Applied Mathematics Seminars

Organisers: Clarice Poon and Ellen Luckins

The Applied Maths Seminars are held on Fridays 12:00-13.00. This year the seminar will be hybrid (at least for Term 1): you can choose to attend in person in room B3.02 or on MS Teams. The team for the seminar is the same as last year, but if you are not a member, you can send a membership request via MS Teams or email the organisers.

Please contact Clarice Poon or Ellen Luckins if you have any speaker suggestions for future terms.

Seminar Etiquette: Here is a set of basic rules for the seminar.

• Please keep your microphone muted throughout the talk. If you want to ask a question, please raise your hand and the seminar organiser will (a) ask you to unmute if you are attending remotely or (b) get the speaker's attention and invite you to ask your question if you are in the room.
• If you are in the room with us, the room microphones capture anything you say very easily, and this is worth keeping in mind ☺️.
• You can choose to keep your camera on or not. Colleagues in the room will be able to see the online audience.
• Please let us know if you would like to meet and/or have lunch with any of the speakers who are coming to visit us so that I can make sure you have a place in the room.

Term 3

Term 2

Term 1

Abstracts

Term 3

Week 1.

Gunnar Peng (UCL) -- Singularity and instability in drop electrohydrodynamics

Electrical manipulation of microscale flows has applications in e.g. printing, coating, microfabrication and microfluidics. Due to the nonlinear coupling between electrical and flow fields, even the simple case of a drop in an electric field can exhibit complex phenomena such as streaming from the poles or equator, and symmetry-breaking Quincke rotation or equatorial vortices. We consider a two-dimensional circular drop, neglecting deformation, and show that the Taylor-Melcher leaky-dielectric model reduces to a nonlinear integro-differential equation for the time evolution of the surface-charge density, which we simulate using a finite-difference scheme and analyse using asymptotic methods. We identify and charactarise a steady-state singularity in which the surface-charge density blows up as x^(-1/3), a finite-time singularity in which it blows up as (t0-t)^(-1/2), and a symmetry-breaking instability to Quincke rotation which can be subcritical, resulting in multi-stability.

Week 2.

Maria Tatulea-Codrean (Cambridge) -- Cooperative dynamics of bacterial flagella: multi-scale modelling in biological fluid mechanics

Cooperation arises at all scales in the natural world, from the cooperative binding of receptors at the supramolecular scale to the migration of animals across continents. In this talk, we will construct and solve mathematical models to understand the mechanisms driving cooperative behaviours at the microscopic scale—namely, cooperative propulsion and spontaneous synchronization. The subject of these models is the bacterium Escherichia coli—one of the best studied model organisms in biology—and the slender helical appendages (flagella) that E. coli uses for propulsion. First, we will show that the hydrodynamic interactions between the flagella, coupled with the limited capacity for torque generation of the bacterial flagellar motor, lead to unexpected trends in the swimming speed of multiflagellated bacteria [1]. Next, we will propose and analyse an elastohydrodynamic mechanism that enables rotating flagella to spontaneously synchronize their phases without the involvement of a central pattern generator [2]. In both studies, we combine numerical and asymptotic techniques with pertinent information about the features of E. coli bacteria to gain new biophysical insights. If time allows, we will conclude by drawing analogies between the elastohydrodynamic mechanism for synchronization and another recently developed model based on load-dependent actuation with time delay [3].

References:

[1] M. Tătulea-Codrean and E. Lauga (2024) Physical mechanism reveals bacterial slowdown above a critical number of flagella. J. R. Soc. Interface, 21:20240283.

[2] M. Tătulea-Codrean and E. Lauga (2022) Elastohydrodynamic synchronization of rotating bacterial flagella. Phys. Rev. Lett., 128:208101.

[3] N. Diederen and M. Tătulea-Codrean (2025) Hydrodynamic synchronization of rotating flagella with load-dependent actuation. In preparation.

Week 3.

