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Optimization approaches are important to design sustainable structures. In structural mechanics, different design objectives can be defined, for example, to minimize the required construction material or to maximize the structural durability. In this paper, the durability of a reinforced concrete (RC) structure is assessed by advanced finite element (FE) models to simulate the cracking behavior and the chloride transport process. The corrosion initiation time is used as durability measure to be maximized within an optimization approach, where the concrete cover is defined as design variable. The variability of structural loads and material parameters and unavoidable construction imprecision leads to a probabilistic reliability and durability assessment, where aleatory as well as epistemic uncertainties are quantified by random variables, intervals and probability‐boxes. The FE simulation models cannot directly be applied to structural analyses and optimizations with polymorphic uncertain parameters and design variables because of the high computational demand of the multi‐loop algorithm (Monte Carlo simulation, interval analysis, global optimization). In this paper, a new surrogate modeling strategy is presented, where artificial neural networks are trained sequentially to speed‐up the coupled mechanical and transport simulation FE models. The new approach is applied to the uncertainty quantification and the structural durability optimization of a RC structure. [ABSTRACT FROM AUTHOR]

In an integrated energy system, source-load multiple uncertainties and correlations lead to an over-limit risk in operating state, including voltage, temperature, and pressure over-limit. Therefore, efficient probabilistic energy flow calculation methods and risk assessment theories applicable to integrated energy systems are crucial. This study proposed a probabilistic energy flow calculation method based on polynomial chaos expansion for an electric-heat-gas integrated energy system. The method accurately and efficiently calculated the over-limit probability of the system state variables, considering the coupling conditions of electricity, heat, and gas, as well as uncertainties and correlations in renewable energy unit outputs and multiple types of loads. To further evaluate and quantify the impact of uncertainty factors on the over-limit risk, a global sensitivity analysis method for the integrated energy system based on the analysis of covariance theory is proposed. This method considered the source-load correlation and aimed to identify the key uncertainty factors that influence stable operation. Simulation results demonstrated that the proposed method achieved accuracy to that of the Monte Carlo method while significantly reducing calculation time. It effectively quantified the over-limit risk under the presence of multiple source-load uncertainties. [ABSTRACT FROM AUTHOR]

Traditional deterministic aerodynamic optimisation cannot consider environmental uncertainty, which may lead to sensitivity issues. The present study proposes a robust design framework for the aerodynamic optimisation of high-speed trains, which accounts for the uncertain wind and its impact on crosswind stability. In this framework, a variance analysis method based on the Non- Intrusive Polynomial Chaos is proposed to determine the deformation area, and a parametric model is subsequently established. The Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is used as the optimiser to minimise the mean and variance of the aerodynamic response. The mean and variance can be quickly predicted by an uncertainty analysis approach combining Monte Carlo simulation and Kriging model. The framework is then applied to the optimisation of a high-speed train under crosswind. The results of the robust optimisation are compared with those of the baseline geometry and deterministic optimisation. The mean and variance of the rolling moment under crosswind are reduced by 2.26% and 3.37% respectively after optimisation, indicating that the performance and robustness are both improved. The proposed framework is effective for the engineering design of high-speed trains and can also provide a reference for the robust design of other aerodynamic shapes. [ABSTRACT FROM AUTHOR]

Reliability analysis of earth slopes considering soil spatial variability has garnered significant attention from researchers. However, previous studies have predominantly focused on long slopes with infinite or ample width, such as dams and subgrades, but rare research was dedicated to slopes with width restricted by boundary constraints. In this regard, this paper proposes an efficient reliability framework suitable for width‐limited slopes in spatially variable soils. The proposed framework employed spatial discretization‐based upper bound limit analysis (UBLA) to conduct the deterministic slope stability analysis. The established failure mechanism can well capture the width‐limited three‐dimensional slip surface characteristics and satisfies the kinematically admissible conditions in spatially variable soils. Two innovative strategies are proposed to boost the efficiency of the discretized mechanism. They reduce the computational time to determine safety factors from over 20 min to less than 2 min. For the uncertainty modeling and slope reliability computing procedure, the Sparse Polynomial Chaos Expansion (SPCE) and Monte Carlo Simulation (MCS) methods are combined to generate the distribution of safety factors and failure probabilities. This combination addresses the high‐dimensional stochastic problem arising from 3D spatial variability. Using the proposed framework, a parametric analysis of a 3D spatially variable slope is conducted to investigate the impact of various factors. The findings of this paper are beneficial to the risk evolution of width‐limited slopes, and the efficiency enhancement strategies have the potential to expedite the limit analysis of other geo‐structures, such as tunnels, retaining walls, and foundations. [ABSTRACT FROM AUTHOR]

