Search

Physics-Informed Neural Networks (PINNs) have revolutionized solving differential equations by integrating physical laws into neural network training. This paper explores PINNs for open-loop optimal control problems (OCPs) with incomplete information, such as unknown initial conditions and sparse boundary data. We derive optimality conditions from the Lagrangian multipliers and use a neural network to predict the state, adjoint, and control variables. In contrast with previous methods, our approach integrates these elements into a single neural network and addresses scenarios with consistently limited data. Specifically, we address the study of partially unknown equations identifying underlying parameters online by searching for the optimal solution. Numerical examples show the effectiveness of the proposed method even in scenarios characterized by a considerable lack of information.

This paper presents a critical analysis of the literature and perspective research ideas for modeling the epidemics caused by the SARS-CoV-2 virus. It goes beyond deterministic population dynamics to consider several key complexity features of the system under consideration. In particular, the multiscale features of the dynamics from contagion to the subsequent dynamics of competition between the immune system and the proliferating virus. Other topics addressed in this work include the propagation of epidemics in a territory, taking into account local transportation networks, the heterogeneity of the population, and the study of social and economic problems in populations involved in the spread of epidemics. The overall content aims to show how new mathematical tools can be developed to address the above topics and how mathematical models and simulations can contribute to the decision making of crisis managers.

The temporal evolution of a contagious viral disease is modelled as the dynamic progression of different classes of population with individuals interacting pairwise. This interaction follows a binary mechanism typical of kinetic theory, wherein agents aim to improve their condition with respect to a mutual utility target. To this end, we introduce kinetic equations of Boltzmann-type to describe the time evolution of the probability distributions of the multi-agent system. The interactions between agents are defined using principles from price theory, specifically employing Cobb-Douglas utility functions for binary exchange and the Edgeworth box to depict the common exchange area where utility increases for both agents. Several numerical experiments presented in the paper highlight the significance of this mechanism in driving the phenomenon toward endemicity.

We introduce a new class of Discontinuous Galerkin (DG) methods for solving nonlinear conservation laws on unstructured Voronoi meshes that use a nonconforming Virtual Element basis defined within each polygonal control volume. The basis functions are evaluated as an L2 projection of the virtual basis which remains unknown, along the lines of the Virtual Element Method (VEM). Contrarily to the VEM approach, the new basis functions lead to a nonconforming representation of the solution with discontinuous data across the element boundaries, as typically employed in DG discretizations. To improve the condition number of the resulting mass matrix, an orthogonalization of the full basis is proposed. The discretization in time is carried out following the ADER (Arbitrary order DERivative Riemann problem) methodology, which yields one-step fully discrete schemes that make use of a coupled space-time representation of the numerical solution. The space-time basis functions are constructed as a tensor product of the virtual basis in space and a one-dimensional Lagrange nodal basis in time. The resulting space-time stiffness matrix is stabilized by an extension of the dof-dof stabilization technique adopted in the VEM framework, hence allowing an element-local space-time Galerkin finite element predictor to be evaluated. The novel methods are referred to as VEM-DG schemes, and they are arbitrarily high order accurate in space and time. The new VEM-DG algorithms are rigorously validated against a series of benchmarks in the context of compressible Euler and Navier-Stokes equations. Numerical results are verified with respect to literature reference solutions and compared in terms of accuracy and computational efficiency to those obtained using a standard modal DG scheme with Taylor basis functions. An analysis of the condition number of the mass and space-time stiffness matrix is also forwarded.

Particle methods based on evolving the spatial derivatives of the solution were originally introduced to simulate reaction-diffusion processes, inspired by vortex methods for the Navier--Stokes equations. Such methods, referred to as gradient random walk methods, were extensively studied in the '90s and have several interesting features, such as being grid free, automatically adapting to the solution by concentrating elements where the gradient is large and significantly reducing the variance of the standard random walk approach. In this work, we revive these ideas by showing how to generalize the approach to a larger class of partial differential equations, including hyperbolic systems of conservation laws. To achieve this goal, we first extend the classical Monte Carlo method to relaxation approximation of systems of conservation laws, and subsequently consider a novel particle dynamics based on the spatial derivatives of the solution. The methodology, combined with asymptotic-preserving splitting discretization, yields a way to construct a new class of gradient-based Monte Carlo methods for hyperbolic systems of conservation laws. Several results in one spatial dimension for scalar equations and systems of conservation laws show that the new methods are very promising and yield remarkable improvements compared to standard Monte Carlo approaches, either in terms of variance reduction as well as in describing the shock structure.

