Data-driven analysis of dynamical systems and complex networks

Supervisor: Stefan Klus

Description: The main focus of this project will be the data-driven analysis of complex dynamical systems exhibiting multiple time scales. Based on simulation or measurement data only, it is possible to extract dominant spatio-temporal patterns, which can then, for instance, be used for dimensionality reduction, the detection of metastable or coherent sets, system identification, or control. Due to the sheer size of the data sets, kernel-based approaches or deep learning techniques might be required to mitigate the curse of dimensionality. The successful candidate will develop, optimize, and implement novel methods to analyze high-dimensional time-series data in order to gain insights into the characteristic properties of the underlying dynamical system. Of particular interest are molecular dynamics problems (analysis of protein folding processes), fluid dynamics problems (detection of coherent sets, ocean dynamics), quantum physics problems (stochastic formulations of quantum mechanics, quantum computing), and graphs and graphons (random walks, transfer operators, spectral clustering).

Non-local operators: Applications and efficient computation

Supervisor: Lehel Banjai

Description: Non-local interactions are ubiquitous in nature and lead to models that are difficult to handle accurately and efficiently. An example of this is the area of fractional differential operators, interest in which has been exploding in recent years among numerical analysts, probabilists, engineers, and mathematical analysts. Applications are wide ranging, including pattern formation in biology, therapeutic ultrasound in medicine, anomalous diffusion in finance and engineering etc. This is a huge and very active field. The project would address efficient computation of the difficult to compute fractional operators and applications to new areas. One interesting possibility is to look into the fractional steady state wave equation that has applications in, e.g., geophysics. Here much is still open including the qualitative behaviour of solutions, appropriate models, analysis of the solutions of the PDE model, and both the efficient computation and the analysis of the numerical schemes.  

Space-time numerical methods for nonlinear acoustics with applications to medical ultrasound 

Supervisor: Lehel Banjai

Description: In this project we would consider the numerical solution of a class of nonlinear wave equations modelling medical ultrasound. One such model is described by the attenuated Westervelt equation. The standard procedure to solve the equation numerically is to first discretize in space by, e.g., the finite element or finite difference method. An explicit and implicit time-stepping applied to this semi-discretization gives rise to a heavily structured space-time discretization. Instead, in this project we will look at fully unstructured space-time meshes that can be adapted in both space and time to the wave travelling through the tissue. Such space-time finite element methods have been investigated since the end of the 80s. However, only lately has there been a surge of interest in them due to the ready availability of high-performance parallel computer infrastructure. Much is still open: optimal formulations, a-posteriori error analysis and adaptivity, efficient construction of space-time elements in 3+1D, solution of resulting linear systems (preconditioning, parallel direct methods) etc. The aim of this project is to look at some of these aspects.  

 Defect interaction in a crystalline lattice 

Supervisor: Julian Braun

Description: Crystalline materials are solids in which the atoms follow the pattern of a periodic lattice. Defects are the imperfections in this lattice structure. As such they are crucial to fully understand the overall behaviour of the material. The aim of this project is an in depth analysis of the interaction of two or more defects on the atomic level. This should lead to the derivation of interaction laws on a larger scale while also giving the opportunity to develop new numerical methods for the computation of defect interaction. 

Modelling the transport of microplastics in the ocean 

Supervisor: Cathal Cummins

Description: There are an estimated 5.25 trillion plastic pieces floating in the global oceans, with approximately 1.5 million tonnes of microplastics polluting the ocean each year. There is an observed size-based preferential loss of this plastic from the ocean surface into the water column, however, we still lack a full understanding of the mechanisms behind this process. This hinders our ability to map the distribution of microplastics in the ocean, monitor their ecological impact or plan for partial removal. However, we have recently made progress in developing a mathematical description of one important process, biofouling (the accumulation of algae on the surface of microplastics) and its role in the vertical movement of floating debris. One of the key findings of the work is that particle properties are the biggest factor in determining the particular excursions that microplastics make beneath the free surface. 

