We study the long-time asymptotics for the solution of the modified Camassa–Holm (mCH) equation with step-like initial data.

\begin{align*} &m_{t}+\left (m\left (u^{2}-u_{x}^{2}\right )\right )_{x}=0, \quad m=u-u_{xx}, \\[3pt] & {u(x,0)=u_0(x)\to \left \{ \begin{array}{l@{\quad}l} 1/c_+, &\ x\to +\infty, \\[3pt] 1/c_-, &\ x\to -\infty, \end{array}\right .} \end{align*}

$c_+$

$c_-$

$(y,t)$

$(\xi, c)$

$c=c_+/c_-$

$\{ (\xi, c)\,:\, \xi \in \mathbb{R}, \ c\gt 0, \ \xi =y/t\}$

$u(y,t)$

$\mathcal{O}(t^{-2})$

$\mathcal{O}(t^{-1})$

We consider the chemotaxis-Navier–Stokes system with generalised fluid dissipation in $\mathbb{R}^3$:

\begin{eqnarray*} \begin{cases} \partial _t n+u\cdot \nabla n=\Delta n- \nabla \cdot (\chi (c)n \nabla c),\\[5pt] \partial _t c+u \cdot \nabla c=\Delta c-nf(c),\\[5pt] \partial _t u +u \cdot \nabla u+\nabla P=-(\!-\Delta )^\alpha u-n\nabla \phi, \\[5pt] \nabla \cdot u=0, \end{cases} \end{eqnarray*}

$\alpha \gt \frac {3}{4}$

$\tilde {\textrm {a}}$

$\alpha \gt \frac {1}{2}$

$\alpha \geq \frac {5}{4}$

$\frac {3}{4}\lt \alpha \lt \frac {5}{4}$

$L^2$

Bioreactor scaffolds must be designed to facilitate adequate nutrient delivery to the growing tissue they support. For perfusion bioreactors, the dominant transport process is determined by the scale of fluid velocity relative to diffusion and the geometry of the scaffold. In this paper, models of nutrient transport in a fibrous bioreactor scaffold are developed using homogenisation via multiscale asymptotics. The scaffold is modelled as an ensemble of aligned strings surrounded by viscous, slowly flowing fluid. Multiple scales analysis is carried out for various parameter regimes which give rise to macroscale transport models that incorporate the effects of advection, reaction and diffusion. Multiple scales in both space and time are employed when macroscale advection balances macroscale diffusion. The microscale model is solved to obtain the effective diffusion coefficient and simple solutions to the macroscale problem are presented for each regime.

Optimal transport tasks naturally arise in gas networks, which include a variety of constraints such as physical plausibility of the transport and the avoidance of extreme pressure fluctuations. To define feasible optimal transport plans, we utilize a $p$-Wasserstein metric and similar dynamic formulations minimizing the kinetic energy necessary for moving gas through the network, which we combine with suitable versions of Kirchhoff’s law as the coupling condition at nodes. In contrast to existing literature, we especially focus on the non-standard case $p \neq 2$ to derive an overdamped isothermal model for gases through $p$-Wasserstein gradient flows in order to uncover and analyze underlying dynamics. We introduce different options for modelling the gas network as an oriented graph including the possibility to store gas at interior vertices and to put in or take out gas at boundary vertices.

In this paper, we consider an optimal distributed control problem for a reaction-diffusion-based SIR epidemic model with human behavioural effects. We develop a model wherein non-pharmaceutical intervention methods are implemented, but a portion of the population does not comply with them, and this non-compliance affects the spread of the disease. Drawing from social contagion theory, our model allows for the spread of non-compliance parallel to the spread of the disease. The quantities of interest for control are the reduction in infection rate among the compliant population, the rate of spread of non-compliance and the rate at which non-compliant individuals become compliant after, e.g., receiving more or better information about the underlying disease. We prove the existence of global-in-time solutions for fixed controls and study the regularity properties of the resulting control-to-state map. The existence of optimal control is then established in an abstract framework for a fairly general class of objective functions. Necessary first–order optimality conditions are obtained via a Lagrangian-based stationarity system. We conclude with a discussion regarding minimisation of the size of infected and non-compliant populations and present simulations with various parameters values to demonstrate the behaviour of the model.

This paper is focused on the existence and uniqueness of nonconstant steady states in a reaction–diffusion–ODE system, which models the predator–prey interaction with Holling-II functional response. Firstly, we aim to study the occurrence of regular stationary solutions through the application of bifurcation theory. Subsequently, by a generalized mountain pass lemma, we successfully demonstrate the existence of steady states with jump discontinuity. Furthermore, the structure of stationary solutions within a one-dimensional domain is investigated and a variety of steady-state solutions are built, which may exhibit monotonicity or symmetry. In the end, we create heterogeneous equilibrium states close to a constant equilibrium state using bifurcation theory and examine their stability.

