which are common in the systems studied here, but scales much better with domain size. Part of this project will be to develop numerical schemes; we already have an extensive, efficient, and robust code base for each approach on which this project can build.

Main Aims: To understand, predict, and possibly control, the equilibria and dynamics of ouzo-like systems through a combination of modelling, numerics, and (if there is interest) experiments.

Useful Background: Interested students should have a strong background in one or more of: mathematical modelling; computation (preferably SDEs and/or PDEs); statistical mechanics. No experimental background is required.

[1] Changing the flow profile and resulting drying pattern of dispersion droplets via contact angle modification, C. Morcillo Perez, M. Rey, B. D. Goddard, and J. H. J. Thijssen, https://arxiv.org/abs/2111.00464

[2] Experimental and theoretical bulk phase diagram and interfacial tension of ouzo, A. J. Archer, B. D. Goddard, D. N. Sibley, J. T. Rawlings, R. Broadhurst, F. F. Ouali, and D .J. Fairhurst, Soft Matt., 2024

[3] Classical dynamical density functional theory: from fundamentals to applications, M. te Vrugt, H. Löwen, and R. Wittkowski, Adv. Phys., 69(2), 121-247, 2020

The Discrete Element Method (DEM) is a computational method, widely used in both industry and academia, primarily to model the dynamics of dry systems containing many small particles [1]. Typical examples include grains, minerals, powders, and pills. There is increasing industrial interest in extending the applicability of DEM to systems where the particles are suspended in a ‘liquid bath’. In particular, systems in which the particle density is high typically form pastes or slurries, which have a wide range of applications in healthcare, electronics, and other formulated products.

The two principle challenges that need to be overcome before DEM can be used to accurately model such systems are:

1. The presence of a liquid bath results in additional forces between the particles. In the systems considered here, the dominant terms arise from ‘lubrication’: for example, if one tries to push two particles together then (on a highly simplified level) one must do more work to also remove the intervening liquid. In many-particle systems, lubrication forces can qualitatively and quantitatively change the dynamics. [2]

2. Current models which couple DEM to a fluid bath through a Computational Fluid Dynamics solver are prohibitively computationally expensive for many applications. They are also generally restricted to the ‘Newtonian’ case where viscous stress is related linearly to strain, whereas slurries exhibit more complex and demanding ‘non-Newtonian’ behaviour, which is crucial to correctly understand their dynamics. [3]

Main Aims: This project can focus on one or both of these related areas. There is also the possibility to collaborate with an industrial partner, who are one of the world-leaders in DEM software development and production. The key applications are in green and sustainable areas, with a strong emphasis on increasing efficiency and decreasing resources, both for DEM itself, and in the application areas.

Useful Background: Interested students should have a strong background in one or more of: mathematical modelling; computation; fluid dynamics; statistical mechanics.

[1] Y. Guo, Yu, and J. S. Curtis, Discrete element method simulations for complex granular flows. Ann. Rev. Fluid Mech., 47(1), 21-46, 2015.

[2] B. D. Goddard, R. D. Mills-Williams, and J. Sun, The singular hydrodynamic interactions between two spheres in Stokes flow, Phys. Fluids, 32, 062001, 2020

[3] R. P. Chhabra, Non-Newtonian fluids: an introduction, In: J. Krishnan, A. Deshpande, P. Kumar (eds), Rheology of Complex Fluids, Springer, New York, NY, 2010

Active matter is a rapidly emerging field in theoretical and statistical physics which concerns itself with systems, such as living systems, that exist far from thermal equilibrium. Indeed, due to their rich behavior, there is a great potential in controlling the emergent collective states of active matter, living or artificial, to design and control optimized physical systems whose functions surpass passive — equilibrium — technology [1,2]. Such functions could include the transportation of material against chemical gradients and the sustaining of cyclic states that perform useful work. However, the non-equilibrium and noisy nature of active matter presents significant challenges to the framework of equilibrium statistical mechanics, and thus makes exploring design and control strategies difficult. To overcome these challenges, this project aims to leverage recent theoretical results [3], machine learning approaches in active matter [4], and automatic differentiation techniques, that have seen success in quantum optimal control problems [5], to optimize particle-based molecular dynamics simulations of active matter.

This project has the scope to cover in-depth theoretical/analytical and computational aspects, and the student can expect to gain significant experience in classical calculus of variations, non-equilibrium thermodynamics, statistical physics, computer simulations, and machine learning approaches.

1. Palacci, J. et al. Science 339, 936 (2013).
2. Falk, M. J. et al. Physical Review Research 3 (2021).
3. Davis, L. K. et al. Physical Review X 14 (2024).
4. Cichos, F. et al. Nat. Mach. Intell. 2, 94 (2020).
5. Leung, N. et al. Physical Review A 95 (2017)

University of Edinburgh (UoE) website

Statistical geometry of active matter

Most students get introduced to what is known as an equation-of-state when they are in high-school, in the form of the ideal gas law: PV = NKT (P is pressure, V is volume, N is particle number, K is Boltzmann’s constant, and T is temperature). Describing equilibrium and non-ideal gases requires additional, though involved, density corrections to the ideal gas equation of state [1]. However, for generic active matter, a wide class of intrinsically far-from-equilibrium physical systems, such as swimming bacteria and flocks of birds, an equation-of-state cannot even be written down in the usual equilibrium manner [2]. Recent attempts to do this often rely heavily on –difficult to obtain — many-body correlation functions or approximating the steady-state distribution [3]. A rather unexplored direction involves leveraging an exact connection between statistical geometry and the equation of state, derived for equilibrium hard-spheres [4], in the context of active systems.

