AMATH Graduate Mini-Conference

Social norms are a powerful process that can shape epidemic dynamics. Most mathematical models of coupled behaviour–disease dynamics treat norms as pre-existing rather than modelling them endogenously. Here, we investigate whether altruistic punishment can sustain a social distancing norm when individuals may defect, cooperate without punishing, or cooperate while paying a cost to punish defectors. We couple a transmission model to an imitation model for these three strategies. Disease prevalence affects behavioural payoffs, while the behavioural composition modifies transmission. We also compare this coupled system with a control where decisions respond to a fixed prevalence. We find a wide parameter regime where an injunctive social norm in support of social distancing is maintained through persistence of the punisher strategy. Disease–behaviour feedback can also create oscillations or tipping points. These effects do not occur in the uncoupled model, although there are still broad parameter regimes where a social norm persists. Our findings show that costly peer punishment can support persistence of social norms that mitigate disease transmission. More broadly, endogenous epidemic feedback can qualitatively change the conditions under which cooperation and punishment are sustained, producing tipping points and long-term behavioural-epidemiological cycles that fixed-payoff models cannot capture. 

The lack of effective clinical biomarkers for predicting patient response to immunotherapy has motivated the development of machine learning (ML) prediction models. These models combine information across features that are correlated with immunotherapy response; however, they fail to incorporate causal relationships. This implicit independence assumption prevents accurate estimation of the effects of interventions. Simulating interventions is a crucial step in answering counterfactual queries, which provide insight into how an outcome would have differed had we intervened on a feature. We demonstrate how to incorporate causality into an ML model and compute relevant causal queries, even with incomplete information about the biological system. This is demonstrated using an existing immunotherapy prediction model, allowing patients who receive unfavourable predictions to access recommendations for minimal, feasible actions to improve their response to treatment. We emphasize that this framework can be applied to any system and ML architecture, and can be strengthened by experimental and clinical knowledge. 

Rising dietary and drinking-water intake of nitrate and nitrite poses a significant public health concern. After ingestion, nitrate enters the enterosalivary circulation, where oral bacteria reduce it to nitrite. When swallowed, nitrite enters the acidic gastric environment, where it can react to form N-nitroso compounds (NOCs), many of which are suspected carcinogens. However, epidemiological evidence for this link remains mixed, likely due to the protective effects of antioxidants such as vitamin C. To better understand and quantify these complex interactions, we develop a dynamic, compartmental quantitative systems pharmacology model of human nitrate and nitrite metabolism and gastric chemistry. The framework tracks nitrate and nitrite fluxes across the stomach, intestine, plasma, and saliva, incorporates postprandial changes in gastric volume and pH, and includes mechanistic nitrosation pathways with vitamin C inhibition. Using this framework, we evaluate NOC formation under different dietary and water-quality contexts, demonstrating the protective effect of dietary vitamin C and examining the role of supplementation. Simulations suggest supplementation is most effective shortly after meals. These findings provide a mechanistic basis for understanding how diet, drinking-water quality, and vitamin C supplementation interact to shape endogenous NOC formation, with implications for nutritional guidelines and risk reduction. 

The first few months of life represent the most critical window for human development, as nearly 47% of under five child deaths occur during the first month. During this time, the infant gut; initially sterile at birth is colonized by trillions of microbes from the mother and the environment. These early colonizers, particularly Bifidobacterium and Lactobacillus are essential for educating the immune system, digesting human milk oligosaccharides (HMOs) and protecting against pathogens. However, when this colonization process is disrupted, a state known as dysbiosis can lead to long-term health issues like Type 1 Diabetes , allergies and inflammatory bowel disease (IBD). While we know which species are generally present, we still don't fully understand the strain level evolutionary dynamics. Different strains of the same species can have vastly different effects; for example, one strain of Bacteroides dorei might be harmless while another carries genes that increase the risk of autoimmunity. My research aims to fill this gap by tracking how these specific microbial lineages evolve, persist and adapt within the infant gut. 

High-fidelity numerical simulations of partial differential equations rely critically on the quality of the underlying spatial discretization (mesh). While meshes composed of hexahedral elements are often required in domains such as aerospace and structural engineering due to their superior numerical accuracy and computational efficiency, generating them for complex geometries remains a significant challenge. Currently, this process requires extensive manual intervention to decompose computer-aided design (CAD) models into sub-domains (blocks) that support the automatic generation of conformal hexahedral meshes. This block decomposition consumes significant engineering time, severely limiting rapid prototyping and design optimization. Consequently, automating block decomposition is an essential prerequisite for achieving fully automated hexahedral mesh generation. To address this bottleneck, we build upon an existing reinforcement learning (RL) approach, introducing a refined sequential decision-making framework to automate CAD decomposition. Unlike traditional heuristic approaches, the RL agent is trained to autonomously explore and execute decomposition actions that facilitate valid, conformal meshing. By training the agent to learn an optimal decomposition policy through autonomous interaction with the CAD environment, this approach provides a data-driven pathway to massively accelerate numerical simulation workflows. 