• A tailored optimal control computational framework is developed by exploiting the sparsity pattern of the sampling-based approach. • Trajectory optimization of a benchmark re-entry problem with complex path and boundary constraints under high-dimensional uncertainties is conducted. • Robustness-optimality tradeoff is studied for different combinations of uncertainties. • A boundary arc consisting of a continuous line formed by the boundary contact points of each sampled trajectory is observed in the ensemble case. • Results are promising in terms of computational efficiency, accuracy and optimality for use in actual multi-phase mission design studies. This paper introduces a sampling-based computational framework with a pseudospectral transcription, designed to investigate performance-robustness tradeoffs for constrained re-entry flights with a non-Gaussian uncertainty propagation. Ensemble of trajectories which are generated with sparse grids with Conjugate Unscented Transformation have been integrated into the nonlinear programming via a vectorized and parallelized formulation. A cross-range maximization problem is formulated to assess the influences of uncertainties in entry interface conditions and model parametric uncertainties. A comprehensive analysis, facilitated by rapid gradient-based optimization convergence and warm-starting, highlights the significant effects of initial speed uncertainties and drag coefficient on problem feasibility, offering insights into optimal control behavior under different degrees and combinations of uncertainties. Proper scaling of controls and the objective function proves pivotal for achieving convergence. Furthermore, a novel phenomenon called "contact arcs" is discovered, composed of contact points that form an arc at the edge of the path constraint in the constrained ensemble optimal control as a counterpart to the boundary arcs in deterministic optimal control problems. Simulation results show excellent agreement with the derived optimality conditions. [ABSTRACT FROM AUTHOR]

Full-scale dome structures intrinsically have numerous sources of irreducible aleatoric uncertainties. A large-scale numerical simulation of the dome structure is required to quantify the effects of these sources on the dynamic performance of the structure using the finite element method (FEM). To reduce the heavy computational burden, a surrogate model of a dome structure was constructed to solve this problem. The dynamic global sensitivity of elastic and elastoplastic structures was analyzed in the uncertainty quantification framework using fully quantitative variance- and distribution-based methods through the surrogate model. The model considered the predominant sources of uncertainty that have a significant influence on the performance of the dome structure. The effects of the variables on the structural performance indicators were quantified using the sensitivity index values of the different performance states. Finally, the effects of the sample size and correlation function on the accuracy of the surrogate model as well as the effects of the surrogate accuracy and failure probability on the sensitivity index values are discussed. The results show that surrogate modeling has high computational efficiency and acceptable accuracy in the uncertainty quantification of large-scale structures subjected to earthquakes in comparison to the conventional FEM. [ABSTRACT FROM AUTHOR]

The paper is focused on recent advances in uncertainty quantification using polynomial chaos expansion (PCE). PCE is a well-known technique for approximation of costly mathematical models with random inputs - surrogate model. Although PCE is a widely used technique and it has several advantages over various surrogate models, it has still several limitations and research gaps. This paper reviews some of the recent theoretical developments in PCE. Specifically a new active learning method optimizing the experimental design and an extension of analytical statistical analysis using PCE will be reviewed. These two topics represent crucial tools for efficient applications: active learning leads generally to a significantly more efficient construction of PCE and improved statistical analysis allows for analytical estimation of higher statistical moments directly from PCE coefficients. Higher statistical moments can be further used for the identification of probability distribution and estimation of design quantiles, which is a crucial task for the probabilistic analysis of structures. Selected applications of the theoretical methods are briefly presented in a context of civil engineering as well as some preliminary results of further research. A part of the paper also presents UQPy package containing state-of-the-art implementation of the PCE theory. [ABSTRACT FROM AUTHOR]