The student will gain significant experience in fundamental non-equilibrium statistical physics, computer simulations of on- and off-lattice active systems, and stochastic processes.

1. Ter Haar, D. “Elements of Statistical Mechanics” (Elsevier, 1995).
2. Solon, A. P. et al. Nature Physics 11, 673–678 (2015).
3. Cameron, S. et al. Physical Review E 108 (2023).
4. Speedy, R. J. Journal of the Chemical Society, Faraday Transactions 2 77, 329 (1981).

University of Edinburgh (UoE) website

This project belongs to the broad field of applied stochastic analysis.

Context: Many systems of interest in the applied sciences share the common feature of possessing multiple scales, either in time or in space, or both.
Some approaches to modelling focus on one scale and incorporate the effect of other scales (e.g. smaller scales) through constitutive relations, which are often obtained empirically.
Multiscale modelling approaches are built on the ambition of treating both scales at the same time, with the aim of deriving (rather than empirically obtaining) efficient coarse grained models which incorporate the effects of the smaller/faster scales.
Multiscale methods have been tremendously successful in applications, as they provide both underpinning for numerics/simulation algorithms and modelling paradigms in an impressive range of fields, such as engineering, material science, mathematical biology, climate modelling (notably playing a central role in Hasselmann’s programme, where climate/ whether are seen as slow/fast dynamics, respectively), to mention just a few.

More detail. In this project, which is in the field of applied stochastic analysis, we will consider systems that are multiscale in time, with particular reference to multiscale interacting particle systems.
We will try to understand how the multiscale approximation interacts with the mean field approximation (produced by letting the number of particles in the systems to infinity so to obtain a PDE for the evolution of the density of the particles).
In the context we will consider the systems at hand will have two scales, so called fast and slow scale, each of them modelled by appropriate stochastic differential equations.
Classical multiscale paradigms consider the setting in which the fast scale has a unique invariant measure (equilibrium). We will consider the often more realistic scenario in which the fast process has multiple invariant measures.
The motivation for this project comes especially from models in mathematical biology, but the applicability of the framework we will investigate is broader.
The candidate that will work on this project will be open to investigate both theoretical and modelling aspects – though developing their own preference in time.

[1] https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2023.0322

[2] https://projecteuclid.org/journals/electronic-journal-of-probability/volume-26/issue-none/Averaging-in-the-case-of-multiple-invariant-measures-for-the/10.1214/21-EJP681.full

This project belongs to the broad field of applied stochastic analysis.

Context. Many systems of interest consist of a large number of particles or agents, (e.g. individuals, animals, cells, robots) that interact with each other.
When the number of agents/particles in the system is very large the dynamics of the full Particle System (PS) can be rather complex and expensive to simulate; moreover, one is quite often more interested in the collective behaviour of the system rather than in its detailed description (e.g. bird flocking).
In this context, the established methodology in statistical mechanics and kinetic theory is to look for simplified models that retain relevant characteristics of the original PS by letting the number N of particles to infinity (so called mean field limit);
the resulting limiting equation for the density of particles is a low dimensional, (in contrast with the initial high dimensional PS) non-linear partial differential equation (PDE), where the non-linearity has a specific structure, commonly referred to as a McKean-Vlasov nonlinearity.
Beyond an intrinsic theoretical interest, such models were proposed with the intent to efficiently direct human traffic, to optimize evacuation times, to study rating systems, opinion formation, etc; and in all these fields they have been incredibly successful.

More detail. In this project we will consider PSs modelled by Stochastic Differential Equations (SDEs) whose limiting behaviour is described by either a deterministic PDE or a stochastic PDE (SPDE).
It is indeed important to notice that, depending on the nature of the stochasticity in the PS, the limiting equation can be either a deterministic PDE – and this would be the most classical framework – or a stochastic PDE.
Either way, the limiting equation is of McKean-Vlasov type. The overall aim of the project is the comparison of the ergodic and dynamic properties of the particle system and of the limiting PDE/SPDE.
These results will help inform modelling decisions for practitioners. From a theoretical standpoint, one of the purposes of this project will be to push forward the ergodic theory for SPDEs.

Keywords for this project are: Stochastic (Partial) Differential equations, McKean Vlasov evolutions, ergodic theory, mean field limits.

Prerequisites: good background in either stochastic analysis/probability or analysis

[1] https://arxiv.org/abs/2211.08004

[2] https://arxiv.org/abs/2404.07488

Many solid materials form a largely regular crystalline lattice at the atomistic scale.
The macroscopic properties of such materials are not just determined by the deformation properties of such a perfect lattice but are critically changed by imperfections in the otherwise regular lattice which we call crystal defects.
In this project we want to study in detail how such defects interact on the atomistic level.

The project consists of two parts.
An analytical part where we use custom tools inspired by elliptic PDE theory to study the problem and its solutions.
There is existing literature that can help us a lot, e.g. [1].
And the second part is a numerical part where we use this acquired knowledge to develop, study, and implement a computational method for a particular case.

[1] Asymptotic Expansion of the Elastic Far-Field of a Crystalline Defect. Julian Braun, Thomas Hudson, Christoph Ortner, Arch Rational Mech Anal 245, 1437–1490, 2022.