We introduce a new family of high order accurate semi-implicit schemes for the solution of non-linear hyperbolic partial differential equations on unstructured polygonal meshes. The time discretization is based on a splitting between explicit and implicit terms that may arise either from the multi-scale nature of the governing equations, which involve both slow and fast scales, or in the context of projection methods, where the numerical solution is projected onto the physically meaningful solution manifold. We propose to use a high order finite volume (FV) scheme for the explicit terms, ensuring conservation property and robustness across shock waves, while the virtual element method (VEM) is employed to deal with the discretization of the implicit terms, which typically requires an elliptic problem to be solved. The numerical solution is then transferred via suitable L2 projection operators from the FV to the VEM solution space and vice-versa. High order time accuracy is achieved using the semi-implicit IMEX Runge-Kutta schemes, and the novel schemes are proven to be asymptotic preserving and well-balanced. As representative models, we choose the shallow water equations (SWE), thus handling multiple time scales characterized by a different Froude number, and the incompressible Navier-Stokes equations (INS), which are solved at the aid of a projection method to satisfy the solenoidal constraint of the velocity field. Furthermore, an implicit discretization for the viscous terms is devised for the INS model, which is based on the VEM technique. Consequently, the CFL-type stability condition on the maximum admissible time step is based only on the fluid velocity and not on the celerity nor on the viscous eigenvalues. A large suite of test cases demonstrates the accuracy and the capabilities of the new family of schemes to solve relevant benchmarks in the field of incompressible fluids.

In this paper, a multiscale constitutive framework for one-dimensional blood flow modeling is presented and discussed. By analyzing the asymptotic limits of the proposed model, it is shown that different types of blood propagation phenomena in arteries and veins can be described through an appropriate choice of scaling parameters, which are related to distinct characterizations of the fluid-structure interaction mechanism (whether elastic or viscoelastic) that exist between vessel walls and blood flow. In these asymptotic limits, well-known blood flow models from the literature are recovered. Additionally, by analyzing the perturbation of the local elastic equilibrium of the system, a new viscoelastic blood flow model is derived. The proposed approach is highly flexible and suitable for studying the human cardiovascular system, which is composed of vessels with high morphological and mechanical variability. The resulting multiscale hyperbolic model of blood flow is solved using an asymptotic-preserving Implicit-Explicit Runge-Kutta Finite Volume method, which ensures the consistency of the numerical scheme with the different asymptotic limits of the mathematical model without affecting the choice of the time step by restrictions related to the smallness of the scaling parameters. Several numerical tests confirm the validity of the proposed methodology, including a case study investigating the hemodynamics of a thoracic aorta in the presence of a stent.

With the rapid advance of Machine Learning techniques and the deep increase of availability of scientific data, data-driven approaches have started to become progressively popular across science, causing a fundamental shift in the scientific method after proving to be powerful tools with a direct impact in many areas of society. Nevertheless, when attempting to analyze dynamics of complex multiscale systems, the usage of standard Deep Neural Networks (DNNs) and even standard Physics-Informed Neural Networks (PINNs) may lead to incorrect inferences and predictions, due to the presence of small scales leading to reduced or simplified models in the system that have to be applied consistently during the learning process. In this Chapter, we will address these issues in light of recent results obtained in the development of Asymptotic-Preserving Neural Networks (APNNs) for hyperbolic models with diffusive scaling. Several numerical tests show how APNNs provide considerably better results with respect to the different scales of the problem when compared with standard DNNs and PINNs, especially when analyzing scenarios in which only little and scattered information is available., Comment: arXiv admin note: text overlap with arXiv:2206.12625