Polynomial chaos expansions (PCEs) have been used in many real-world engineering applications to quantify how the uncertainty of an output is propagated from inputs by decomposing the output in terms of polynomials of the inputs. PCEs for models with independent inputs have been extensively explored in the literature. Recently, different approaches have been proposed for models with dependent inputs to expand the use of PCEs to more real-world applications. Typical approaches include building PCEs based on the Gram–Schmidt algorithm or transforming the dependent inputs into independent inputs. However, the two approaches have their limitations regarding computational efficiency and additional assumptions about the input distributions, respectively. In this paper, we propose a data-driven approach to build sparse PCEs for models with dependent inputs without any distributional assumptions. The proposed algorithm recursively constructs orthonormal polynomials using a set of monomials based on their correlations with the output. The proposed algorithm on building sparse PCEs not only reduces the number of minimally required observations but also improves the numerical stability and computational efficiency. Four numerical examples are implemented to validate the proposed algorithm. The source code is made publicly available for reproducibility. [ABSTRACT FROM AUTHOR]

Uncertainties in the atmosphere and flight conditions can drastically impact the performance of an aircraft and result in certification delays. However, uncertainty propagation in high-fidelity simulations, which have become integral to the design process, can pose intractably high computational costs. This study presents a non-intrusive, parametric reduced order modeling (ROM) method to enable the prediction of uncertain fields with thousands of random variables and nonlinear features under limited sampling budgets. The methodology combines linear dimensionality reduction with sparse polynomial chaos expansions and is assessed in a variety of CFD-based test cases, including 3D supersonic flow over a passenger aircraft with uncertain flight conditions. Each problem has strong nonlinearities, such as shocks, to investigate the effectiveness of models in real-world aerodynamic simulations that may arise during conceptual or preliminary design. The performance is assessed by comparing the uncertain mean, variance, point predictions, and integrated quantities of interest obtained using the ROMs to Monte Carlo simulations. It is observed that if the flow is entirely supersonic or subsonic, then the method can predict the pressure field accurately and rapidly. Moreover, it is also seen that statistical moments can be efficiently obtained using closed-form analytical expressions and closely match Monte Carlo results. [ABSTRACT FROM AUTHOR]

Many probability-based uncertainty quantification (UQ) schemes require a large amount of sampled data to build credible probability density function (PDF) models for uncertain parameters. Unfortunately, the amounts of data collected as to compressor blades of aero-engines are mostly limited due to the expensive and time-consuming tests. In this paper, we develop a preconditioner-based data-driven polynomial chaos (PDDPC) method that can efficiently deal with uncertainty propagation of limited amounts of sampled data. The calculation accuracy of a PDDPC method is closely related to the sample size of collected data. Therefore, the influence of sample size on this PDDPC method is investigated using a nonlinear test function. Subsequently, we consider the real manufacturing errors in stagger angles for compressor blades. Under three different operating conditions, the PDDPC method is applied to investigate the effect of stagger-angle error on UQ results of multiple aerodynamic parameters of a two-dimensional compressor blade. The results show that as the sample-size of measured data increases, UQ results regarding aerodynamic performance obtained by the PDDPC method gradually converge. There exists a critical sample size that ensures accurate UQ analysis of compressor blades. The probability information contained in the machining error data is analyzed through Kullback–Leibler divergence, and the critical sample size is determined. The research results can serve as a valuable reference for the fast and cheap UQ analysis of compressor blades in practical engineering. [ABSTRACT FROM AUTHOR]

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In this work, we describe an algorithm based on Polynomial-Chaos (PC) expansions for the study of uncertainty quantification problems involving grating filters. The proposed method adaptively builds anisotropic PC models for the quantities of interest, considering varying polynomial orders. In addition, optimal experimental designs are constructed that exploit the local variance of the samples, further increasing the reliability of the computations. The method is applied to the uncertainty quantification of a typical resonant grating filter, where the efficiency of the proposed approach over standard techniques is demonstrated.

This paper proposes a video security transmission enhancement algorithm based on block permutation and dynamic multiplexing sequences encryption based on 4D autonomous hyperchaotic system. Firstly, we employ the block permutation encryption and diffusion confusion encryption module, which is based on dynamic multiplexing chaotic sequences, to encrypt the plaintext and obtain the ciphertext. Subsequently, the hash value of this round's ciphertext is utilized to generate the chaotic key, produced by the multiplexing sequence of this round after mathematical processing. Then, the key is used to generate the chaotic sequence to confuse the N-th of the multiplexed sequence, and the next round of multiplexed sequence is obtained. If the current round of chaotic sequence has been completely confused, the chaotic sequence is re-generated to generate a new multiplex sequence by using the key generated by the current round key and the initial key. Finally, the above steps are repeated for the encryption of each frame of the video. Compared with the traditional permutation coding algorithm, it increases the difficulty of estimation or recognition while ensuring efficiency, and effectively improves the avalanche effect of the algorithm. Through frame by frame ciphertext closed-loop feedback, it has the ability to resist known plaintext attack and selected plaintext attack. The results show that the scheme has high security and significant diffusion characteristics, and can effectively resist various common cryptographic attacks. [ABSTRACT FROM AUTHOR]