The project investigates new approaches combining Discrete Differential Geometry and finite element exterior calculus / discrete calculus. Applications include discrete representations and discretisation of fluids, such as investigating cohomological and nonlinear behaviour of fluids on manifolds with general topology. New ideas and methods from the project can also be applied to other areas such as multiphysics problems from solar and plasma physics and geometric processing.

The candidate will enjoy an active and friendly scientific environment with peer PhD students, postdocs and world-wide networks created by an ERC Starting Grant “Geometric Finite Elements (GeoFEM)”.

Potential co-supervisors and collaborators include Amir Vaxman (Informatics, UoE).

1. Arnolʹd, V.I. and Khesin, B.A., 2009. Topological methods in hydrodynamics (Vol. 19). New York: Springer.

2. Arnold, D.N., Falk, R.S. and Winther, R., 2006. Finite element exterior calculus, homological techniques, and applications. Acta numerica, 15, pp.1-155.

3. Hang Yin, Mohammad Sina Nabizadeh, Baichuan Wu, Stephanie Wang, and Albert Chern. 2023. Fluid Cohomology. ACM Trans. Graph. 42, 4, Article 126 (August 2023).

4. Hu, K., Lee, Y.J. and Xu, J., 2021. Helicity-conservative finite element discretization for incompressible MHD systems. Journal of Computational Physics, 436, p.110284.

5. Gu, D.X. and Saucan, E., 2023. Classical and Discrete Differential Geometry: Theory, Applications and Algorithms. CRC Press.

6. Christiansen, S. H. (2011). On the linearization of Regge calculus. Numerische Mathematik, 119, 613-640.

7. Stephanie Wang, Mohammad Sina Nabizadeh, and Albert Chern. 2023. Exterior Calculus in Graphics: Course Notes for a SIGGRAPH 2023 Course. In ACM SIGGRAPH 2023 Courses (SIGGRAPH ’23).

Multiple positions will be funded by an ERC Starting Grant “Geometric Finite Element Methods (GeoFEM)”. The project provides an active and friendly scientific environment with peer PhD students, postdocs and worldwide networks.

Partial differential equations (PDEs) describe important models in science and engineering. Many of these PDE-based models encode fundamental geometric and topological principles. For general relativity, gravity is described as the curvature of spacetime governed by the Einstein equations. For materials, defects and microstructures can be modelled as geometric quantities such as curvature. Since controlled experiments and analytical solutions are only available in very special cases, it is essential to simulate these equations on computers. Despite significant progress in the past decades, cutting-edge applications still call for reliable numerical methods. In numerical relativity, codes may break down or significantly lose precision in the long-term simulation of black holes due to the violation of geometric constraints. For continua with microstructures, convergence may degenerate as multiple length scales are present. The common challenge behind these examples is to find an intrinsic way to discretise high-order tensors in geometry with certain symmetries.

The research will address the fundamental problem of discretising high-order tensors by bringing together geometry, algebra, PDEs and numerical analysis. GeoFEM develops an algebraic framework and a systematic construction of tensorial finite elements with symmetries. By clarifying mathematical structures at both continuous and discrete levels,
GeoFEM will investigate reliable methods for discretising the Einstein equations and continuum models with microstructures. The new framework will also inspire the development of fundamental concepts and models, and establish novel connections between numerical schemes, discrete differential geometry, measure-valued solutions of PDEs, and discrete physics, e.g., quantum gravity and lattice gauge theory. Hilbert complexes will be the key tool for discovering mathematical and computational structures.

Topics can be tailored to candidates’ interests. The project is multidisciplinary and connects various fields (continuous and discrete differential geometry, algebraic topology, numerical analysis, PDEs, network sciences, continuum mechanics, numerical relativity, computer graphics etc.). Candidates with different backgrounds (applied and pure mathematics, theoretical and computational physics, continuum mechanics and computer sciences) are encouraged to apply. Possible directions include:

1. Finite Element Tensor Calculus, 2. Discrete Differential Geometry and Discrete Physics. 3. Numerical Relativity and the Einstein Equations. 4. Generalized Continuum Mechanics, Microstructures, and Geometric Mechanics. 5. Graphs and Networks.

The candidates will have access to an international research network. Collaborators include Douglas Arnold (Uni. Minnesota), Andreas Čap (Uni. Vienna), Snorre Christiansen (Uni. Oslo), Patrick Farrell (Uni. Oxford), Jay Gopalakrishnan (Portland State Uni.), Johnny Guzmán (Brown), Ralf Hiptmair (ETH Zürich), Anil Hirani (Uni. Illinois at Urbana-Champaign), Buyang Li (Hong Kong Polytechnic Uni.), Michael Neunteufel (Portland State Uni.), Joachim Schöberl (TU Wien), Amir Vaxman (Uni. Edinburgh), Ragnar Winther (Uni. Oslo).

Candidates are encouraged to contact Dr. Hu to discuss directions and ideas.

1. Arnold, D. N., Falk, R. S., & Winther, R. (2006). Finite element exterior calculus, homological techniques, and applications. Acta numerica, 15, 1-155.

2. Arnold, D. N., & Hu, K. (2021). Complexes from complexes. Foundations of Computational Mathematics, 21(6), 1739-1774.

3. Čap, A., & Hu, K. (2024). BGG sequences with weak regularity and applications. Foundations of Computational Mathematics, 24(4), 1145-1184.

