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We propose a novel surrogate modelling approach to efficiently and accurately approximate the response of complex dynamical systems driven by time-varying exogenous excitations over extended time periods. Our approach, namely manifold nonlinear autoregressive modelling with exogenous input (mNARX), involves constructing a problem-specific exogenous input manifold that is optimal for constructing autoregressive surrogates. The manifold, which forms the core of mNARX, is constructed incrementally by incorporating the physics of the system, as well as prior expert- and domain- knowledge. Because mNARX decomposes the full problem into a series of smaller sub-problems, each with a lower complexity than the original, it scales well with the complexity of the problem, both in terms of training and evaluation costs of the final surrogate. Furthermore, mNARX synergizes well with traditional dimensionality reduction techniques, making it highly suitable for modelling dynamical systems with high-dimensional exogenous inputs, a class of problems that is typically challenging to solve. Since domain knowledge is particularly abundant in physical systems, such as those found in civil and mechanical engineering, mNARX is well suited for these applications. We demonstrate that mNARX outperforms traditional autoregressive surrogates in predicting the response of a classical coupled spring-mass system excited by a one-dimensional random excitation. Additionally, we show that mNARX is well suited for emulating very high-dimensional time- and state-dependent systems, even when affected by active controllers, by surrogating the dynamics of a realistic aero-servo-elastic onshore wind turbine simulator. In general, our results demonstrate that mNARX offers promising prospects for modelling complex dynamical systems, in terms of accuracy and efficiency.

A comprehensive mathematical model of the multiphysics flow of blood and Cerebrospinal Fluid (CSF) in the brain can be expressed as the coupling of a poromechanics system and Stokes' equations: the first describes fluids filtration through the cerebral tissue and the tissue's elastic response, while the latter models the flow of the CSF in the brain ventricles. This model describes the functioning of the brain's waste clearance mechanism, which has been recently discovered to play an essential role in the progress of neurodegenerative diseases. To model the interactions between different scales in the porous medium, we propose a physically consistent coupling between Multi-compartment Poroelasticity (MPE) equations and Stokes' equations. In this work, we introduce a numerical scheme for the discretization of such coupled MPE-Stokes system, employing a high-order discontinuous Galerkin method on polytopal grids to efficiently account for the geometric complexity of the domain. We analyze the stability and convergence of the space semidiscretized formulation, we prove a-priori error estimates, and we present a temporal discretization based on a combination of Newmark's $\beta$-method for the elastic wave equation and the $\theta$-method for the other equations of the model. Numerical simulations carried out on test cases with manufactured solutions validate the theoretical error estimates. We also present numerical results on a two-dimensional slice of a patient-specific brain geometry reconstructed from diagnostic images, to test in practice the advantages of the proposed approach.

Prognostic Health Management aims to predict the Remaining Useful Life (RUL) of degrading components/systems utilizing monitoring data. These RUL predictions form the basis for optimizing maintenance planning in a Predictive Maintenance (PdM) paradigm. We here propose a metric for assessing data-driven prognostic algorithms based on their impact on downstream PdM decisions. The metric is defined in association with a decision setting and a corresponding PdM policy. We consider two typical PdM decision settings, namely component ordering and/or replacement planning, for which we investigate and improve PdM policies that are commonly utilized in the literature. All policies are evaluated via the data-based estimation of the long-run expected maintenance cost per unit time, using monitored run-to-failure experiments. The policy evaluation enables the estimation of the proposed metric. We employ the metric as an objective function for optimizing heuristic PdM policies and algorithms' hyperparameters. The effect of different PdM policies on the metric is initially investigated through a theoretical numerical example. Subsequently, we employ four data-driven prognostic algorithms on a simulated turbofan engine degradation problem, and investigate the joint effect of prognostic algorithm and PdM policy on the metric, resulting in a decision-oriented performance assessment of these algorithms.

Time-Aware Shaper (TAS) is a time-triggered scheduling mechanism that ensures bounded latency for time-critical Scheduled Traffic (ST) flows. The Linux kernel implementation (a.k.a TAPRIO) has limited capabilities due to varying CPU workloads and thus does not offer tight latency bound for the ST flows. Also, currently only higher cycle times are possible. Other software implementations are limited to simulation studies without physical implementation. In this paper, we present $\mu$TAS, a MicroC-based hardware implementation of TAS onto a programmable SmartNIC. $\mu$TAS takes advantage of the parallel-processing architecture of the SmartNIC to configure the scheduling behaviour of its queues at runtime. To demonstrate the effectiveness of $\mu$TAS, we built a Time-Sensitive Networking (TSN) testbed from scratch. This consists of multiple end-hosts capable of generating ST and Best Effort (BE) flows and TSN switches equipped with SmartNICs running $\mu$TAS. Time synchronization is maintained between the switches and hosts. Our experiments demonstrate that the ST flows experience a bounded latency of the order of tens of microseconds.