However, this study neglected inertial effects, such as added mass and history effects, which result from the lagging boundary layer development of an accelerating particle. It also did not consider the effects of non-spherical geometries. In a recent review, we discovered that history effects could only be neglected for microplastics of 55micron diameter and less in regular ocean conditions. Given that microplastics are defined as any plastic debris with diameter 1micron – 5mm, there remains a considerable range of microplastics whose dynamics should include an analysis of the history force. This project aims to investigate the influence of inertial and geometric effects on the migration and ultimate fate of biofouled particles in the ocean.

Optimal algorithms for nonlinear partial differential equations. 

Supervisor: Sebastien Loisel

Description: The efficient solution of partial differential equations plays an important role in all fields of application. For example, the stationary heat equation (the Laplacian or Poisson problem) asks for a solution $u$ to the partial differential equation $u_{xx} + u_{yy} = f$. One can discretize this problem, e.g. replacing the derivatives by finite differences, which yields a finite-dimensional linear problem. I am interested in nonlinear problems, such as the $p$-Laplacian. These problems are much harder, and published solvers often fail to converge, or converge very slowly. 

If the PDE is discretized on a grid with $n$ points, it is obviously impossible to solve a PDE in less than $O(n)$ time. In this project, we will investigate algorithms for solving nonlinear PDEs in almost $O(n)$ time. 

 Modelling Epithelial Wound Healing 

Supervisor: Jonathan Sherratt 

Description: The term epithelium refers to the surface layer of an organ, and it is the first line of defense to injury. The healing of epithelial wounds has been studied in great detail in the skin and the cornea of the eye – including a significant body of mathematical modelling. I am keen to develop these models to apply specifically to epithelia in other tissues, which can show significant points of difference such as a close interplay with the immune system. Work in this area is well suited to a student keen to apply partial differential equation models to a specific biological system with potential medical implications. 

Modelling Vegetation Patterns in Semi-Arid Regions 

Supervisor: Jonathan Sherratt 

Description: In regions where water is the limiting resource, plants often cluster together, forming large-scale spatial patterns. Mathematical models of this process have been studied for 20 years, and have contributed hugely to our understanding of the patterns. I am keen to develop these models, making them more realistic by including factors such as long-range. This will involve development of new mathematical methodologies, for example to construct numerical bifurcation diagrams for integro-partial differential equations.  The aims include identification of new signatures for imminent ecosystem collapse, and the development of optimal strategies for replanting of degraded landscapes. 

Ecological and epidemiological models of wildlife and livestock systems 

Supervisor: Andrew White

Description: Mathematical models are key tools to understand the population and infectious disease dynamics of natural systems. Results from model studies have been used to guide policy decisions and shape conservation strategies to protect endangered species. Models typically focus on pairwise interactions, such as predator-prey or host-disease dynamics, but there is now evidence that the ecological community composition emerges through complex interactions where, for example, the interplay between competition, predation, disease transmission, seasonality and spatial structure can all play a key role. Examples include how the shared pathogen, squirrelpox, and the shared predator, the pine marten, can alter the outcome of species competition between red and grey squirrels and how the re-introduction of a native predator species, wolves, can reduce the prevalence of tuberculosis in wildlife prey species such as wild boar and deer and thereby reduce the chance of disease spillover to livestock populations.

This project would aim to develop new models and theory that capture the complexity of the real world by examining complex species interactions which integrate the effects of competition, predation and disease across trophic levels. The models will be developed in collaboration with biologists with expertise in the red and grey squirrel, squirrelpox and pine marten case study system in the UK and Ireland, with biologists who examine pathogen diversity at the interface between wildlife and livestock populations in Spain and in collaboration with the theoretical ecology group at UC Berkeley, USA.

Bilevel optimisation for inverse problems: analysis, fast computations, and Bayes.