The spectral element method when enriched with the concept of fictitious domain constitutes the spectral cell method which is appropriately able to model geometrically complicated problems by simplifying the mesh generation. In this paper, we propose a new computational framework called stochastic spectral cell method (SSCM) as a stochastic development of the spectral cell method for uncertainty quantification applied to structural dynamics and wave propagations. The stochastic form of spectral cell method has not been developed yet. Hence, the SSCM is a novel stochastic method comprising the advantages of spectral cell method for computational mechanics considering the geometrical complexity resulting from computer‐aided design. The SSCM is established by incorporating the polynomial chaos and Karhunen‐Loève expansions for the uncertainty quantification, and the interpolation functions of spectral elements for the spatial discretization. In the SSCM, the numerical solution of the Fredholm integral equation of the second kind required in Karhunen‐Loève expansion is obtained using the spectral cell method. The use of Cartesian mesh, diagonal mass matrix, and higher order interpolation functions of spectral elements for stochastic dynamic analysis lead to desirable computational efficiency and accuracy of the SSCM. The capability, effectiveness, and accuracy of the proposed SSCM are demonstrated with several numerical examples of wave propagation as well as dynamic analysis of structures. [ABSTRACT FROM AUTHOR]

The ensemble of surrogate models is commonly used to replace computationally expensive simulations due to their superior prediction accuracy and robustness compared to individual models. This paper proposes a new pointwise ensemble of surrogate models, namely, a pointwise-optimal ensemble of surrogate models (POEMs). To address the limitations of the cross-validation (CV) error in evaluating the performance of regression surrogate models, this paper introduces the compensated cross-validation error, which is more reliable in selecting better individual surrogate models and improving the accuracy of surrogate model ensembles. To overcome the limitations of CV error in calculating pointwise weight factors, this paper designs and solves an optimization problem at training points to obtain corresponding pointwise weight factors. Additionally, this paper proposes two weight calculation methods to be applied in the interpolation and extrapolation regions, respectively, to reduce the instability of ensembles caused by extrapolation. Thirty test functions are employed to investigate the appropriate hyperparameters of POEM and the Friedman test is used to verify the rationality of the a value. The thirty test functions are also used to examine the performance of POEM and compare it with state-of-the-art ensemble surrogate models. Furthermore, POEM is applied to a large-aperture mirror holder optimization case to verify its superiority. The results demonstrate that POEM presents better accuracy and robustness than individual surrogates and other compared ensembles of surrogate models. [ABSTRACT FROM AUTHOR]

The application of an interference fit in the assembly process of thin-walled aerospace composite parts is prone to stress stiffness effects, namely the structural stiffness change caused by the inplate-induced stress. Limited applicability has been available in this regard when using the compliant assembly analysis based on the stiffness-invariant assumption, such as the method of influence coefficient. In addition, these materials possess a hierarchical structure that necessitates the use of material uncertainty in the analysis. However, it is costly and complicated to develop uncertainty analysis and modeling calculations at different scales. In this study, a deviation propagation model of the composite structure (DPMoCS) considering the stress-stiffening effect is proposed to improve the analysis efficiency. Based on the geometric nonlinear theory, the factors of interest affecting the stress-stiffening effect in the interference assembly of a thin-walled composite material are investigated, which are the material elastic and stress field. Our simulations are integrated with material uncertainty quantization and propagation across scales to characterize the uncertainty of the factors of interest as well as reduce the computational cost based on the equivalent model. The method is combined with the Monte Carlo-based stochastic finite element method and applied to the assembly analysis of a composite panel subassembly. The results show that consideration of the stress-hardening effect has a significant effect on the DPMoCS and affects the assembly accuracy as the fiber layup angle deviation increases. [ABSTRACT FROM AUTHOR]

Computational knee models have shown that predicted condylar reactions are sensitive to the utilized ligament mechanical parameters. These models, however, are computationally expensive with multiple sources of uncertainty. Traditional uncertainty analysis using Monte-Carlo (MC) inspired methods are costly to perform. The purpose of this study was to use two example calibrated knee models to compare quasi-MC versus polynomial chaos expansion (PCE) sensitivity analyses of predicted condylar reactions that included uncertainty in the mechanical parameters of the ligaments. PCE was practically identical versus quasi-MC with 95% and 98% reductions in model evaluations for analyses with 10 and 6 uncertain variables, respectively. [ABSTRACT FROM AUTHOR]