4. Hu, K., Lin, T., & Zhang, Q. (2023). Distributional Hessian and divdiv complexes on triangulation and cohomology. arXiv preprint arXiv:2311.15482.

5. Christiansen, S. H., Hu, K., & Lin, T. (2023). Extended Regge complex for linearized Riemann-Cartan geometry and cohomology. arXiv preprint arXiv:2312.11709.

6. Lim, L. H. (2020). Hodge Laplacians on graphs. Siam Review, 62(3), 685-715.

7. Christiansen, S. H. (2011). On the linearization of Regge calculus. Numerische Mathematik, 119, 613-640.

8. Gu, D. X., & Saucan, E. (2023). Classical and Discrete Differential Geometry: Theory, Applications and Algorithms. CRC Press.

9. Alcubierre, M. (2008). Introduction to 3+ 1 numerical relativity (Vol. 140). OUP Oxford.

10. Gopalakrishnan, J., Neunteufel, M., Schöberl, J., & Wardetzky, M. (2022). Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics. arXiv preprint arXiv:2206.09343.

Discontinuous Galerkin (dG) methods, a class of finite element methods, have received astounding popularity over the last 20 years as a framework of numerical approximation to PDE problems, especially in the contexts of solid mechanics and fluid dynamics. Within the last decade, dG methods have been successfully generalised to be able to admit computational meshes of arbitrary shapes, allowing for unprecedented potential in computational complexity reduction [dG1,dG2].

To harvest this potential, a key broad challenge is the development of efficient solution algorithms for the, typically vast in size, linear algebraic systems resulting from the computer implementation of dG methods, including preconditioned iterative methods for such systems [Pr1,Pr2]. This PhD project aims to address exactly this challenge for a number of industrially/practically relevant stationary and time-dependent problems, and so bridge a key gap to even wider applicability of dG schemes for these classes of problems.

The PhD project is relatively flexible and open in terms of the profile and background of the selected PhD student and can range from theoretical analysis of dG methods, numerical linear algebra for the resulting systems of equations, to computer implementation, or even code development in parallel computing architectures. It is anticipated that there will be opportunity to work with an industry partner, should this be of interest to the student.

[dG1] Cangiani, A., Dong, Z., Georgoulis, E. H., Houston, P. hp-version discontinuous Galerkin methods on polygonal and polyhedral meshes. SpringerBriefs Math. Springer, Cham, 2017. viii+131 pp.

[dG2] Cangiani, A., Dong, Z., Georgoulis, E. H. hp-version discontinuous Galerkin methods on essentially arbitrarily-shaped elements. Math. Comp. 91(2021), no.333, pp.1–35.

[Pr1] Pearson, J. W., Pestana, J. Preconditioners for Krylov subspace methods: an overview.
GAMM-Mitt.43(2020), no.4, e202000015, 35 pp.

[Pr2] Leveque, S., Pearson, J. W. Parameter-robust preconditioning for Oseen iteration applied to stationary and instationary Navier-Stokes control. SIAM J. Sci. Comput.44(2022), no.3, pp. B694–B722.

In recent years computational mathematics, including numerical linear algebra, has been subject to a “randomized revolution”, enabling classical methods for solving huge-scale problems to be made significantly more efficient. For example, one key technique is that of randomized sketching, where the dimension of the matrices arising in a linear algebra problem is reduced by projection with random matrices, followed by the solution of a much smaller problem with deterministic methods. Rigorous results from random matrix theory and compressive sensing lead to guarantees that randomized computations are accurate with overwhelmingly high probability, while the intermediate dimension reduction substantially reduces the computational complexity.

The goal of this project is to apply state-of-the-art randomized techniques to enhance the numerical solution of discretized PDEs and optimization problems. This gives rise to a range of theoretical and computational questions, which we will resolve using fundamental numerical linear algebra kernels from PDEs and optimization, such as eigenvalue problems, linear systems and associated iterative solvers, and singular value decompositions.

The student working on this project should have experience in numerical mathematics for scientific applications, including numerical linear algebra, as well as solving PDEs and/or optimization problems. There is some flexibility in the research directions of the project, depending on the interests of the student.

The subject matter of this PhD project is related to the EPSRC grant on “Randomized Numerical Linear Algebra for Optimization and Control of PDEs”, recently-awarded to the supervisor. This means that the student will have an opportunity to work within a team, including Stefan Güttel (University of Manchester), the Co-Investigator of the grant, and the postdoctoral researchers recruited through the grant. There is also the potential to work with the industry partners involved with this grant, if this is of interest to the student.

Some relevant recent work on randomized numerical methods, in particular by the project team, as well as some ideas which link with the topics of the proposed research, is provided in the references below.

S. Güttel, I. Simunec. A sketch-and select Arnoldi process, SIAM Journal on Scientific Computing 46(4), A2774-A2797, 2024.

L. Burke, S. Güttel. Krylov subspace recycling with randomized sketching for matrix functions, to appear in SIAM Journal on Matrix Analysis and Applications, 2024.

P.-G. Martinsson, J. A. Tropp. Randomized numerical linear algebra: Foundations and algorithms, Acta Numerica 29, 403-572, 2020.

T. Wagner, J.W. Pearson, M. Stoll. A preconditioned interior point method for support vector machines using an ANOVA-decomposition and NFFT-based matrix-vector products, arXiv preprint arXiv:2312.00538, 2024.