Supervisors: Dr Abdul-Lateef Haji-Ali

Description: Inverse problems concern the estimation of parameters of mathematical models given real world data: we estimate the permeability of a groundwater reservoir using measurements of the hydrostatic pressure in the reservoir, we reconstruct the position and shape of a tumour using attenuated X-rays in medical imaging, and we train the weight and bias matrices in a deep neural network that aims at distinguishing cats and dogs. Inverse problems are usually not uniquely solvable or their solution is brittle with respect to small perturbations in the data -- they are ill-posed.

Two ways that can often overcome ill-posedness are regularisation on the one hand and the Bayesian approach on the other hand. The regularisation approach consists in minimising a certain functional that is a sum of the negative log-likelihood of the observed data given the unknown parameter and an additional term that has favourable properties. The Bayesian approach is probabilistic: we model the unknown parameter as a random variable being distributed according to the so-called prior distribution. Using the aforementioned likelihood and Bayes' formula, we can then obtain the posterior distribution - that is the conditional distribution of the parameter given the data observation. The posterior can be used for point estimation and uncertainty quantification.

Regulariser and prior can have a large influence on the solution of the inverse problem and an appropriate choice is hard. In bilevel optimisation, we aim to `learn' the regulariser based on available data. Such a parameter can be a simple prefactor of a usual regulariser [Reyes et. al; Journal of Mathematical Imaging and Vision 57: 1–25 (2017)] that needs to be determined or the regulariser can be completely determined by a neural network [Mukherjee et al.; ArXiv 2008.02839 (2020)], where then weights and biases need to be learned.

We commence this project by looking at the method by [Antil et al.; Inverse Problems, 36: 064001 (2020)], which presents an interesting method to estimate the fraction of a fractional Laplacian that is used for regularisation. Here, the primary goal is to find a more scalable version of this algorithm through modern techniques in numerical linear algebra, allowing us to reconstruct large scale medical images. Future work may include: bilevel optimisation of prior distributions in Bayesian inversion, learning of sparse dictionaries through optimisation on manifolds, other fractional operators (such as total variation), and a continuous-time analysis of stochastic gradient descent in bilevel optimisation [Jin et al.; ArXiv 2112.03754 (2021)].

Numerical analysis for multiscale bulk-surface PDEs

Supervisor: Mariya Ptashnyk

Description: Coupled systems of nonlinear partial differential equations defined in bulk domains and on surfaces are naturally arising in modelling of many biological and physical systems. Popular examples are models for intercellular signalling processes, crucial to all biological processes in living tissues. In such models the dynamics of signalling molecules in intercellular and/or intracellular spaces (bulk domain) are coupled to the dynamics of receptors on cell membranes (surface). In this project we will consider the design and analysis (a priori and a posteriori error analysis) of numerical schemes for the multiscale bulk-surface problems, when considering processes on two different spatial scales (e.g. on the level of a single cell and on a tissue level). (joint with C. Venkataraman, University of Sussex)

Numerical analysis for nonlocal cross-diffusion systems

Supervisor: Mariya Ptashnyk (joint with Lehel Banjai, Heriot-Watt University)

Description: Cross-diffusion systems arise in modelling many different biological and physical processes, e.g. the movement of cells, bacteria or animals, transport through ion-channels in cells, tumour growth, gas dynamics, carrier transport in semiconductors, with the chemotaxis system being one of the most important examples of a cross-diffusion system. The motivation for considering nonlocal cross-diffusion systems, where the Laplacian, modelling random walk, is replaced by the fractional Laplacian is derived from the experimental observation that both in the context of cell motility and population dynamics in certain situations organisms move according to Lévy processes. In this project we will consider the design, analysis and implementation of the efficient numerical schemes for the simulation of nonlocal cross-diffusion systems. There two main challenges in the numerical simulations of the fractional cross-diffusion systems: the cross-diffusion terms and the nonlocality of the fractional Laplacian.

