Enrique Zuazua

Friedrich-Alexander-Universität

CONTROL AND MACHINE LEARNING

In this lecture, we will discuss recent results from our group that explore the relationship between control theory and machine learning, specifically supervised learning and universal approximation. We will take a novel approach by considering the simultaneous control of systems of Residual Neural Networks (ResNets). Each item to be classified corresponds to a different initial datum for the ResNet's Cauchy problem, resulting in an ensemble of solutions to be guided to their respective targets using the same control. We will introduce a nonlinear and constructive method that demonstrates the attainability of this ambitious goal, while also estimating the complexity of the control strategies. This achievement is uncommon in classical dynamical systems in mechanics, and it is largely due to the highly nonlinear nature of the activation function that governs the ResNet dynamics. This perspective opens up new possibilities for developing hybrid mechanics-data driven modeling methodologies. Throughout the lecture, we will also address some challenging open problems in this area, providing an overview of the exciting potential for further research and development.

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Mohammed Hachama

National Higher School of Mathematics

Nonlocal approaches in image processing

Partial differential equations and variational approaches are powerful tools for modeling physical phenomena. They are used in several image processing applications such as restoration, enhancement, combination, ... Their study is important to demonstrate the existence and uniqueness of solutions, study their properties and asymptotic behaviors, and propose techniques for their resolution. On the other hand, non-local image processing techniques allow to take into account the relationship and interaction between spatially distant pixels. This includes techniques using non-local differential operators, non-local functionals, or those based on fractional derivatives. These relatively recent approaches have shown superior performance at the cost of higher computational times. The aim of this presentation is to present non-local models (PDEs and functionals) used in image processing. We will focus on the existence and uniqueness of solutions, as well as the design of fast and efficient algorithms. These models will be applied to shadow removal and image fusion problems.

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Antonio Di Teodoro

Universidad San Francisco de Quito

Two-Sided Fractional Monogenic Functions

In this talk we study two-sided (left and right) axially symmetric solutions of a generalized fractional Cauchy-Riemann operator. We present a method based on the Cauchy-Kowalevski extension theorem to obtain two-sided fractional monogenic polynomial.

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Anotida Madzvamuse

University of British Columbia

Analysis and simulations of bulk-surface reaction-diffusion systems with
applications to cell motility

In this talk, I will derive Turing diffusion-driven instability conditions for bulk-surface reaction-diffusion systems which describe naturally the spatio-temporal inteactions between active and inactive Rho GTPases in cell biology. Bulk-surface reaction-diffusion equations couple bulk reaction-diffusion process, those taking place inside the cell cytosol, with surface reaction-diffusion processes, those taking place near or on or in the plasma membrane. The mathematical formalism couples Laplace and Laplace Beltrami operators which brings several challenges for their analysis. Under appropriate geometric assumptions
that allow compartibility between these two spatial operators, I will derive generalised conditions for Turing diffusion-driven instability. These conditions show that under appropriate bulk and surface diffusion
coefficients, the bulk reaction-diffusion system is capable of generating patterns on the surface, even though the surface reaction-diffusion system is not able to do so. This is due to the Robin-type boundary conditions coupling the two systems. On the other hand, the surface reaction-diffusion system is only able to induce patterns in a small region close to the surface (similar to the epidermis), but not everywhere in the bulk. To support these theoretical findings, a bulk-surface finite element method is employed for the
space-discretisation, while a fractional-step Theta scheme is employed for marching forward in time.

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Jorge Honles

Instituto de Investigación para el Desarrollo

Modeling pesticide pollution risk at a small geographic scale and the relationship with human health: case of Peru

Peru is one of the main food providers in South America. With approximately 11.6 million hectares of cultivated land, the stakes for managing pesticide pollution are high. As a result of such intensive agricultural practices, pesticide residues pose a significant threat to the environment and to public health. According to the Peruvian health authorities, cancer is the second leading cause of death. Despite the implementation of health surveillance programs in Peru, there is still a lack of tangible information that prevents an accurate description of environmental risk factors and cancer burden in the country. This research aims to offer insights into the dynamics of pesticide dispersion and the implications for public health in Peru.Using pesticide chemical and physical properties and georeferenced seed data bases such as pesticide use, agricultural areas, production of permanent and transitory crops, hydrographic areas, slope, precipitation, organic carbon, and soil textures, we are currently calculating the percentage of pesticide residues in soil and surface water. Crop land is being classified using artificial intelligence and all results are being analyzed interannually and seasonally according to the crop cycle of each region of Peru. Moreover, the epidemiological dynamics of cancer in Peru is being investigated using the registries of the National Institute for Neoplastic Diseases (INEN) from the period of 2007-2020 (n>150,000). All cases are being merged to georeferenced sociodemographic data using Structured Query Language (SQL) for relational database management systems and GBU R environment. The result data is being analyzed using Integrated Nested Laplace Approximation (INLA) for Bayesian geospatial modelling. Over the period 2007-2020, there has been an overall increase in cancer incidence. However, this trend does not apply to all types of cancer and varies according to geographic region. There were areas of concern in Peru regarding cancer risk and pesticide use. Junín and Pasco have areas of high pesticide use related to hepatocellular cancer.A number of areas in Peru have an increased risk of cancer due to the use of pesticides. It is urgently necessary to implement pesticide management measures in Peru and to establish a national cancer information system to promote more effective policy planning.