S. Güttel, J.W. Pearson. A spectral-in-time Newton-Krylov method for nonlinear PDE-constrained optimization, IMA Journal of Numerical Analysis 42(2), 1478-1499, 2022.

J.W. Pearson, A. Potschka. On symmetric positive definite preconditioners for multiple saddle-point systems, IMA Journal of Numerical Analysis 44(3), 1731-1750, 2024.

S. Leveque, J.W. Pearson. Parameter-robust preconditioning for Oseen iteration applied to stationary and instationary Navier–Stokes control, SIAM Journal on Scientific Computing 44(3), B694-B722, 2022.

Optimization problems where PDEs are posed as physical constraints, form a class of problems with significant practical relevance, for example to fluid flow control, image processing including medical imaging, and chemical and biological systems. Such problems are also highly applicable to industrial processes, including model predictive control problems from transport or thermodynamics, and optimal sensor placement on mechanical structures.

The key challenge this project seeks to address is the fast and robust numerical solution of problems which, upon discretization, lead to huge-scale systems of equations to be solved. Substantial theoretical and algorithmic challenges arise from these problems, as they are dependent on effective discretization strategies in space and time variables, followed by efficient and reliable computations with the resulting discretized systems. A particular focus of this work is that of iterative methods accelerated by powerful preconditioners, to enable us to reliably tackle much larger problems than would otherwise be possible. Some particular challenges, which could be incorporated into the PhD project if of interest to the student, include:

– Investigating how iterative solution methods for selected time-stepping schemes may be applied in parallel over many computational units, in particular working alongside or within parallel-in-time methods. Such methods include multiple shooting, domain decomposition, and space–time multigrid methods.

– The combination of the above techniques with a recently-devised multiple saddle point framework, which has to date been focused on steady optimization problems and PDEs, and with which there is an excellent opportunity to tackle complicated time-dependent systems.

– The incorporation of cutting-edge linear algebra within solvers for discretizations using space–time Galerkin finite element methods, including discontinuous Galerkin methods.

Some examples of work previously undertaken on effective discretizations and numerical methods for PDE-constrained optimization, with PhD students at Edinburgh, is provided in the references below.

The student working on this project should have experience in numerical mathematics for scientific applications, including numerical methods for PDEs and/or optimization problems, as well as numerical linear algebra. It is anticipated that this work will have a computational focus, but more theoretical directions are also possible depending on the interests of the student. Interest in high-performance computing is desirable but not essential. There is some flexibility in the research directions of the project, depending on the interests of the student.

S. Pougkakiotis, J.W. Pearson, S. Leveque, J. Gondzio. Fast solution methods for convex quadratic optimization of fractional differential equations, SIAM Journal on Matrix Analysis and Applications 41(3), 1443-1476, 2020.

S. Leveque, J.W. Pearson. Parameter-robust preconditioning for Oseen iteration applied to stationary and instationary Navier–Stokes control, SIAM Journal on Scientific Computing 44(3), B694-B722, 2022.

M. Aduamoah, B.D. Goddard, J.W. Pearson, J.C. Roden. Pseudospectral methods and iterative solvers for optimization problems from multiscale particle dynamics, BIT Numerical Mathematics 62(4), 1703-1743, 2022.

J. Gondzio, S. Pougkakiotis, J.W. Pearson. General-purpose preconditioning for regularized interior point methods, Computational Optimization and Applications 83(3), 727-757, 2022.

A. Miniguano-Trujillo, J.W. Pearson, B.D. Goddard. Efficient nonlocal linear image denoising: Bilevel optimization with Nonequispaced Fast Fourier Transform and matrix-free preconditioning, arXiv preprint arXiv:2407.06834, 2024.

B. Heinzelreiter, J.W. Pearson, Diagonalization-based parallel-in-time preconditioners for instationary fluid flow control problems, arXiv preprint arXiv:2405.18964, 2024.

Haji-Ali, Abdul-Lateef, and Andreas Stein. “An Antithetic Multilevel Monte Carlo-Milstein Scheme for Stochastic Partial Differential Equations.” arXiv preprint arXiv:2307.14169 (2023).

The main objective of the Ph.D. project is the development of advanced numerical methods for solving systems governed by multiscale partial differential equations (PDEs) that depend on various uncertain parameters, such as initial and boundary conditions or external sources. These uncertainties are particularly prominent in models derived from empirical data rather than first principles, such as in environmental modeling, epidemiology, finance, and social sciences. In these cases, the challenge lies in estimating how uncertainty in the parameters influences the solution, a problem that becomes more complex with the increasing dimensionality—often referred to as the “curse of dimensionality.”

To tackle this, the project will explore both deterministic numerical methods and stochastic particle-based approaches, such as Monte Carlo or particle-in-cell methods, which rely on random sampling to efficiently handle high-dimensional problems. Additionally, machine learning techniques will be employed to build surrogate models that can approximate solutions quickly by leveraging experimental data. These models offer a promising alternative for reducing computational costs, particularly in scenarios where real-time or repeated simulations are required.

The project spans classical fields such as fluid dynamics and kinetic theory, while also focusing on modern applications with pronounced uncertainty. Prerequisites include knowledge of numerical analysis, with a focus on numerical methods for ordinary differential equations, familiarity with partial differential equations (PDEs), and a solid foundation in probability theory.

– Giacomo Dimarco, Lorenzo Pareschi, Multi-scale variance reduction methods based on multiple control variates for kinetic equations with uncertainties, Multiscale Model. Simul. 18 (2020), no. 1, 351-382.