Matt Hennessy (Bristol) -- Drying colloidal suspensions: a poromechanics perspective

The evaporation of liquid drops that contain solid particles has captivated mathematical scientists for decades. The majority of research aims to resolve the intricate fluid mechanics of drying. However, the aggregation of solid particles due to drying can transform liquid-like drops into soft porous solids. Shrinkage of the solid alongside flow-driven deformation can induce mechanical instabilities such as buckling, fracture, and delamination. The aim of this talk is to shed light on the solid mechanics of drying through the use of modelling, asymptotics, and numerics. Using a poroelastic analogue of lubrication theory, we show that the stresses in a drop are controlled by the shape of the drop when it solidifies, which explains the wide variety of fracture patterns observed in experiments. An asymptotically reduced model for a poroelastic drop that dries on an elastic plate is then presented and validated against nonlinear finite element simulations. By comparing our theory to experimental measurements of the plate deflection, we find that the data can only be explained if the model accounts for the simultaneous growth and deformation of the solid. Capturing both aspects in the model leads to an interesting coupling, as the reference configuration needed for poroelasticity is set by the drying problem for the liquid part of the drop. The talk will conclude with a summary of ongoing work and future directions.

Week 4.

Gabriel Peyré (ENS Paris) -- Diffusion Flows and Optimal Transport in Machine Learning

In this talk, I will review how concepts from optimal transport can be applied to analyze seemingly unrelated machine learning methods for sampling and training neural networks. The focus is on using optimal transport to study dynamical flows in the space of probability distributions. The first example will be sampling by flow matching, which regresses advection fields. In its simplest case (diffusion models), this approach exhibits a gradient structure similar to the displacement seen in optimal transport. I will then discuss Wasserstein gradient flows, where the flow minimizes a functional within the optimal transport geometry. This framework can be employed to model and understand the training dynamics of the probability distribution of neurons in two-layer networks. The final example will explore modeling the evolution of the probability distribution of tokens in deep transformers. This requires modifying the optimal transport structure to accommodate the softmax normalization inherent in attention mechanisms.

Week 5.

Shekar Venkataraman (Sussex) -- Coupled bulk-surface PDE problems in cell biology

Term 2

Week 1.

Julien Landel (Lyon) -- Modelling transport phenomena for the decontamination of porous and adsorbent surfaces

In this presentation I will discuss problems associated with the decontamination of surfaces. Typical scenarios are found in the decontamination of harmful chemicals spilled onto a solid surface after an accident or a terrorist attack, or in military context. The basic scenario considers the interaction between a cleansing liquid shear flow applied onto the surface to remove an unwanted miscible liquid substance. The substance is deposited onto the surface in the form of a drop. After reviewing briefly the case of impermeable surfaces, I will focus on the more challenging cases or porous and absorbent surfaces. As the cleansing liquid flows over the surface, a convective mass transfer enables transport of the substance away from the original contamination spot. However, in the case of porous and absorbent surfaces, the transport processes are limited by the permeability of the material, as the substance generally absorbs in the bulk of the material. In addition, the washing flow can potentially increase the contaminated area through bulk spreading or redistribution via the washing flow. I will present some models of the transport processes for low and high permeability porous material, as well as for non-porous but absorbent media such as swelling hydrogel layers. To validate these models, I will show our latest progress in developing laboratory techniques to probe transport processes in situ and in real time in the bulk of porous or absorbent material.

Week 2.

Jared Tanner (Oxford) -- Deep neural network initialisation: Nonlinear activations impact on the Gaussian process

Abstract: Randomly initialised deep neural networks are known to generate a Gaussian process for their pre-activation intermediate layers. We will review this line of research with extensions to deep networks having structured random entries such as block-sparse or low-rank weight matrices. We will then discuss how the choice of nonlinear activations impacts the evolution of the Gaussian process. Specifically we will discuss why sparsifying nonlinear activations such as soft thresholding are unstable, we will show conditions to overcome such issues, and we will show how non-sparsifying activations can be improved to be more stable when acting on a data manifold.

This work is joint with Michael Murray (UCLA), Vinayak Abrol (IIIT Delhi), Ilan Price (DeepMind), and from Oxford: Alireza Naderi, Thiziri Nait Saada, Nicholas Daultry Ball, Adam C. Jones, and Samuel C.H. Lam.