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Montaz Ali

University of the Witwatersrand

Hybrid Renewable Energy Systems Modelling, Solution Approach and Applications

The critical challenge of sizing hybrid renewable energy systems over their lifespan lies in the uncertainties in energy sources and load demands. By leveraging two-stage stochastic programming, we introduce novel modelling techniques that ensures robust optimization without compromising numerical tractability. In our first modelling approach the stochastic program is converted into a deterministic three-block separable optimization problem, and then solve it using ADMM. The theoretical convergence of ADMM to the optimal solution based on the concept of lower semi-continuity and the Kurdyka-Lojasiewicz property is established. In the second approach, we reformulate   the two-stage stochastic program as a quasi-optimal control.  Unlike the first approach or any conventional method, which determine system parameters for the entire project duration upfront, the optimal control enables annual adjustments to the renewable components' sizing. This dynamic strategy alleviates early-stage under-utilization, aligning system capacity with evolving demand patterns. This reduces the number of optimization variables significantly, simplifying the solution process. Moreover, thousands of constraints are replaced with a system of differential equations, enhancing computational efficiency. By minimizing capital costs and dynamic operating expenses, we achieve an optimal system size. To demonstrate the practical applicability and superiority of our approach, we conduct a comprehensive case study in a rural area of South Africa. Our modelling approaches are then compared against traditional methods such as progressive hedging and Monte Carlo techniques, giving significant improvements.

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 TAUBER Pierre Clovis

University of Tours

Vector-Valued Medical Image Processing

Alzheimer’s Disease (AD) is one of the most important neurodegenerative diseases worldwide with 38.5 million cases in 2023. Being able to identify the people who will develop AD is a major challenge to face its evolution and react in time. Although numerous studies focused on predicting the individual who will face AD conversion or diagnose current and future patients’ AD states, the challenge of AD is not solved and is still under close investigation. In this work, we proposed a model that exploits the multidimensional aspect of the data acquired in clinical routine along with a longitudinal Bayesian Nonlinear Mixed effect model to estimate population and individual disorder trajectories at an entire life timescale. The evolution and pattern of an entire population is characterized by a unique continuous curve, reflecting average disorder progression steps. More importantly, this allows one to estimate the disease state for the patient and predict its evolution within a given period of time. Using 5033 observations constituted of MP-RAGE MRI scans, Cognitive Scores, and background data, from 883 individuals who had at least 4 observations and no regression of their theoretical diagnosis, a complete model has been trained. Generated population curves exhibit coherent trends and keys events in comparison to AD state of the art and allow to characterize individual disease development by deformation. Individual deformation parameters have been proved as relevant indicators for individual AD progression, predicting 3 years in advance theoretical diagnosis with a 0.952 ± 0.026 accuracy and AD conversion in this period of time with a 0.916 ± 0.044 accuracy. Facing the challenging AD classification and predicting the future evolution of the diagnosis illustrates the potential of the proposed approach and its usefulness for clinicians and researchers to detect abnormal disease progression in advance.

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OMRANE Abdennebi

University of French Guyane

Mathematical study of some models in ecology and agroecology

Here, we present some PDE mathematical models in application to ecology and agroecology.
The goal of modeling is to understand some phenomena. In these studies the target is to
preserve biodiversity and/or to reduce pollution.
The first work deals with a mathematical model of a forest harvesting problem using a first
order nonlinear PDE. An optimal control function representing wood cutting and a criterion to
maximize do complete the model. In the cost function, a term for forest regeneration is added.
We use a fixed point method for the existence of a solution, and we show that the optimal
control problem admits a bang-bang type control.
Secondly, we present a problem of absorption of nutrients by agricultural plants in polluted
soils. This problem is modeled by an advection-diffusion PDE type with missing data (the
pollution term). We use a weak formulation here to show the existence of a solution. For the
optimal control problem, we choose the low regret control method of J.-L. Lions which shows
the existence of an optimal solution.