When investigating epidemic dynamics through differential models, the parameters needed to understand the phenomenon and to simulate forecast scenarios require a delicate calibration phase, often made even more challenging by the scarcity and uncertainty of the observed data reported by official sources. In this context, Physics-Informed Neural Networks (PINNs), by embedding the knowledge of the differential model that governs the physical phenomenon in the learning process, can effectively address the inverse and forward problem of data-driven learning and solving the corresponding epidemic problem. In many circumstances, however, the spatial propagation of an infectious disease is characterized by movements of individuals at different scales governed by multiscale PDEs. This reflects the heterogeneity of a region or territory in relation to the dynamics within cities and in neighboring zones. In presence of multiple scales, a direct application of PINNs generally leads to poor results due to the multiscale nature of the differential model in the loss function of the neural network. To allow the neural network to operate uniformly with respect to the small scales, it is desirable that the neural network satisfies an Asymptotic-Preservation (AP) property in the learning process. To this end, we consider a new class of AP Neural Networks (APNNs) for multiscale hyperbolic transport models of epidemic spread that, thanks to an appropriate AP formulation of the loss function, is capable to work uniformly at the different scales of the system. A series of numerical tests for different epidemic scenarios confirms the validity of the proposed approach, highlighting the importance of the AP property in the neural network when dealing with multiscale problems especially in presence of sparse and partially observed systems.

Uncertainty in data is certainly one of the main problems in epidemiology, as shown by the recent COVID-19 pandemic. The need for efficient methods capable of quantifying uncertainty in the mathematical model is essential in order to produce realistic scenarios of the spread of infection. In this paper, we introduce a bi-fidelity approach to quantify uncertainty in spatially dependent epidemic models. The approach is based on evaluating a high-fidelity model on a small number of samples properly selected from a large number of evaluations of a low-fidelity model. In particular, we will consider the class of multiscale transport models recently introduced in Bertaglia, Boscheri, Dimarco & Pareschi, Math. Biosci. Eng. (2021) and Boscheri, Dimarco & Pareschi, Math. Mod. Meth. App. Scie. (2021) as the high-fidelity reference and use simple two-velocity discrete models for low-fidelity evaluations. Both models share the same diffusive behavior and are solved with ad-hoc asymptotic-preserving numerical discretizations. A series of numerical experiments confirm the validity of the approach.

In this survey we report some recent results in the mathematical modeling of epidemic phenomena through the use of kinetic equations. We initially consider models of interaction between agents in which social characteristics play a key role in the spread of an epidemic, such as the age of individuals, the number of social contacts, and their economic wealth. Subsequently, for such models, we discuss the possibility of containing the epidemic through an appropriate optimal control formulation based on the policy maker's perception of the progress of the epidemic. The role of uncertainty in the data is also discussed and addressed. Finally, the kinetic modeling is extended to spatially dependent settings using multiscale transport models that can characterize the impact of movement dynamics on epidemic advancement on both one-dimensional networks and realistic two-dimensional geographic settings.

In this paper we introduce a space-dependent multiscale model to describe the spatial spread of an infectious disease under uncertain data with particular interest in simulating the onset of the COVID-19 epidemic in Italy. While virus transmission is ruled by a SEIAR type compartmental model, within our approach the population is given by a sum of commuters moving on a extra-urban scale and non commuters interacting only on the smaller urban scale. A transport dynamic of the commuter population at large spatial scales, based on kinetic equations, is coupled with a diffusion model for non commuters at the urban scale. Thanks to a suitable scaling limit, the kinetic transport model used to describe the dynamics of commuters, within a given urban area coincides with the diffusion equations that characterize the movement of non-commuting individuals. Because of the high uncertainty in the data reported in the early phase of the epidemic, the presence of random inputs in both the initial data and the epidemic parameters is included in the model. A robust numerical method is designed to deal with the presence of multiple scales and the uncertainty quantification process. In our simulations, we considered a realistic geographical domain, describing the Lombardy region, in which the size of the cities, the number of infected individuals, the average number of daily commuters moving from one city to another, and the epidemic aspects are taken into account through a calibration of the model parameters based on the actual available data. The results show that the model is able to describe correctly the main features of the spatial expansion of the first wave of COVID-19 in northern Italy.