Week 3.

Andrew Duncan (Imperial) -- Learning complicated probability densities using energy discrepancies.

Abstract: The problem of fitting un-normalised densities to data is a common challenge across many different fields of science and engineering. Solutions to this either involve score-matching or its variants, which require computation of score functions and its derivatives; or alternatively require expensive Markov Chain Monte Carlo Simulations. I present a very simple alternative approach called Energy Discrepancy (ED), which does not require either. I show that, as a divergence, ED interpolates between score matching and KL divergence. As a result of this, minimum ED estimation overcomes the problem of “near-sightedness” encountered in score-based estimation methods, while also enjoying theoretical guarantees. I also show how this can be adapted beyond the continuous variable setting, using heat kernels on discrete spaces.

Finally, I will demonstrate how this approach can be effectively used to solve some challenging PDE-based Bayesian inverse problems and associated experimental design problems.

Week 4.

Giovanni Montana (Warwick WMG) -- Developments in Offline Reinforcement Learning

Abstract: Reinforcement learning (RL) has emerged as a powerful framework for learning sequential decision policies through interaction and experimentation. However, in critical domains such as healthcare and autonomous driving, direct real-world interactions can be prohibitively costly, unsafe, or ethically problematic. Offline (or batch) RL addresses these limitations by learning policies from historical data, even when collected under suboptimal conditions. This talk will begin with an accessible overview of Markov decision processes and traditional RL fundamentals. I will then explore the distinct challenges and opportunities in offline RL, presenting two recent advances: First, our "learning from one mode" approach, which employs exponentially weighted behaviour cloning to extract optimal policies from multimodal data. Second, we introduce a novel extension to Q-learning that leverages state reachability analysis to relax constraints on observed state-action pairs. Both methods achieve state-of-the-art performance on continuous control benchmarks. This research represents joint work with Mianchu Wang, Yue Jin, and Charlie Hepburn.

Week 5.

Lois Baker (Edinburgh) -- Lagrangian filtering for wave-mean flow decomposition

In geophysical and astrophysical flows, we are often interested in understanding the impact of internal waves on the non-wavelike flow. For example, oceanic internal waves generated at the surface and the seafloor transfer energy from the large scale flow to dissipative scales, thereby influencing the global ocean state. A primary challenge in the study of wave-flow interactions is how to separate these processes – since waves and non-wavelike flows can vary on similar spatial and temporal scales in the Eulerian frame. However, in a Lagrangian flow-following frame, temporal filtering offers a convenient way to isolate waves. In this talk I’ll present and discuss some recently developed methods for evolving Lagrangian mean fields alongside the governing equations in a numerical simulation, allowing effective filtering of waves from non-wavelike processes.

Week 6.

a. Kawa Manmi (Warwick) -- Simplified Models for Understanding Battery SEI Layer Growth

The solid electrolyte interphase (SEI) is a crucial protective layer that forms primarily on the negative electrode of lithium-ion batteries during early cycling, particularly in the manufacturing formation and aging process. This layer continues to grow slowly throughout the battery's life as a major degradation mechanism. The properties of the SEI fundamentally impact battery performance, including irreversible capacity loss, rate capability, cycling stability, and safety. Due to the layer's inherent complexity - with its heterogeneous composition, spatiotemporal evolution, and multiple length scales - both experimental investigation and mathematical modelling of SEI formation and growth remain challenging. Zero-dimensional models have emerged as a computationally efficient approach to capture the essential physics while reducing this complexity. This talk will first compare common zero-dimensional SEI growth models in the context of both formation and normal cycling conditions. The second part will introduce our ongoing work with collaborators at WIAS Berlin on developing new mathematical frameworks based on non-equilibrium thermodynamics to better describe SEI evolution. This work aims to bridge the gap between simplified zero-dimensional models and the underlying physical processes governing SEI formation and growth.

b. Oscar Holroyd (Warwick) -- Feedback control of thin liquid films falling down inclined planes

