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Physical systems such as water and gas networks are usually operated in a state of equilibrium and feedback control is employed to damp small perturbations over time. We consider flow problems on networks, described by hyperbolic balance laws, and analyze the stability of the linearized systems. Sufficient conditions for exponential stability in the continuous and discretized setting are presented. The analysis is extended to arbitrary Sobolev norms. Computational experiments illustrate the theoretical findings.

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In the past decades, numerous experiments have emerged to unveil the nature of dark matter, one of the most discussed open questions in modern particle physics. Among them, the CRESST experiment, located at the Laboratori Nazionali del Gran Sasso, operates scintillating crystals as cryogenic phonon detectors. In this work, we present first results from the operation of two detector modules which both have 10.46 g LiAlO 2 targets in CRESST-III. The lithium contents in the crystal are 6 Li, with an odd number of protons and neutrons, and 7 Li, with an odd number of protons. By considering both isotopes of lithium and 27 Al, we set the currently strongest cross section upper limits on spin-dependent interaction of dark matter with protons and neutrons for the mass region between 0.25 and 1.5 GeV/c 2 .

Recently, neural networks (NN) with an infinite number of layers have been introduced. Especially for these very large NN the training procedure is very expensive. Hence, there is interest to study their robustness with respect to input data to avoid unnecessarily retraining the network. Typically, model-based statistical inference methods, e.g. Bayesian neural networks, are used to quantify uncertainties. Here, we consider a special class of residual neural networks and we study the case, when the number of layers can be arbitrarily large. Then, kinetic theory allows to interpret the network as a dynamical system, described by a partial differential equation. We study the robustness of the mean-field neural network with respect to perturbations in initial data by applying UQ approaches on the loss functions.

We investigate the propagation of uncertainties in the Aw-Rascle-Zhang model, which belongs to a class of second order traffic flow models described by a system of nonlinear hyperbolic equations. The stochastic quantities are expanded in terms of wavelet-based series expansions. Then, they are projected to obtain a deterministic system for the coefficients in the truncated series. Stochastic Galerkin formulations are presented in conservative form and for smooth solutions also in the corresponding non-conservative form. This allows to obtain stabilization results, when the system is relaxed to a first-order model. Computational tests illustrate the theoretical results.