We consider a fully discrete scheme for stochastic Allen-Cahn equation in a multi-dimensional setting. Our method uses a polynomial based spectral method in space, so it does not require the elliptic operator A and the covariance operator Q of noise in the equation commute, and thus successfully alleviates a restriction of Fourier spectral method for stochastic partial differential equations pointed out by Jentzen, Kloeden and Winkel [Ann. Appl. Probab. 21 (2011), pp. 908–950]. The discretization in time is a tamed semi-implicit scheme which treats the nonlinear term explicitly while being unconditionally stable. Under regular assumptions which are usually made for SPDEs, we establish strong convergence rates in the one spatial dimension for our fully discrete scheme. We also present numerical experiments which are consistent with our theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the adequacy of radial basis function (RBF)-based models as surrogates in uncertainty quantification (UQ) and CFD shape optimization; for the latter, problems with and without uncertainties are considered. In UQ, these are used to support the Monte Carlo, as well as, the non-intrusive, Gauss Quadrature and regression-based polynomial chaos expansion methods. They are applied to the flow around an isolated airfoil and a wing to quantify uncertainties associated with the constants of the γ − R ˜ e θ t transition model and the surface roughness (in the 3D case); it is demonstrated that the use of the RBF-based surrogates leads to an up to 50% reduction in computational cost, compared with the same UQ method that uses CFD computations. In shape optimization under uncertainties, solved by stochastic search methods, RBF-based surrogates are used to compute statistical moments of the objective function. In applications with geometric uncertainties which are modeled through the Karhunen–Loève technique, the use on an RBF-based surrogate reduces the turnaround time of an evolutionary algorithm by orders of magnitude. In this type of applications, RBF networks are also used to perform mesh displacement for the perturbed geometries. [ABSTRACT FROM AUTHOR]

A wind turbines' power curve is an easily accessible form of damage-sensitive data, and as such is a key part of structural health monitoring (SHM) in wind turbines. Power curve models can be constructed in a number of ways, but the authors argue that probabilistic methods carry inherent benefits in this use case, such as uncertainty quantification and allowing uncertainty propagation analysis. Many probabilistic power curve models have a key limitation in that they are not physically meaningful – they return mean and uncertainty predictions outside of what is physically possible (the maximum and minimum power outputs of the wind turbine). This paper investigates the use of two bounded Gaussian processes (GPs) in order to produce physically meaningful probabilistic power curve models. The first model investigated was a warped heteroscedastic Gaussian process, and was found to be ineffective due to specific shortcomings of the GP in relation to the warping function. The second model – an approximated GP with a Beta likelihood was highly successful and demonstrated that a working bounded probabilistic model results in better predictive uncertainty than a corresponding unbounded one without meaningful loss in predictive accuracy. Such a bounded model thus offers increased accuracy for performance monitoring and increased operator confidence in the model due to guaranteed physical plausibility. [ABSTRACT FROM AUTHOR]

Landing of supersonic transport (SST) suffers from a large uncertainty due to its highly sensitive aerodynamic properties in the subsonic domain, as well as the wind gusts around runways. At the vehicle design stage, a landing trajectory optimization under wind uncertainty in a multi-objective solution space is desired to explore the possible trade-off in its key flight performance metrics. The proposed algorithm solves this robust constrained multi-objective optimal control problem by integrating non-intrusive polynomial chaos expansion into a constrained evolutionary algorithm. The computationally tractable optimization is made possible through the conversion of a probabilistic problem into an equivalent deterministic representation while maintaining a form of the multi-objective problem. The generated guidance trajectories achieve a significant reduction of the uncertainty in their terminal states with a marginal modification in the control history of the deterministic solutions, validating the importance of the consideration of robustness in trajectory optimization. [ABSTRACT FROM AUTHOR]