Since the seminal work of Raissi, Perdikaris and Karniadakis [1], there has been an explosion in developing physics informed neural network (PINNs) approaches ranging form forward and inverse problems and a variety of linear and nonlinear partial differential equations (PDEs). Current state-of the-art PINNs approaches are capable of solving parametrized problems and significant headway has been made in solving for weak solutions [3, 5]. Moreover, recently remarkable improvements have been achieved by the addition of physical constraints such as entropy inequalities to PINNs loss functions [4]. Nevertheless, many PINNs approaches remain costly to train and typically do not provide surrogates with industrially relevant error tolerances. This is particularly true in the context of nonlinear PDEs on complex geometries and when weak solutions are of interest (e.g., shocked problems). Alternatively, a number of works have leveraged numerical methods frameworks, such as finite volume [6] and finite-element [2, 7] methods, to train neural networks. The advantage of using numerical methods frameworks is that the training procedure implicitly leverages the mathematical sophistication of such approaches and often leads to less expensive training procedures and surrogates with better error properties. For example, for methods that employ the weak-form of PDEs, the resulting loss function provides a natural scaling for initial and boundary conditions. In this talk, we look to leverage the summation-by-parts (SBP) framework as a means of training machine learning surrogates. The advantage of the SBP approach is that it results in numerical methods that have provable properties. We take advantage of this to construct loss functions that not only appropriately scale initial and boundary conditions but also provides a principled approach to adding physical constraints like conservation and entropy inequalities. We will examine this approach for a variety of linear and nonlinear PDES (e.g., linear advection and compressible Euler equations) on complex geometry using a variety of ML architectures. [1] M. Raissi, P. Perdikaris, and G.E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378, pp. 686–707 (2019). [2] M. Abda, M. Hamedi, E. Piollet, C. Blake, and F.P. Gosselin, The finite element neural network method: One-dimensional study, Next Research 2(4), 100885 (2025). DOI: 10.1016/j.nexres.2025.100885. [3] E. Kharazmi, Z. Zhang, G. E. M. Karniadakis, hp-VPINNs: Variational physics-informed neural networks with domain decomposi tion, Computer Methods in Applied Mechanics and Engineering, 374 (2021), 113547. DOI: 10.1016/j.cma.2020.113547. [4] T. De Ryck, S. Mishra, R. Molinaro, wPINNs: Weak Physics Informed Neural Networks for Approximating Entropy Solutions of Hyperbolic Conservation Laws, SIAM Journal on Numerical Analysis, 62(2) (2024), 811–841. DOI: 10.1137/22M1522504. [5] C. Liu, H. Wu, cv-PINN: Efficient learning of variational physics-informed neural network with domain decomposition, Extreme Mechanics Letters, 63 (2023), 102051. DOI: 10.1016/j.eml.2023.102051. [6] T. Li, Y. Zou, S. Zou, X. Chang, L. Zhang, X. Deng, Learning to solve PDEs with finite volume-informed neural networks in a data-free approach, Journal of Computational Physics, 530 (2025), 113919. DOI: 10.1016/j.jcp.2025.113919. [7] R. E. Meethal, A. Kodakkal, M. Khalil et al., Finite element method-enhanced neural network for forward and inverse problems, Advances in Modeling and Simulation in Engineering Sciences, 10 (2023), 6. DOI: 10.1186/s40323-023-00243-1. 

Energy functions are generalizations of controllability and observability Gramians to nonlinear systems and as such find applications in both nonlinear balanced truncation and feedback control. These energy functions are solutions to the Hamilton-Jacobi-Bellman (HJB) equations---partial differential equations defined over spatial dimensions determined by the number of state variables in the nonlinear system. Thus, they cannot be resolved with local basis functions, even for problems of modest dimension. In this paper, we extend recent results that utilize Kronecker products to generate polynomial approximations to HJB equations. Specifically, we consider the addition of linear constraints that exhibit a Stokes-type differential-algebraic equation (DAE) structure. This extension leverages the so-called strangeness framework for DAEs to create separate sets of algebraic and differential variables with differential equations that only involve the differential variables and algebraic equations that relate them both. At this point, the existing polynomial approximations using Kronecker products can be applied to find approximations to the energy functions. This approach is demonstrated on two polynomial feedback control problems. While the standard transformation destroys the sparsity in the original system, we present a formulation that preserves the original sparsity structure. 