Multiscale methods and Multiscale Interacting particle systems

Supervisor: Michela Ottobre

Description: 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.

Interacting Particle systems and Stochastic Partial Differential Equations

Supervisor: Michela Ottobre

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

Mathematical Models for modern approaches to Ecosystem Restoration

Supervisor: Michela Ottobre

Description: Achieving sustainable human-wildlife coexistence in well-functioning ecosystems is a vitally important and major challenge under global change. In response, novel approaches to Ecosystem Restoration (sometimes generally referred to as Rewilding) have emerged in the past 20 years, and are now starting to establish in the ecological community. Such approaches have shifted the attention from e.g. recovery of single species to ecosystem health as a whole. Efforts in this direction need the support of mathematical and statistical models to help monitor and forecast not just the behaviour of single species but of e.g. ecosystem functions and processes. This project will deal with modelling aspects related to monitoring, modelling and forecasting for Ecosystem Restoration projects and will be in connection with relevant non-academic stakeholders, such as Forest Research, the Center for Ecology and Hydrology, or the High Weald National Landscape. I am more than happy to be contacted for informal enquiries.

Factors influencing the time to disease fade-out.

Supervisor: Damian Clancy

Description: The spread of infectious disease through a population is an inherently random process, and can be studied using stochastic models. For diseases which become endemic in a population, one object of interest is the time until fade-out of infection (a random variable). The expected time to fade-out may be computed straightforwardly through Monte-Carlo simulation, or more exactly from general Markov process theory. For more complicated models, implementing these approaches becomes less straightforward, and approximation methods may also be needed. In this project, you will investigate a variety of approaches, with the aim of understanding the effects of particular disease features upon time to fade-out. There are many different features of different diseases that you could study - for instance, you might examine the impact of environmental transmission upon disease persistence, or the effects of changes in the birth and death rates of the susceptible population.  

References: 

"Approximating time to extinction for endemic infection models" by Damian Clancy and Elliott Tjia (2018), Methodology and Computing in Applied Probability volume 20, pages 1043–1067 10.1007/s11009-018-9621-8

"The Influence of Latent and Chronic Infection on Pathogen Persistence" by A. O'Neill, A. White, D. Clancy, F. Ruiz-Fons & C. Gortázar (2021), Mathematics volume 9, article number 1007 https://doi.org/10.3390/math9091007

Trustworthy Deep Learning Strategies for Inverse Problems in Imaging

Supervisor: Audrey Repetti (HWU)

Potential Co-Supervisors: Julie Delon (Université Paris-Cité), Nelly Pustelnik (ENS Lyon), Ulugbek Kamilov (Washington University)

Description: 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.

Innovative approaches to uncertainty quantification for multiscale kinetic equations

Supervisor(s): Lorenzo Pareschi (HWU) Co-supervisor: Emmanuil Georgoulis (HWU)

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.

Related references

– 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.

Advanced stochastic particle optimization methods and applications to machine learning

Supervisor: Lorenzo Pareschi (HWU) Co-supervisor: Michela Ottobre (HWU)

Description: 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.

Related references:

– 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

Non-commutative integrable systems

Supervisor: Simon Malham

Description: This project concerns the integrability of non-commutative nonlinear partial differential equations. In particular, the non-commutative Kadomtsev-Petviashvili (KP) hierarchies, and their modified forms, are very much of interest. Establishing their integrability by direct linearisation would be one goal. An underlying operator algebra, the pre-Poppe algebra, provides a natural context for establishing direct linearisation for these hierarchies. These hierarchies have an incredibly rich structure and many applications, for example, in nonlinear optics, ferromegnetism and Bose-Einstein condensates. They have intimate connections to: string theory and D-branes; Jacobians of algebraic curves and theta functions; Fredholm Grassmannians; the KPZ equation for the random growth off a one-dimensional substrate; and so forth. In addition to these aspects, there are many further directions that could also be explored, for example: the log-potential form; super-symmetric extensions; establishing efficient numerical methods via the direct linearisation approach; etc.