This paper presents a new weak Galerkin (WG) method for elliptic interface problems on general curved polygonal partitions. The method's key innovation lies in its ability to transform the complex interface jump condition into a more manageable Dirichlet boundary condition, simplifying the theoretical analysis significantly. The numerical scheme is designed by using locally constructed weak gradient on the curved polygonal partitions. We establish error estimates of optimal order for the numerical approximation in both discrete $H^1$ and $L^2$ norms. Additionally, we present various numerical results that serve to illustrate the robust numerical performance of the proposed WG interface method.

In situations where both extreme and non-extreme data are of interest, modelling the whole data set accurately is important. In a univariate framework, modelling the bulk and tail of a distribution has been extensively studied before. However, when more than one variable is of concern, models that aim specifically at capturing both regions correctly are scarce in the literature. A dependence model that blends two copulas with different characteristics over the whole range of the data support is proposed. One copula is tailored to the bulk and the other to the tail, with a dynamic weighting function employed to transition smoothly between them. Tail dependence properties are investigated numerically and simulation is used to confirm that the blended model is sufficiently flexible to capture a wide variety of structures. The model is applied to study the dependence between temperature and ozone concentration at two sites in the UK and compared with a single copula fit. The proposed model provides a better, more flexible, fit to the data, and is also capable of capturing complex dependence structures.

We aim to establish Bowen's equations for upper capacity invariance pressure and Pesin-Pitskel invariance pressure of discrete-time control systems. We first introduce a new invariance pressure called induced invariance pressure on partitions that specializes the upper capacity invariance pressure on partitions, and then show that the two types of invariance pressures are related by a Bowen's equation. Besides, to establish Bowen's equation for Pesin-Pitskel invariance pressure on partitions we also introduce a new notion called BS invariance dimension on subsets. Moreover, a variational principle for BS invariance dimension on subsets is established.

Learning and predicting the dynamics of physical systems requires a profound understanding of the underlying physical laws. Recent works on learning physical laws involve generalizing the equation discovery frameworks to the discovery of Hamiltonian and Lagrangian of physical systems. While the existing methods parameterize the Lagrangian using neural networks, we propose an alternate framework for learning interpretable Lagrangian descriptions of physical systems from limited data using the sparse Bayesian approach. Unlike existing neural network-based approaches, the proposed approach (a) yields an interpretable description of Lagrangian, (b) exploits Bayesian learning to quantify the epistemic uncertainty due to limited data, (c) automates the distillation of Hamiltonian from the learned Lagrangian using Legendre transformation, and (d) provides ordinary (ODE) and partial differential equation (PDE) based descriptions of the observed systems. Six different examples involving both discrete and continuous system illustrates the efficacy of the proposed approach.

We explore a linear inhomogeneous elasticity equation with random Lam\'e parameters. The latter are parameterized by a countably infinite number of terms in separated expansions. The main aim of this work is to estimate expected values (considered as an infinite dimensional integral on the parametric space corresponding to the random coefficients) of linear functionals acting on the solution of the elasticity equation. To achieve this, the expansions of the random parameters are truncated, a high-order quasi-Monte Carlo (QMC) is combined with a sparse grid approach to approximate the high dimensional integral, and a Galerkin finite element method (FEM) is introduced to approximate the solution of the elasticity equation over the physical domain. The error estimates from (1) truncating the infinite expansion, (2) the Galerkin FEM, and (3) the QMC sparse grid quadrature rule are all studied. For this purpose, we show certain required regularity properties of the continuous solution with respect to both the parametric and physical variables. To achieve our theoretical regularity and convergence results, some reasonable assumptions on the expansions of the random coefficients are imposed. Finally, some numerical results are delivered.

The emergence of complex structures in the systems governed by a simple set of rules is among the most fascinating aspects of Nature. The particularly powerful and versatile model suitable for investigating this phenomenon is provided by cellular automata, with the Game of Life being one of the most prominent examples. However, this simplified model can be too limiting in providing a tool for modelling real systems. To address this, we introduce and study an extended version of the Game of Life, with the dynamical process governing the rule selection at each step. We show that the introduced modification significantly alters the behaviour of the game. We also demonstrate that the choice of the synchronization policy can be used to control the trade-off between the stability and the growth in the system.