In real-world systems, the exact parameter values governing system dynamics may be unknown or difficult to estimate. As such, it is important to determine how this uncertainty in the state-space model affects the controllability properties of the system, such as small-controllability. The concept of structural controllability, introduced in the 1970s, directly addresses this issue. In this talk I describe some recent work on using an existing Lie-algebraic sufficient condition for small-controllability of single-input bilinear systems with a drift term to formulate a structural sufficient condition for small-controllability of such systems. The main technique used to accomplish this task is to convert the Lie-algebraic condition into linear-algebraic terms. 

In this talk, we study uniform stabilization of strongly continuous semigroups on general Banach spaces through a perturbation theory for quasi-compact semigroups. We first show that quasi-compactness is preserved under suitable θ-subordinate perturbations of generators of analytic semigroups, provided that the perturbation is compact along the trajectories of the unperturbed semigroup. Building on this preservation result, we establish sufficient conditions under which uniform exponential stability is preserved under θ-subordinate perturbations on arbitrary Banach spaces. The argument relies on the spectral structure of quasi-compact semigroups and, in particular, on the interplay between quasi-compactness and stability properties of semigroups.  

Turbulence, the phenomena of random and chaotic fluid motion over a wide range of length scales, is a key property of the Earth's atmosphere. Turbulence acts to cascade kinetic energy from its generation at large scales, down to its dissipation at small scales. However, turbulence is hard to reproduce in numerical models of the atmosphere due to its effects at all scales. We present a study of the mean square vorticity, or enstrophy, to investigate the physical mechanisms driving this cascade within the Earth's convective boundary layer (CBL). Analysis of turbulence in the CBL is necessary to better understanding the formation of eddies and weather phenomena in the lowest layer of the atmosphere and also to understand how turbulence is represented numerically. The Weather Research and Forecasting Model (WRF) is used to perform large eddy simulation of an idealized CBL. We simulate a rectangular prism with periodic boundary conditions, heated by a mean constant temperature flux at the surface, in order to mimic the solar heating of the Earth's surface. Then, we calculate the terms in the enstrophy budget to investigate the roles of vortex stretching, baroclinic vorticity generation and eddy viscosity in the energy cascade. With thorough analysis of this idealization, we present our findings on the enstrophy budget in the boundary layer to better understand the physical mechanisms at work in atmospheric turbulence.  

Excessive total phosphorus (TP) loading drives severe eutrophication and hypolimnetic anoxia in Hamilton Harbour, a designated Great Lakes Area of Concern. We investigate a reduced-order two-box mass balance model tracking TP dynamics across a 14-year monitoring record (2010–2024). By modulating vertical exchange and internal sediment release via temperature-dependent regime switches, the model captures the primary seasonal transitions governing nutrient dynamics. Sparse multi-station grab samples are mapped into the state space through 3D Voronoi tessellation to evaluate model skill and are assimilated via an Ensemble Kalman Filter (EnKF) to constrain state trajectories and forecast uncertainty. This study demonstrates the utility of low-order dynamical systems in extracting clear, mechanistic insights from long-term environmental monitoring data within a complex aquatic system. 

In this talk I will present a recently developed non-perturbative framework for studying the dynamics of particle detectors coupled to quantum fields. In the literature, continuous finite duration interactions are solved perturbatively and mostly up to leading order, that is, under the assumption of weak coupling. In this work, we focused on the ubiquitous Unruh–DeWitt particle detector model where a finite-duration interaction is approximated by a sequence of instantaneous couplings. From there we derived the resulting detector evolution non perturbatively. In particular, we can derive analytic expressions for Gaussian field states. We then analyzed the detector transition probability and coherence for arbitrary initial qubit states as the coupling strength is increased. Finally, I will discuss some of the numerical difficulties and potential workarounds as the computation gets more difficult with stronger coupling. 

Suppose you are given infinitely many number of qubits, each getting erased with a probability p. Can you make one high quality logical qubit out of them? The answer is yes if and only if p is strictly less than 1/2. The reason why it is not possible for p >= 1/2 is exactly the no-cloning theorem. However, can you make a slightly better logical qubit when p >= 1/2? More precisely, can you simulate a q-erasure channel with asymptotically many copies of p-erasure channels if 1/2 <= q < p? In 2008, Matthew Hastings conjectured that the answer is no. In a recent work, we prove this conjecture by establishing a framework called fractional extendibility. Therein we present the distributed monogamy of entanglement, a fundamental limit to entangled pair extraction. With these tools, we prove the aforementioned conjecture and more.