– Giacomo Dimarco, Lorenzo Pareschi, Numerical methods for kinetic equations. Acta Numerica 23, 369-520, 2014.

– Andrea Medaglia, Lorenzo Pareschi, Mattia Zanella, Stochastic Galerkin particle methods for kinetic equations of plasmas with uncertainties, J. Comp. Phys. Volume 479, 112011, 2023

– Lorenzo Pareschi, An introduction to uncertainty quantification for kinetic equations and related problems, in Trails in kinetic theory: foundational aspects and numerical methods, SEMA-SIMAI Springer Series 25:141-181, 2021.

The relationship between optimization processes and systems of interacting stochastic particles has its roots in the field of “Swarm Intelligence”, where the coordinated behavior of agents interacting locally with their environment leads to the emergence of global patterns.
Stochastic particle dynamics are typically guided by heuristics, and the resulting methods often excel at solving complex optimization problems where conventional deterministic methods fall short.
Gradient-based optimizers are effective at finding local minima for high-dimensional, convex problems; however, most gradient-based optimizers struggle with noisy, discontinuous functions and are not designed to handle discrete and mixed discrete-continuous variables.
Such high-dimensional problems arise in many areas of interest in machine learning applied to fields like engineering, finance, healthcare, and more.
The main goal of this Ph.D. project lies at different levels, ranging from the development of a robust mathematical framework for popular metaheuristics like simulated annealing, particle swarm optimization, genetic algorithms, and ant colony optimization, to their convergence analyses and applications to machine learning problems.
Furthermore, the relationships between well-known optimizers like SGD, Adam, RMSprop and existing metaheuristics will be studied by means of their continuous representation.

Prerequisites for the Ph.D. project include knowledge of at least one course in numerical analysis, covering multidimensional optimization methods, and some familiarity with PDEs, such as a course on the mathematical analysis of PDEs or one on the fundamental equations of mathematical physics. Knowledge of probability theory is also recommended.

– Alessandro Benfenati, Giacomo Borghi, Lorenzo Pareschi, Binary interaction methods for high dimensional global optimization and machine learning, Applied Math. Optim. 86(9):1-41, 2022.

– Giacomo Borghi, Michael Herty, Lorenzo Pareschi, Constrained consensus-based optimization, SIAM J. Optimization 33(1):10.1137, 2023.

– Giacomo Borghi, Lorenzo Pareschi, Kinetic description and convergence analysis of genetic algorithms for global optimization, Comm. Math. Sci. to appear. Preprint arXiv:2310.08562, 2023.

– Lorenzo Pareschi, Optimization by linear kinetic equations and mean-field Langevin dynamics, Math. Mod. Meth. App. Sci. to appear. Preprint arXiv:2401.05553, 2024

The past quarter century has seen a dramatic increase in the study of computational methods for high dimensional statistical inference, mostly based on the broad framework of Markov Chain Monte Carlo (MCMC) methods.    Examples include Hamiltonian Monte Carlo (HMC) methods, Langevin-based sampling algorithms (which solve stochastic differential equations), piecewise deterministic Markov processes and thermostats, such as Nos\’{e}-Hoover and variants .  These algorithms can allow efficient generation of statistical samples which can help to underpin reliable study of molecular conformations [1], uncertainty quantification [2] and explainable AI [3].  In general, after burn-in, these methods can be viewed as drawing samples from a stationary (invariant) distribution.  Some recent work on sampling in neural networks can be found in [4,5]. Of particular relevance to this project is the proposer’s recent development of Langevin dynamics methods to explore distributions in high dimensional spaces with applications to logistic regression and neural networks, see [6,7,8].

By and large, these methodological developments emanate directly from computational statistical physics, but the methods of that field extend beyond traditional statistical equilibrium and include nonequilibrium or driven systems in which systems are subject to continual forcing by external inputs [9,10].  Computational methods of this field may resemble or build on equilibrium sampling methods, but without the assumptions of global stationarity (or ergodicity of a unique invariant state) that are often invoked in statistical computations.  The ideas of nonequilibrium modelling can also be relevant in real-time machine learning applications where dynamical data acquisition continually perturbs a posterior parameter distribution, but one can also think of using artificial perturbations in order to explore rare event transitions and difficult-to-sample modes in a complex application.

This project will involve the study of computational methods for nonequilibrium modelling and their exploitation in  machine learning problems.  A target application is the computation of statistical properties and parameter inference bounds for neural networks.

This project will include potential collaboration with Daniel Paulin, University of Edinburgh, and Christoph Dellago, University of Vienna.

[1] Hénin, J. ., Lelièvre, T., Shirts, M. R., Valsson, O., & Delemotte, L. (2022). Enhanced Sampling Methods for Molecular Dynamics Simulations [Article v1.0]. Living Journal of Computational Molecular Science, 4(1), 1583, 2022. https://doi.org/10.33011/livecoms.4.1.1583

[2] Latz,J., Schneider, D. and Wacker, P., Nested Sampling for Uncertainty Quantification and Rare Event Estimation, 2023. https://arxiv.org/pdf/2310.03040

[3] Linardatos, P.; Papastefanopoulos, V.; Kotsiantis, S. Explainable AI: A Review of Machine Learning Interpretability Methods. Entropy, 23, 18, 2021. https://doi.org/10.3390/e23010018

[4] Wiese, J., Wimmer, L. Papamarkou, T., Bischl, B., Guennemann, S. and Ruegamer, D., Towards efficient MCMC sampling in Bayesian neural networks by exploiting symmetry, 2023. https://arxiv.org/abs/2304.02902.