We outline methods to control a thin liquid film falling down an inclined plane towards an unstable flat solution by injecting or removing fluid from the base. The two-phase Navier-Stokes equations that govern the dynamics of a falling liquid film pose a challenging control problem: it is an infinite-dimensional, nonlinear system with complex boundary conditions, and we are limited to a finite-dimensional boundary control. By using a hierarchy of successively simplifying assumptions, we show that a linear quadratic regulator (LQR) control can be used to stabilize the otherwise unstable flat (or Nusselt) solution. We demonstrate that applying the LQR controls to the Navier-Stokes problem is successful well outside the parameter regime that the simplified models are designed for, and also in cases where observations of the system are restricted.

Week 7.

Matthias Sachs (Birmingham) -- From Machine Learning Interatomic Potentials to Dynamics-preserving Coarse-graining Strategies

Recent progress in the development of equivariant neural network architectures predominantly used for machine learning interatomic potentials (MLIPs) has opened new possibilities in the development of data-driven coarse-graining strategies. In this talk, I will first present our work on the development of learning potential energy surfaces and other physical quantities, namely the Hyperactive Learning framework[1], a Bayesian active learning strategy for automatic efficient assembly of training data in MLIP and ACEfriction [2], a framework for equivariant model construction based on the Atomic Cluster Expansion (ACE) for learning of configuration-dependent friction tensors in the dynamic equations of molecule surface interactions and Dissipative Particle Dynamics (DPD). In the second part of my talk, I will provide an overview of our work on the simulation and analysis of Generalized Langevin Equations [3,4] as obtained from systematic coarse-graining of Hamiltonian Systems via a Mori-Zwanzig projection and present an outlook on our ongoing work on developing data-driven approaches for the construction of dynamics-preserving coarse-grained representations.

 References:

[1] van der Oord, C., Sachs, M., Kovács, D.P., Ortner, C. and Csányi, G., 2023. Hyperactive learning for data-driven interatomic potentials. npj Computational Materials

[2] Sachs, M., Stark, W.G., Maurer, R.J. and Ortner, C., 2024. Equivariant Representation of Configuration-Dependent Friction Tensors in Langevin Heatbaths. to appear in Machine Learning: Science & Technology

[3] Leimkuhler, B. and Sachs, M., 2022. Efficient numerical algorithms for the generalized Langevin equation. SIAM Journal on Scientific Computing

[4] Leimkuhler, B. and Sachs, M., 2019. Ergodic properties of quasi-Markovian generalized Langevin equations with configuration-dependent noise and non-conservative force. In Stochastic Dynamics Out of Equilibrium: Institut Henri Poincaré, 2017 

Week 8.

Alexandra Tzella (Birmingham) -- Diffusion in arrays of obstacles: beyond homogenisation

We examine the diffusion of a chemical or heat released in a homogeneous medium interrupted by an infinite number of impermeable obstacles arranged in a periodic lattice. We extend classical results due to Maxwell, Rayleigh and Keller by applying ideas of large-deviation theory to describe the concentration or temperature distribution at large distances from the point of release. We use matched asymptotics to obtain explicit results in the case of nearly touching obstacles, when the transport is strongly inhibited. The technique developed can be applied to complex systems including porous media and composite materials. This is based on joint work with Y. Farah, D. Loghin and J. Vanneste.

Week 9.

Nikhil Desai (Cambridge) -- Self-propulsion of chemically active drops along a rigid surface

Active drops are synthetic, micron-sized ''swimmers'' that convert chemical energy from their surroundings into mechanical motion. These drops are physico-chemically isotropic and emit a chemical solute whose concentration gradients cause interfacial flows which drive the solute's own transport via advection. This (nonlinear) coupling between fluid flow and solute transport around the drop causes a spontaneous symmetry-breaking, leading to self-propulsion of the inertialess drop, if the ratio of convective-to-diffusive solute transport, or Péclet number (Pe), is large enough.