Land surface models rely on a multitude of parameters to simulate land‐atmosphere interactions, but the parameter uncertainty can limit the reliability of model predictions. This study utilizes a parameter uncertainty quantification (UQ) framework to quantify and reduce the parameter uncertainty of the Noah‐MP land surface model in a grassland and sandy soil region in the Midwest of the USA. First, the sparse polynomial chaos expansion method which can quantify the interaction effect of parameters, is employed. A relatively small parameter sample size (i.e., 20 times of the number of parameters) was sufficient to identify the sensitive parameters; an additional sensitive parameter, the saturated soil hydraulic conductivity, was screened out compared to previous study. Then, based on the selected sensitive parameters, the weighted multi‐objective adaptive surrogate modeling‐based optimization algorithm is used as the parameter optimization method. The optimization results showed that the root mean square error of flux of latent heat (FLH) on about 82% of the total grids was reduced, and the number was about 57% for gross primary production (GPP) compared to the results using the original parameter settings, indicating that the Pareto parameter set by the UQ framework improved the Noah‐MP model in simulating FLH and GPP in a grassland and sandy soil region in the Midwest of the USA. Plain Language Summary: Land surface model is a crucial component of earth system model and it can have a significant impact on the accuracy and reliability of climate projections, as it directly affects the representation of land surface processes such as vegetation dynamics, soil moisture, and land use change. However, the uncertain values of certain sensitive parameters can affect the accuracy of these models that can lead to differences between model predictions and real‐world observations. This study uses an uncertainty quantification (UQ) framework to reduce parameter uncertainty in the Noah‐MP model, which is used to simulate flux of latent heat (FLH) and gross primary production (GPP) in a grassland and sandy soil region in the Midwest of the United States. We identify sensitive parameters using a sensitivity analysis method that considers parameter interaction effects and then use a multi‐objective optimization method to find optimal values for those parameters. The results show that this UQ framework significantly improves the accuracy of FLH and GPP simulations. The study provides a parameter UQ framework for land surface models, which will help improve predictions of water, energy and carbon cycles in grassland ecosystems. Key Points: Sensitive parameters of the Noah‐MP land surface model are identified by the sparse polynomial chaos expansionA novel sensitive parameter, saturated soil hydraulic conductivity, is screened outPareto parameter set by the multi‐objective adaptive surrogate modeling‐based optimization improves the simulation of the Noah‐MP model [ABSTRACT FROM AUTHOR]

Evaluating the failure probability of a system is essential in order to assess its reliability. This probability may significantly depend on deterministic parameters such as distribution parameters or design parameters. The sensitivity of the failure probability with regard to these parameters is then critical for the reliability analysis of the system or in reliability-based design optimization. Here, we introduce a new approach to estimate the failure probability derivatives with respect to deterministic inputs, where the bias can be controlled and the simulation budget is kept low. The sensitivity estimate is obtained as a byproduct of a heteroscedastic polynomial regression with a database built with simulation methods. The polynomial comes from a Taylor series expansion of the approximated sensitivity domain integral obtained with the Weak approach. This new methodology is applied to two engineering use cases with the importance sampling strategy. [ABSTRACT FROM AUTHOR]

This article presents an uncertainty analysis method for systems with hybrid stochastic and fuzzy uncertainty parameters based on polynomial chaos expansion (PCE). Parameters in the system are described by probability boxes, interval numbers, and fuzzy sets, respectively, based on the differences in their limited stochastic knowledge. First, this method transforms the uncertain parameters into standard normal distribution and interval variables through equal probability transformation or α$$ \alpha $$‐cut operations. Second, the Legendre and Hermite polynomials are used as the PCE model's primary functions, and the expansion coefficients are calculated by the Galerkin projection method based on sparse grid numerical integration. Then, the system response bounds under the pre‐defined confidence level can be obtained using a genetic algorithm to solve the optimization problem constructed based on PCE models. Finally, the feasibility and effectiveness of the method are illustrated by taking the tank bi‐directional stabilized system and the double‐pendulum‐controlled system as examples. The numerical results show that the system response bounds obtained by the PCE model optimization algorithm are consistent with the Monte Carlo simulation. Still, the computational efficiency is much higher. The proposed method effectively combines fuzzy sets and probability boxes and dramatically simplifies the analysis process of uncertain systems. The method exhibits fine precision even in high‐dimensional uncertainty analysis problems. [ABSTRACT FROM AUTHOR]