[5] Blundell, C., Cornebise, J. Kavukcuoglu, K. and Wierstra, D., Weight uncertainty in neural networks, Proceedings of the 32 nd International Conference on Machine
Learning, Lille, France, 2015. https://arxiv.org/pdf/1505.05424

[6] Leimkuhler, B., Paulin, D. and Whalley, P., Contraction rate estimates of stochastic gradient kinetic Langevin integrators, 2023. https://arxiv.org/abs/2306.08592

[7] Chada, N., Leimkuhler, B., Paulin, D. and Whalley, P., Unbiased kinetic Langevin Monte Carlo with inexact gradients, 2024. https://arxiv.org/abs/2311.05025

[8] Paulin, D., Whalley, P., Chada, N. and Leimkuhler, B., Sampling from Bayesian neural network posteriors with symmetric minibatch splitting Langevin dynamics, https://arxiv.org/abs/2410.19780

[9] Evans DJ, Morriss G. Statistical Mechanics of Nonequilibrium Liquids. 2nd ed. Cambridge University Press, 2008.

[10] Todd BD, Daivis PJ. Nonequilibrium Molecular Dynamics: Theory, Algorithms and Applications, Cambridge University Press, 2017.

Modern data science relies strongly on probability theory to solve challenging problems. In this context, probabilistic models represent the raw data observation process and the prior knowledge available; and solutions are then obtained by performing (often Bayesian) statistical inference analyses. From a computation viewpoint, these analyses are conducted using Markov chain Monte Carlo (MCMC) algorithms, stemming from the theory of Markov processes. Constructing efficient MCMC algorithms for high-dimensional problems is difficult (e.g., problems related to image processing and machine learning), and this has stimulated a lot of research on efficient high-dimensional MCMC algorithms and theoretical structures (relying mostly on the theory of Markov processes) to understand, analyse and quantify their efficiency. In particular, MCMC algorithms based on SDEs have received a lot of attention lately, leading to some major developments in highly efficient MCMC methodology.

In this project, we will be looking to combine ideas from different areas of applied probability and machine learning to develop further highly efficient MCMC algorithms for solving high dimensional arising in computational imaging and machine learning.

In addition to joining the Maxwell Institute this PhD is also related to the Probabilistic AI hub a collaboration between 6 leading UK universities.

A. Durmus, G. O. Roberts, G. Vilmart, K. C. Zygalakis, Fast Langevin based algorithm for MCMC in high
dimensions Ann. App. Prob., 27(4), 2195-2237, (2017).

J. M. Sanz Serna, K. C. Zygalakis, Wasserstein distance estimates for the distributions of numerical
approximations to ergodic stochastic differential equations. J. Mach. Learn. Res., 22, 1–37, (2021).

T. Klatzer, P. Dobson, Y. Altmann, M. Pereyra, J. M. Sanz-Serna, and K. C. Zygalakis Accelerated Bayesian
imaging by relaxed proximal-point Langevin sampling. SIAM J. Imaging Sci. 17(2) 1078-1117, (2024).

In areas as diverse as climate modelling and medicine, mathematical models and computer simulations are routinely used to inform decisions and assess risk. However, the parameters appearing in the mathematical models are often impossible to measure fully and accurately, due to limitations in for example sensor technology, time or budget. This project is concerned with the inverse problem of determining the unknown parameters in the model, given some measurements of the process governed by the model.

In the Bayesian framework, the solution to this inverse problem is the posterior probability distribution of the unknown parameters, resulting from conditioning the prior on the observed measurements. Markov chain Monte Carlo (MCMC) methods are often considered the gold standard for sampling from the posterior distribution. However, for complex mathematical models such as those based on partial differential equations, these methods become prohibitively expensive, since the generation of each sample requires the evaluation of the model and typically many such samples are required. Therefore, there is a pressing need to make the methodology more efficient.

A PhD project in this area will combine ideas from numerical analysis and Bayesian statistics, and could focus on computational or analytical aspects. A background in Bayesian statistics (e.g. prior/posterior distributions, Markov chain Monte Carlo methods) or numerical analysis (e.g. optimisation, numerical methods for PDEs) would be useful, depending on the precise project agreed between the student and the supervisor.

-T.J. Dodwell, C. Ketelsen, R. Scheichl, A.L. Teckentrup. Multilevel Markov Chain Monte Carlo. SIAM Review, 61(3), 509-545, 2019.

– A. Istratuca, A.L. Teckentrup. Smoothed Circulant Embedding with Applications to Multilevel Monte Carlo Methods for PDEs with Random Coefficients. Available at https://arxiv.org/abs/2306.13493 .

– R. Scheichl, A.M. Stuart, A.L. Teckentrup. Quasi-Monte Carlo and Multilevel Monte Carlo Methods for Computing Posterior Expectations in Elliptic Inverse Problems. SIAM/ASA Journal on Uncertainty Quantification, 5(1), 493–518, 2017.

Many problems in science and engineering involve an unknown complex process, which it is not possible to observe fully and accurately. The goal is then to reconstruct the unknown process, given a small number of direct or indirect observations. Mathematically, this problem can be reformulated as reconstructing a function from limited information available, such as a small number of function evaluations. Statistical approaches, such as interpolation or regression using Gaussian processes, provide us with a best guess of the unknown function, as well as a measure of how confident we are in our reconstruction. There are many open questions related to efficient computations with and convergence properties of these methodologies, including challenges in high dimensional input or output spaces, goal-oriented experimental design, the use of deep Gaussian processes, and improving the methodology by incorporating information about the process such as partial differential equation (PDE) constraints.