The importance of spatial networks in the spread of an epidemic is an essential aspect in modeling the dynamics of an infectious disease. Additionally, any realistic data-driven model must take into account the large uncertainty in the values reported by official sources, such as the amount of infectious individuals. In this paper we address the above aspects through a hyperbolic compartmental model on networks, in which nodes identify locations of interest, such as cities or regions, and arcs represent the ensemble of main mobility paths. The model describes the spatial movement and interactions of a population partitioned, from an epidemiological point of view, on the basis of an extended compartmental structure and divided into commuters, moving on a suburban scale, and non-commuters, acting on an urban scale. Through a diffusive rescaling, the model allows us to recover classical diffusion equations related to commuting dynamics. The numerical solution of the resulting multiscale hyperbolic system with uncertainty is then tackled using a stochastic collocation approach in combination with a finite-volume IMEX method. The ability of the model to correctly describe the spatial heterogeneity underlying the spread of an epidemic in a realistic city network is confirmed with a study of the outbreak of COVID-19 in Italy and its spread in the Lombardy Region.

A noteworthy aspect in blood flow modeling is the definition of the mechanical interaction between the fluid flow and the biological structure that contains it, namely the vessel wall. It has been demonstrated that the addition of a viscous contribution to the mechanical characterization of vessels brings positive results when compared to in-vivo measurements. In this context, the numerical implementation of boundary conditions able to keep memory of the viscoelastic contribution of vessel walls assumes an important role, especially when dealing with large circulatory systems. In this work, viscoelasticity is taken into account in entire networks via the Standard Linear Solid Model. The implementation of the viscoelastic contribution at boundaries (inlet, outlet and junctions), is carried out considering the hyperbolic nature of the mathematical model. A non-linear system is established based on the definition of the Riemann Problem at junctions, characterized by rarefaction waves separated by contact discontinuities, among which the mass and the total energy are conserved. Basic junction tests are analyzed, such as a trivial 2-vessels junction, for both a generic artery and vein, and a simple 3-vessels junction, considering an aortic bifurcation scenario. The chosen asymptotic-preserving IMEX Runge-Kutta Finite Volume scheme is demonstrated to be second-order accurate in the whole domain and well-balanced, even when including junctions. Two different benchmark models of the arterial network are implemented, differing in number of vessels and in viscoelastic parameters. Comparison of the results obtained in the two networks underlines the high sensitivity of the model to the chosen viscoelastic parameters. The conservation of the contribution provided by the viscoelastic characterization of vessel walls is assessed in the whole network, including junctions and boundary conditions., Comment: 42 pages, 19 figures

We consider the development of hyperbolic transport models for the propagation in space of an epidemic phenomenon described by a classical compartmental dynamics. The model is based on a kinetic description at discrete velocities of the spatial movement and interactions of a population of susceptible, infected and recovered individuals. Thanks to this, the unphysical feature of instantaneous diffusive effects, which is typical of parabolic models, is removed. In particular, we formally show how such reaction-diffusion models are recovered in an appropriate diffusive limit. The kinetic transport model is therefore considered within a spatial network, characterizing different places such as villages, cities, countries, etc. The transmission conditions in the nodes are analyzed and defined. Finally, the model is solved numerically on the network through a finite-volume IMEX method able to maintain the consistency with the diffusive limit without restrictions due to the scaling parameters. Several numerical tests for simple epidemic network structures are reported and confirm the ability of the model to correctly describe the spread of an epidemic.

This work aims at identifying and quantifying uncertainties related to elastic and viscoelastic parameters, which characterize the arterial wall behavior, in one-dimensional modeling of the human arterial hemodynamics. The chosen uncertain parameters are modeled as random Gaussian-distributed variables, making stochastic the system of governing equations. The proposed methodology is initially validated on a model equation, presenting a thorough convergence study which confirms the spectral accuracy of the stochastic collocation method and the second-order accuracy of the IMEX finite volume scheme chosen to solve the mathematical model. Then, univariate and multivariate uncertain quantification analyses are applied to the a-FSI blood flow model, concerning baseline and patient-specific single-artery test cases. A different sensitivity is depicted when comparing the variability of flow rate and velocity waveforms to the variability of pressure and area, the latter ones resulting much more sensitive to the parametric uncertainties underlying the mechanical characterization of vessel walls. Simulations performed considering both the simple elastic and the more realistic viscoelastic constitutive law show that the great uncertainty of the viscosity parameter plays a major role in the prediction of pressure waveforms, enlarging the confidence interval of this variable. In-vivo recorded patient-specific pressure data falls within the confidence interval of the output obtained with the proposed methodology and expectations of the computed pressures are comparable to the recorded waveforms.