Minimizing an adversarial surrogate risk is a common technique for learning robust classifiers. Prior work showed that convex surrogate losses are not statistically consistent in the adversarial context – or in other words, a minimizing sequence of the adversarial surrogate risk will not necessarily minimize the adversarial classification error. We connect the consistency of adversarial surrogate losses to properties of minimizers to the adversarial classification risk, known as adversarial Bayes classifiers. Specifically, under reasonable distributional assumptions, a convex surrogate loss is statistically consistent for adversarial learning iff the adversarial Bayes classifier satisfies a certain notion of uniqueness.

Machine learning has exhibited substantial success in the field of natural language processing (NLP). For example, large language models have empirically proven to be capable of producing text of high complexity and cohesion. However, at the same time, they are prone to inaccuracies and hallucinations. As these systems are increasingly integrated into real-world applications, ensuring their safety and reliability becomes a primary concern. There are safety critical contexts where such models must be robust to variability or attack and give guarantees over their output. Computer vision had pioneered the use of formal verification of neural networks for such scenarios and developed common verification standards and pipelines, leveraging precise formal reasoning about geometric properties of data manifolds. In contrast, NLP verification methods have only recently appeared in the literature. While presenting sophisticated algorithms in their own right, these papers have not yet crystallised into a common methodology. They are often light on the pragmatical issues of NLP verification, and the area remains fragmented. In this paper, we attempt to distil and evaluate general components of an NLP verification pipeline that emerges from the progress in the field to date. Our contributions are twofold. First, we propose a general methodology to analyse the effect of the embedding gap – a problem that refers to the discrepancy between verification of geometric subspaces, and the semantic meaning of sentences which the geometric subspaces are supposed to represent. We propose a number of practical NLP methods that can help to quantify the effects of the embedding gap. Second, we give a general method for training and verification of neural networks that leverages a more precise geometric estimation of semantic similarity of sentences in the embedding space and helps to overcome the effects of the embedding gap in practice.

This work investigates the online machine learning problem of prediction with expert advice in an adversarial setting through numerical analysis of, and experiments with, a related partial differential equation. The problem is a repeated two-person game involving decision-making at each step informed by $n$ experts in an adversarial environment. The continuum limit of this game over a large number of steps is a degenerate elliptic equation whose solution encodes the optimal strategies for both players. We develop numerical methods for approximating the solution of this equation in relatively high dimensions ($n\leq 10$) by exploiting symmetries in the equation and the solution to drastically reduce the size of the computational domain. Based on our numerical results we make a number of conjectures about the optimality of various adversarial strategies, in particular about the non-optimality of the COMB strategy.

Deep neural networks and other modern machine learning models are often susceptible to adversarial attacks. Indeed, an adversary may often be able to change a model’s prediction through a small, directed perturbation of the model’s input – an issue in safety-critical applications. Adversarially robust machine learning is usually based on a minmax optimisation problem that minimises the machine learning loss under maximisation-based adversarial attacks. In this work, we study adversaries that determine their attack using a Bayesian statistical approach rather than maximisation. The resulting Bayesian adversarial robustness problem is a relaxation of the usual minmax problem. To solve this problem, we propose Abram – a continuous-time particle system that shall approximate the gradient flow corresponding to the underlying learning problem. We show that Abram approximates a McKean–Vlasov process and justify the use of Abram by giving assumptions under which the McKean–Vlasov process finds the minimiser of the Bayesian adversarial robustness problem. We discuss two ways to discretise Abram and show its suitability in benchmark adversarial deep learning experiments.

In this paper, we report the spatiotemporal dynamics of an intraguild predation (IGP)-type predator–prey model incorporating harvesting and prey-taxis. We first discuss the local and global existence of the classical solutions in N-dimensional space. It is found that the model has a global classical solution when controlling the prey-taxis coefficient in a certain range. Thereafter, we focus on the existence of the steady-state bifurcation. Moreover, we theoretically investigate the properties of the bifurcating solution near the steady-state bifurcation critical threshold. As a consequence, the spatial pattern formation of this model can be theoretically confirmed. Importantly, by means of rigorous theoretical derivation, we provide discriminant criteria on the stability of the bifurcating solution. Finally, the complicated patterns are numerically displayed. It is demonstrated that the harvesting and prey-taxis significantly affect the pattern formation of this IGP-type predator–prey model. Our main results of this paper reveal that (i) The repulsive prey-taxis could destabilize the spatial homogeneity, while the attractive prey-taxis effect and self-diffusion will stabilize the spatial homogeneity of this model. (ii) Numerical results suggest that over-harvesting for prey or predators is not advisable, it can lead to an ecological imbalance due to a significant reduction in population numbers. However, harvesting within a certain range is a feasible approach.

Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source

\begin{align*} |x|^{-2}\partial _tu=\Delta u+|x|^{\sigma }u^p, \quad (x,t)\in {\mathbb {R}}^N\times (0,T), \end{align*}

$p\gt 1$

$\sigma \ge -2$

$t\to \infty$

$\sigma =-2$

Precise knowledge of magnetic fields is crucial in many medical imaging applications such as magnetic resonance imaging (MRI) or magnetic particle imaging (MPI), as they form the foundation of these imaging systems. Mathematical methods are essential for efficiently analysing the magnetic fields in the entire field-of-view. In this work, we propose a compact and unique representation of the magnetic fields using real solid spherical harmonic expansions, which can be obtained by spherical t-designs. To ensure a unique representation, the expansion point is shifted at the level of the expansion coefficients. As an application scenario, these methods are used to acquire and analyse the magnetic fields of an MPI system. Here, the field-free-point of the spatial encoding field serves as the unique expansion point.