A PhD project in this area will combine ideas from machine learning, numerical analysis and statistics, and could focus on computational or analytical aspects. A background in Bayesian statistics (e.g. prior/posterior distributions, Markov chain Monte Carlo methods), applied probability (e.g. Gaussian processes, Gaussian measures) or numerical analysis (e.g. interpolation, numerical methods for PDEs) would be useful, depending on the precise project agreed between the student and the supervisor.

– T. Bai, A.L. Teckentrup, K.C. Zygalakis. Gaussian processes for Bayesian inverse problems associated with linear partial differential equations. Statistics and Computing, 34(4), p.139, 2024. Available as arXiv preprint arXiv:2307.08343.

– A.L. Teckentrup. Convergence of Gaussian process regression with estimated hyper-parameters and applications in Bayesian inverse problems. SIAM/ASA Journal on Uncertainty Quantification, 8(4), p. 1310-1337, 2020. Available as arXiv preprint arXiv:1909.00232.

– M.M. Dunlop, M. Girolami, A.M. Stuart, A.L. Teckentrup. How deep are deep Gaussian processes? Journal of Machine Learning Research, 19, 1-46, 2018. Available as arXiv preprint arXiv:1711.11280.

This project aims to advance neural operator learning and AI4Science by tackling theoretical and algorithmic challenges in the training, fine-tuning, and robustness of deep learning models for scientific simulations, particularly for systems governed by partial differential equations (PDEs). These equations are essential for modelling complex systems like climate dynamics and fluid mechanics. While neural surrogate models have demonstrated impressive performance, achieving accuracy comparable to traditional numerical solvers with a fraction of the computational cost [1], they still face fundamental challenges. Key limitations include sensitivity to noise and adversarial perturbations, limited ability to generalize to out-of-distribution events [2], and difficulty in effectively unrolling long trajectories or incorporating historical information [3,4]. Our focus will be on deepening the understanding of these models in these challenging contexts.

The research will combine theoretical insights with practical experimentation, focusing on robustness in neural operators, advanced training strategies such as adversarial training, and methods to enhance stability and improve long-term rollout in simulations. Through a blend of deep learning and generative modelling techniques [5,6], mathematical analysis, and linear algebra [7,8,9], we will seek solutions to enhance our understanding of neural operators and their accuracy, efficiency, and stability. Ultimately, the goal is to advance our ability to create data-driven models capable of generalizing to unseen conditions, managing large datasets, and accurately forecasting critical events in PDE-governed systems, such as weather and climate phenomena, with greater reliability.

[1] Aurora: A Foundation Model of the Atmosphere, https://arxiv.org/abs/2405.13063

[2] Do AI models produce better weather forecasts than physics-based models? A quantitative evaluation case study of Storm Ciarán. npj Clim Atmos Sci 7, 93 (2024). https://doi.org/10.1038/s41612-024-00638-w

[3] Exploring the design space of deep-learning-based weather forecasting systems, https://arxiv.org/abs/2410.07472v1

[4] Mixture of Neural Operators: Incorporating Historical Information for Longer Rollouts, https://openreview.net/forum?id=rYZjCmIlmx

[5] Heavy-tailed diffusion models, https://arxiv.org/abs/2410.14171

[6] GenCast: Diffusion-based ensemble forecasting for medium-range weather, https://arxiv.org/abs/2312.15796

[7] A Mathematical Guide to Operator Learning, https://arxiv.org/abs/2312.14688

[8] Generative AI for fast and accurate Statistical Computation of Fluids, https://arxiv.org/abs/2409.18359

[9] Error analysis for deep neural network approximations of parametric hyperbolic conservation laws, https://arxiv.org/abs/2207.07362

Inverse problems are central to many imaging applications, including medical imaging, remote sensing, and astronomy. For a few decades, iterative optimisation methods have been state-of-the art for solving such problems. However, they often face limitations in terms of computational and reconstruction efficiency. Recent advances in deep learning offer powerful tools for solving inverse problems, but concerns about the trustworthiness, interpretability, and reliability of these methods hinder their broader adoption in critical applications.

This project aims to develop trustworthy deep learning strategies for inverse problems by leveraging powerful mathematical tools such as optimisation, Bayesian and optimal transport theories. Specifically, we will focus on hybrid novel strategies mixing the power of deep learning and neural networks, with the theoretical guarantees of mathematics. In this context, three main research directions are of great interest: building powerful models for imaging inverse problems; investigating robustness and convergence guarantees of data-driven methods; and exploring frugal learning strategies to tackle computational complexity challenges. 

Ultimately, the goal is to bridge the gap between data-driven deep learning approaches and traditional model-based methods, offering a framework that is both practically effective and theoretically grounded for solving complex inverse problems in imaging.

Such a project englobes research directions at the core of multiple communities including inverse problems, computational imaging, optimisation/OR, computational mathematics, machine learning.

Related recent works can be found here: https://sites.google.com/view/audreyrepetti/research/publications 

Pre-requisites: Background on topics related to optimisation, OR, optimal transport, or foundation of machine learning would be appreciated, as well as knowledge of Python.