arXiv subjects

G. Visconti

Publications and source records attributed to G. Visconti.

9 recordsLinked to original sources

arXiv2022

Continuous limits of residual neural networks in case of large input data

Residual deep neural networks (ResNets) are mathematically described as interacting particle systems. In the case of infinitely many layers the ResNet leads to a system of coupled system of ordinary differential equations known as neural differential equations. For large scale input data we derive a mean--field limit and show well--posedness of the resulting description. Further, we analyze the existence of solutions to the training process by using both a controllability and an optimal control point of view. Numerical investigations based on the solution of a formal optimality system illustrate the theoretical findings.

math.AP↗

arXiv2021

Quinpi: integrating conservation laws with CWENO implicit methods

Many interesting applications of hyperbolic systems of equations are stiff, and require the time step to satisfy restrictive stability conditions. One way to avoid small time steps is to use implicit time integration. Implicit integration is quite straightforward for first order schemes. High order schemes instead need also to control spurious oscillations, which requires limiting in space and time also in the implicit case. We propose a framework to simplify considerably the application of high order non oscillatory schemes through the introduction of a low order implicit predictor, which is used both to set up the nonlinear weights of a standard high order space reconstruction, and to achieve limiting in time. In this preliminary work, we concentrate on the case of a third order scheme, based on DIRK integration in time and CWENO reconstruction in space. The numerical tests involve linear and nonlinear scalar conservation laws.

math.NA↗

arXiv2021

Mean-Field and Kinetic Descriptions of Neural Differential Equations

Nowadays, neural networks are widely used in many applications as artificial intelligence models for learning tasks. Since typically neural networks process a very large amount of data, it is convenient to formulate them within the mean-field and kinetic theory. In this work we focus on a particular class of neural networks, i.e. the residual neural networks, assuming that each layer is characterized by the same number of neurons $N$, which is fixed by the dimension of the data. This assumption allows to interpret the residual neural network as a time-discretized ordinary differential equation, in analogy with neural differential equations. The mean-field description is then obtained in the limit of infinitely many input data. This leads to a Vlasov-type partial differential equation which describes the evolution of the distribution of the input data. We analyze steady states and sensitivity with respect to the parameters of the network, namely the weights and the bias. In the simple setting of a linear activation function and one-dimensional input data, the study of the moments provides insights on the choice of the parameters of the network. Furthermore, a modification of the microscopic dynamics, inspired by stochastic residual neural networks, leads to a Fokker-Planck formulation of the network, in which the concept of network training is replaced by the task of fitting distributions. The performed analysis is validated by artificial numerical simulations. In particular, results on classification and regression problems are presented.

math.OC↗

arXiv2021

Model of vehicle interactions with autonomous cars and its properties

We study a hierarchy of models based on kinetic equations for the descriptions of traffic flow in presence of autonomous and human--driven vehicles. The autonomous cars considered in this paper are thought of as vehicles endowed with some degree of autonomous driving which decreases the stochasticity of the drivers' behavior. Compared to the existing literature, we do not model autonomous cars as externally controlled vehicles. We investigate whether this feature is enough to provide a stabilization of traffic instabilities such as stop and go waves. We propose two indicators to quantify traffic instability and we find, with analytical and numerical tools, that traffic instabilities are damped as the penetration rate of the autonomous vehicles increases.

physics.soc-ph↗

arXiv2020

Qualitative Properties of Mathematical Model For Data Flow

In this paper, properties of a recently proposed mathematical model for data flow in large-scale asynchronous computer systems are analyzed. In particular, the existence of special weak solutions based on propagating fronts is established. Qualitative properties of these solutions are investigated, both theoretically and numerically.

math.AP↗

arXiv2020

Identifying trending coefficients with an ensemble Kalman filter

This paper extends the ensemble Kalman filter (EnKF) for inverse problems to identify trending model coefficients. This is done by repeatedly inflating the ensemble while maintaining the mean of the particles. As a benchmark serves a classic EnKF and a recursive least squares (RLS) on the example of identifying a force model in milling, which changes due to the progression of tool wear. For a proper comparison, the true values are simulated and augmented with white Gaussian noise. The results demonstrate the feasibility of the approach for dynamic identification while still achieving good accuracy in the static case. Further, the inflated EnKF shows a remarkably insensitivity on the starting set but a less smooth convergence compared to the classic EnKF.

math.OC↗

arXiv2020

From kinetic to macroscopic models and back

We study kinetic models for traffic flow characterized by the property of producing backward propagating waves. These waves may be identified with the phenomenon of stop-and-go waves typically observed on highways. In particular, a refined modeling of the space of the microscopic speeds and of the interaction rate in the kinetic model allows to obtain weakly unstable backward propagating waves in dense traffic, without relying on non-local terms or multi--valued fundamental diagrams. A stability analysis of these waves is carried out using the Chapman-Enskog expansion. This leads to a BGK-type model derived as the mesoscopic limit of a Follow-The-Leader or Bando model, and its macroscopic limit belongs to the class of second-order Aw-Rascle and Zhang models.

math.AP↗

arXiv2019

Reconstruction of traffic speed distributions from kinetic models with uncertainties

In this work we investigate the ability of a kinetic approach for traffic dynamics to predict speed distributions obtained through rough data. The present approach adopts the formalism of uncertainty quantification, since reaction strengths are uncertain and linked to different types of driver behaviour or different classes of vehicles present in the flow. Therefore, the calibration of the expected speed distribution has to face the reconstruction of the distribution of the uncertainty. We adopt experimental microscopic measurements recorded on a German motorway, whose speed distribution shows a multimodal trend. The calibration is performed by extrapolating the uncertainty parameters of the kinetic distribution via a constrained optimisation approach. The results confirm the validity of the theoretical set-up.

nlin.AO↗

arXiv2016

CWENO: uniformly accurate reconstructions for balance laws

In this paper we introduce a general framework for defining and studying essentially non-oscillatory reconstruction procedures of arbitrarily high order accuracy, interpolating data in a central stencil around a given computational cell ($\CWENO$). This technique relies on the same selection mechanism of smooth stencils adopted in $\WENO$, but here the pool of candidates for the selection includes polynomials of different degrees. This seemingly minor difference allows to compute an analytic expression of a polynomial interpolant, approximating the unknown function uniformly within a cell, instead of only at one point at a time. For this reason this technique is particularly suited for balance laws for finite volume schemes, when averages of source terms require high order quadrature rules based on several points; in the computation of local averages, during refinement in h-adaptive schemes; or in the transfer of the solution between grids in moving mesh techniques, and in general when a globally defined reconstruction is needed. Previously, these needs have been satisfied mostly by ENO reconstruction techniques, which, however, require a much wider stencil then the $\CWENO$ reconstruction studied here, for the same accuracy.

math.NA↗