A mathematical construction of the digital twin of bread leavening

One of the main concepts of the fourth industrial revolution is the digital twin (DT), which helps build a bridge between the physical and the digital world and is classically defined as the virtual replica of a real system that is continuously updated with data from its physical counterpart. The main idea of our project, which leads to the development of this doctoral thesis is the mathematical building of a DT of bread leavening to avoid energy waste. The focus of this thesis is the use, development and connection of mathematical theories, such as continuum mechanics, machine learning and source estimation via inverse problems, to formulate and solve mathematical problems with an industrial application. In this doctoral thesis, we develop a mathematical model of bread leavening in a warm chamber by coupling heat transfer, yeast growth, and the presence of carbon dioxide with the deformation of bread dough. Specifically, the above-mentioned continuum model involves a heat equation to describe the evolution of temperature in the bread, an ordinary differential equation to mirror the life cycle of yeasts and their breeding, then a diffusive equation for the carbon dioxide production and propagation and finally, the volumetric expansion represented by considering the elastic energy related to the bread, seen as a hyperelastic material. We analyze the corresponding system of partial differential equations and we discretize the problem, by using a semi-implicit Euler method and FEM for the time and space variables, respectively. Numerical simulations allow a sensitivity analysis to identify the energy consumption necessary to achieve a target volume under different settings of the leavening chamber and different concentrations of yeast in the bread dough, thereby providing a tool for the identification of cost-effective energy usage protocols [1]. At this point, we study operator networks that emerged as promising deep learning tools for approximating the solution of PDEs. These networks map input functions that describe material properties, forcing functions and boundary data to the solution of a PDE. Therefore, we build a surrogate model, based on the numerical one, which simulates the physical behavior, is computationally cheaper and needs few data as input. We exploit the physical-mathematical model, especially its space-time weak formulation, to define an ad-hoc variationally mimetic operator network (VarMiON) by following and adapting the procedure presented in [2] to the PDEs that characterize our problem. In the application considered here, the presence of a surrogate model is motivated by the fact that it has to run online with the real process to monitor it and to estimate the energy consumption thus avoiding waste [3]. Finally, we present a general observation on how to replace changes of material properities in limited regions within a domain with fictitious forcing terms in initial- and boundary-value problems associated with linear wave propagation and diffusion [4]. Then, by considering a paradigmatic heat conduction problem on a domain with a cavity, we prove that the presence of the void can be replaced by a fictitious heat source with support that coincides with the cavity. We illustrate this fact in a situation where the source term can be analytically recovered from the values of the temperature and heat flux at the boundary of the cavity. Our result provides a strategy to map the nonlinear geometric inverse problem of void identification into a more manageable one that involves the estimate of forcing terms given the knowledge of external boundary data. To set the stage for a systematic study of the inverse problem, we present algebraic reconstructions, based on FEM, that give an approximation of the fictitious source from different sets of temperature measurements. In future work, we want to use the latter strategy to rebuild the porous texture of bread by solving the resulting source inverse problem.