A wide and critical comparison of the capability of Method of Characteristics, Explicit Path-Conservative Finite Volume Method and Semi-Implicit Staggered Finite Volume Method is presented and discussed, in terms of accuracy and efficiency. The viscoelastic behaviour of the pipe wall, the effects of the unsteadiness of the flow on the friction losses, cavitation and cross-sectional changes are considered. The analyses are performed comparing numerical solutions obtained using the three models against experimental data and analytical solutions. Water hammer studies in high density polyethylene pipes, for which laboratory data have been provided, are used as test cases. Considering the viscoelastic mechanical behaviour of plastic materials, a 3-parameter and a multi-parameter linear viscoelastic rheological model are adopted and implemented in each numerical scheme. Original extensions of existing techniques for the numerical treatment of such viscoelastic models are introduced in this work for the first time. After a focused calibration of the viscoelastic parameters, the different performance of the numerical models is investigated. A comparison of the results is presented considering the unsteady wall-shear stress, with a new approach proposed for turbulent flows, or simply considering a quasi-steady friction model. A predominance of the damping effect due to viscoelasticity with respect to the damping effect related to the unsteady friction is confirmed in these contexts. All the numerical methods show a good agreement with the experimental data and a high efficiency of the Method of Characteristics in standard configuration is observed. Three Riemann Problems are chosen and run to stress the numerical methods, considering cross-sectional changes, more flexible materials and cavitation cases. In these demanding scenarios, the weak spots of the Method of Characteristics are depicted.

Mathematical models are widely recognized as a valuable tool for cardiovascular diagnosis and the study of circulatory diseases, especially to obtain data that require otherwise invasive measurements. To correctly simulate body hemodynamics, the viscoelastic properties of vessel walls are a key aspect to be taken into account as they play an essential role in cardiovascular behavior. The present work aims to apply the augmented fluid-structure interaction system of blood flow to real case studies to assess the validity of the model as a valuable resource to improve cardiovascular diagnostics and the treatment of pathologies. First, the ability of the model to correctly simulate pulse waveforms in single arterial segments is verified using literature benchmark test cases. Such cases are designed taking into account a simple elastic behavior of the wall in the upper thoracic aorta and in the common carotid artery. Furthermore, in-vivo pressure waveforms, extracted from tonometric measurements performed on four human common carotid arteries and two common femoral arteries, are compared to numerical solutions. It is highlighted that the viscoelastic damping effect of arterial walls is required to avoid an overestimation of pressure peaks. An effective procedure to estimate the viscoelastic parameters of the model is herein proposed, which returns hysteresis curves of the common carotid arteries dissipating energy fractions in line with values calculated from literature hysteresis loops in the same vessel., Comment: submitted to Journal of Biomechanics

Mathematical models and numerical simulations are widely used in the field of hemodynamics, representing a valuable resource to better understand physiological and pathological processes. The theory behind the phenomenon is closely related to the study of incompressible flow through compliant thin-walled tubes. The mechanical interaction between blood flow and vessel wall must be properly described by the model. Recent works show the benefits of characterizing the rheology of the vessel wall through a viscoelastic law. Considering the viscous contribution of the wall material and not simply the elastic one leads to a more realistic representation of the vessel behavior, which manifests not only an instantaneous elastic strain but also a viscous damping effect on pulse pressure waves, coupled to energy losses. The aim of this work is to propose an easily extensible 1D mathematical model able to accurately capture the FSI. The originality lies in the introduction of a viscoelastic tube law in PDE form, valid for both arterial and venous networks, leading to an augmented FSI system. In contrast to well established mathematical models, the proposed one is natively hyperbolic. The model is solved with a 2nd-order numerical scheme based on an IMEX RK scheme conceived for applications to hyperbolic systems with stiff relaxation terms. The validation of the model is performed on different tests. Results obtained in Riemann problems, adopting a simple elastic tube law for the characterization of the vessel wall, are compared with available exact solutions. To validate the contribution given by the viscoelastic term, the Method of Manufactured Solutions is applied. Specific tests are designed to verify the well-balancing and the AP property of the scheme. A specific test with an inlet pulse pressure wave is designed to assess the effects of viscoelasticity., Comment: to be published in Computer Methods in Applied Mechanics and Engineering