Geometric parameters in general and curvature in particular play a fundamental role in our understanding of the structure and functioning of real-world networks. Here, the discretisation of the Ricci curvature proposed by Forman is adapted to capture the global influence of the network topology on individual edges of a graph. This is implemented mathematically by assigning communicability distances to edges in the Forman–Ricci definition of curvature. We study analytically both the edge communicability curvature and the global graph curvature and give mathematical characterisations of them. The Forman–Ricci communicability curvature is interpreted ‘physically‘ on the basis of a non-conservative diffusion process taking place on the graph. We then solve analytically a toy model that allows us to understand the fundamental differences between edges with positive and negative Forman–Ricci communicability curvature. We complete the work by analysing three examples of applications of this new graph-theoretic invariant on real-world networks: (i) the network of airport flight connections in the USA, (ii) the neuronal network of the worm Caenorhabditis elegans and (iii) the collaboration network of authors in computational geometry, where we strengthen the many potentials of this new measure for the analysis of complex systems.

This paper is concerned with a predator–prey system with hunting cooperation and prey-taxis under homogeneous Neumann boundary conditions. We establish the existence of globally bounded solutions in two dimensions. In three or higher dimensions, the global boundedness of solutions is obtained for the small prey-tactic coefficient. By using hunting cooperation and prey species diffusion as bifurcation parameters, we conduct linear stability analysis and find that both hunting cooperation and prey species diffusion can drive the instability to induce Hopf, Turing and Turing–Hopf bifurcations in appropriate parameter regimes. It is also found that prey-taxis is a factor stabilizing the positive constant steady state. We use numerical simulations to illustrate various spatiotemporal patterns arising from the abovementioned bifurcations including spatially homogeneous and inhomogeneous time-periodic patterns, stationary spatial patterns and chaotic fluctuations.

We introduce a general class of transport distances $\mathrm {WB}_{\Lambda }$ over the space of positive semi-definite matrix-valued Radon measures $\mathcal {M}(\Omega, \mathbb {S}_+^n)$, called the weighted Wasserstein–Bures distance. Such a distance is defined via a generalised Benamou–Brenier formulation with a weighted action functional and an abstract matricial continuity equation, which leads to a convex optimisation problem. Some recently proposed models, including the Kantorovich–Bures distance and the Wasserstein–Fisher–Rao distance, can naturally fit into ours. We give a complete characterisation of the minimiser and explore the topological and geometrical properties of the space $(\mathcal {M}(\Omega, \mathbb {S}_+^n),\mathrm {WB}_{\Lambda })$. In particular, we show that $(\mathcal {M}(\Omega, \mathbb {S}_+^n),\mathrm {WB}_{\Lambda })$ is a complete geodesic space and exhibits a conic structure.

This work is concerned with the exponential turnpike property for optimal control problems of particle systems and their mean-field limit. Under the assumption of the strict dissipativity of the cost function, exponential estimates for both optimal states and optimal control are proven. Moreover, we show that all the results for particle systems can be preserved under the limit in the case of infinitely many particles.

We propose and unify classes of different models for information propagation over graphs. In a first class, propagation is modelled as a wave, which emanates from a set of known nodes at an initial time, to all other unknown nodes at later times with an ordering determined by the arrival time of the information wave front. A second class of models is based on the notion of a travel time along paths between nodes. The time of information propagation from an initial known set of nodes to a node is defined as the minimum of a generalised travel time over subsets of all admissible paths. A final class is given by imposing a local equation of an eikonal form at each unknown node, with boundary conditions at the known nodes. The solution value of the local equation at a node is coupled to those of neighbouring nodes with lower values. We provide precise formulations of the model classes and prove equivalences between them. Finally, we apply the front propagation models on graphs to semi-supervised learning via label propagation and information propagation on trust networks.

Contemporary epidemiological models often involve spatial variation, providing an avenue to investigate the averaged dynamics of individual movements. In this work, we extend a recent model by Vaziry, Kolokolnikov, and Kevrekidis [Royal Society Open Science 9 (10), 2022] that included, in both infected and susceptible population dynamics equations, a cross-diffusion term with the second spatial derivative of the infected population density. Diffusion terms of this type occur, for example, in the Keller–Siegel chemotaxis model. The presented model corresponds to local orderly commute of susceptible and infected individuals and is shown to arise in two dimensions as a limit of a discrete process. The present contribution identifies and studies specific features of the new model’s dynamics, including various types of infection waves and buffer zones protected from the infection. The model with vital dynamics additionally exhibits complex spatio-temporal behaviour that involves the generation of quasiperiodic infection waves and emergence of transient strongly heterogeneous patterns.