[1] Doikou, A., Malham, S.J.A. and Stylianidis, I. 2021, Grassmannian flows and applications to non-commutative non-local and local integrable systems, Physica D 415, 132744.

[2] Malham, S.J.A. 2022, Integrability of local and nonlocal non-commutative fourth order quintic NLS equations, IMA J. Appl. Math. 87(2), 231-259.

[3] Malham, S.J.A. 2022, The non-commutative Korteweg-de Vries hierarchy and combinatorial Poppe algebra, Physica D 434.

[4] Blower, G., Malham, S.J.A. 2023, The algebraic structure of the non-commutative nonlinear Schrodinger and modified Korteweg-de Vries hierarchy, Physica D 456, 133913.

[5] Blower, G., Malham, S.J.A. 2024, Direct linearisation of the non-commutative Kadomtsev-Petviashvili equations, submitted.

Coagulation sol-gel phenomena

Supervisor: Simon Malham

Description: This project concerns Smoluchowski coagulation and sol-gel models. We consider scenarios where particles of different sizes coalesce to form larger particles, including possibly a "gel" state. There are many applications including: aerosols, clouds/smog, clustering of stars/galaxies, schooling/flocking, genealogy, nanostructures on substrates such as ripening or island coarsening, blood clotting and polymer growth, for example, in biopharmaceuticals. The goal is to find analytical solutions as well as construct efficient numerical simulation methods. Determining the dual sol-gel state is an important aspect of these models. Planar tree structures play an important role as well, both at the nonlinear partial differential equation model level, and at the particle model level where, naturally, coalescent stochastic processes represent their overall evolution. Including spatial diffusivity in the model at both these levels, for example to model colloids, adds another complexity. There are many directions to explore, for example: the particle interactions can be much more complex; there is a natural Hopf algebra of planar trees that is likely useful in optimising numerical simulation; investigating the dual reverse time, branching Brownian motion, perspective; and so forth.

[1] Doikou, A., Malham, S.J.A., Stylianidis, I. and Wiese A., 2023, Applications of Grassmannian flows to coagulation equations, Physica D 451, 133771.

[2] Malham, S.J.A. 2024, Coagulation, non-associative algebras and binary trees, Physica D 460, 134054.

Model-Based Generative Particles: Leveraging Bayesian Network Structures for Generative Modeling

Supervisor: Panagiota Birmpa

Description: Generative models--such as Generative Adversarial Networks (GANs) and diffusion models--are powerful modern tools for learning the underlying data distribution of complex datasets. They have achieved remarkable success in applications such as sample generation, image denoising, and domain adaptation. However, most of Generative models are model-free, meaning that they do not exploit any prior knowledge about the structure of the data distribution. In many practical scenarios, though, additional information is available, such as the independence structure of variables represented by a Bayesian network or Markov Random Field.

This project investigates how such structural information can be incorporated into the design of model-based Generative models, with a particular focus on a specific class of implicit generative models known as generative particles (as introduced in [1]). We propose a theoretical framework grounded in the study of learning local marginals--those corresponding to neighborhoods within the given graphical model.

These theoretical insights motivate a principled approach to constructing model-based generative particles that employ a collection of local discriminators, each operating on the neighborhoods of the Bayesian network rather than a single global discriminator acting on the entire system. Empirical results on both synthetic and real-world datasets will be conducted to evaluate and validate the effectiveness of our proposed model-based generative model.

References

[1] Gu, H., Birmpa, P., Pantazis, Y., Rey-Bellet, L. and Katsoulakis, M.A., 2024. Lipschitz-regularized gradient flows and generative particle algorithms for high-dimensional scarce data. SIAM Journal on Mathematics of Data Science, 6(4), pp.1205-1235.

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