As a result of their net buoyancy, active drops evolve at small finite distances from boundaries. Yet, most theoretical studies on drop propulsion focus on unbounded domains, where problems are more tractable due to a simpler geometry. We use numerical simulations to address this gap in understanding and provide physical insight on the propulsion of active drops along a rigid wall. We first model the drop as a rigid sphere that isotropically emits a solute and develops a 'slip velocity' over its surface in response to solute concentration gradients. We describe how this drop hovers stationarily over a rigid wall due to a balance between its weight (which causes the drop to settle normally toward the wall) and a slip-velocity-induced hydrodynamic force (which causes the drop to move away from the wall). We then analyze the linear stability of this hovering state to show that a reduction in the drop-to-wall separation promotes the self-propulsion of the drop, due to an efficient rearrangement of the solute distribution around the drop. This is further confirmed by nonlinear simulations of the drop's steady-state motion.

Finally, we relax the ''isotropic active particle'' assumption and consider a general propulsion mechanism: concentration gradients of the solute emitted by the drop result in diffusiophoretic interfacial flows as well as Marangoni stresses. We show that if their weights are kept the same, then a less viscous drop swims closer to the wall than a more viscous drop. This leads to stronger concentration gradients driving the drop's motion and hence faster swimming. We also investigate the influence of the relative strengths of diffusiophoretic and Marangoni flows--called the mobility ratio--and show that, for moderate values of Pe, the swimming speed of the drop depends very weakly on its mobility ratio.

Week 10.

Alice Vanel (Institut Fresnel) -- Modal approximations for plasmonic resonators in the time domain and application to super-localisation

We present recent results on the spectral analysis of the singular integral operator associated with Maxwell's equation. The operator is neither compact nor self-adjoint in the general case. However, in the electrostatic case (zero frequency), it is possible to obtain a modal decomposition for the singular integral operator. We use Kato's perturbative spectral theory and recent results on the analysis of the Neumann-Poincaré operator to show that the decomposition remains valid outside the electrostatic case and how to treat the essential spectrum. We also show how these theoretical results can be applied to solve practical problems arising in nanophotonic imaging experiments.

Term 1

Week 1.

(a) Daniel Booth (Warwick) -- A Hele-Shaw Newton's Cradle

In this talk, I study the motion of bubbles in a Hele-Shaw cell under a uniform background flow. I focus on a distinguished limit in which the bubble is approximately circular in plan view. Each bubble’s velocity is determined by a net force balance incorporating the Hele-Shaw viscous pressure and drag due to the thin films separating the bubble from the cell walls. The qualitative behaviour of the system is found to depend on a dimensionless parameter $\delta \propto \mathrm{Ca}^{1/3}R/h$, where $\mathrm{Ca}$ is the capillary number, $R$ is the bubble radius and $h$ is the cell height. First, it is found that an isolated bubble travels faster than the external fluid if $\delta>1$ or slower if $\delta<1$, and the theoretical dependence of the bubble velocity on $\delta$ is found to agree well with experimental observations. Then, in a system of two bubbles of different radii on different streamlines of the background flow, it is found that the larger bubble can overtake the smaller one, and they avoid contact by rotating around each other while passing. Finally, in a train of three identical bubbles travelling along the centre line, the middle bubble either catches up with the one in front (if $\delta >1$) or is caught by the one behind (if $\delta <1$), forming what we term a Hele-Shaw Newton's cradle.

(b) Ellen Jolley (Warwick) -- Modelling fluid-particle interactions for aircraft icing applications

As an aircraft flies through cloud at temperatures below freezing, it encounters ice particles and supercooled droplets, which results in the accretion of ice onto its surfaces and hence deformation of its aerodynamic shape. This can, in worst cases, cause series accidents. Mathematical modelling can inform new predictive codes and improved safety tests. In this talk I will introduce a model for the motion of individual ice particles near a surface in a high Reynolds number flow, revealing an array of different possible motions including collision with the wall, departing far from the wall and some others in between. I will also discuss a model for a particle interacting with a surface coated in water, as is common in icing conditions, enabling the particle to ‘skim’ along the water surface.

See also:
Mathematics Research Centre
Mathematical Interdisciplinary Research at Warwick (MIR@W)
Past Events 
Past Symposia