Accurate analysis of the dynamic response of the muzzle shows a vital role in improving the accuracy of small-caliber artillery burst fire. However, the complex small-caliber artillery system and the uncertainty of parameters make the muzzle dynamic response analysis more difficult. To solve the problem that it is difficult to analyze the dynamic response of small-caliber artillery containing uncertain parameters by traditional methods, the critical uncertain parameters are regarded as interval variables, and the bonded space theory is used to establish the muzzle dynamic response model of small-caliber artillery with continuous firing, and combined with Chebyshev polynomial expansion method to solve it, and finally the muzzle dynamic response interval is obtained. Through computational analysis and experimental testing, 84.6 % and 95 % of the tested muzzle dynamic response curves in the horizontal and vertical directions fall into the simulation interval, respectively. Meanwhile, the error of the mean, the variance, the most value, and the amplitude of the test and simulation are less than 10 %, which verifies the method's accuracy. In addition, by adjusting the optimized parameters, the amplitude of the dynamic response of the muzzle was reduced by 78.67 %. We conclude that the muzzle dynamic response interval method works well for analyzing and controlling the muzzle dynamic response problem with the desired accuracy. [ABSTRACT FROM AUTHOR]

The uncertainty of natural frequencies of composite plates due to the uncertainty of high-temperature distribution on the plate is investigated. Mechanical properties are assumed to be uncertain linear functions of temperature with assuming a stochastic linear dependency function (SLDF). The temperature gradient is assumed to be a Gaussian stochastic field with an exponential kernel. Assuming first-order shear deformation theory, the displacement field is defined. The stochastic assumed modes (SAM) method is proposed and discretized stochastic equations of motion are derived. The probability space is studied using an intrusive polynomial chaos approach. Stochastic simulations are verified using the stochastic finite element method (SFEM) based on the Monte Carlo approach. Results show the proposed semi-analytical method is an efficient alternative to time-consuming SFEM with resulting accurate statistical moments. Uncertainty in natural frequencies is quantified for symmetric and anti-symmetric composite plates with different side to thickness ratios. The applicability of linear dependency of composite material properties to the temperature is evaluated by uncertainty propagation in SLDF. Uncertainties in SLDF and temperature distribution have significant effects on predicted natural frequencies. Based on the results, the application of linear dependency of material properties to the temperature can lead to significant errors in predicted natural frequencies. [ABSTRACT FROM AUTHOR]

Determination of relevant model parameters is crucial for accurate mathematical modelling and efficient numerical simulation of a wide spectrum of applications in geosciences. The conventional method of choice is the global sensitivity analysis (GSA). Unfortunately, at least the classical Monte-Carlo based GSA requires a high number of model runs. Response surfaces based techniques, e.g. arbitrary Polynomial Chaos (aPC) expansion, can reduce computational effort, however, they suffer from the Gibbs phenomena and high hardware requirements for higher accuracy. We introduce GSA for arbitrary Multi-Resolution Polynomial Chaos (aMR-PC) which is a localized aPC based data-driven polynomial discretization. The aMR-PC allows to reduce the Gibbs phenomena by construction and to achieve higher accuracy by means of localization also for lower polynomial degrees. We apply these techniques to perform the sensitivity analysis for the Stokes–Darcy problem which describes fluid flow in coupled free-flow and porous-medium systems. We consider the Stokes equations in the free-flow region, Darcy's law in the porous-medium domain and the classical interface conditions across the fluid–porous interface including the conservation of mass, the balance of normal forces and the Beavers–Joseph condition for the tangential velocity. This coupled problem formulation contains four uncertain parameters: the exact location of the interface, the permeability, the Beavers–Joseph slip coefficient and the uncertainty in the boundary conditions. We carry out the sensitivity analysis of the coupled model with respect to these parameters using the Sobol indices on the aMR-PC expansion and conduct the corresponding numerical simulations. [ABSTRACT FROM AUTHOR]

The accurate simulation of complex turbulent flow has remained a challenging issue in computational fluid dynamics and plays a crucial role in engineering practice. Due to computational limitations, the Reynolds Averaged Navier-Stokes (RANS) simulation method continues to be the preferred choice for engineers to solve practical problems. Nonetheless, there are concerns regarding the applicability of RANS models to complex flows such as those with large-scale separation or jet interaction, as recommended coefficients may not be suitable. This work aims to perform a specified analysis of parameters' uncertainty in the Speziale-Sarkar-Gatsk/Launder-Reece-Rodi (SSG/LRR)-ω turbulence model. This work employs the non-intrusive polynomial chaos (NIPC) method to establish a surrogate model between parameters and computational results for uncertainty quantification. Sensitivity analysis is conducted via the Sobol index to identify key parameters critical for different flow structures. Then, Bayesian inference is applied for calibrating parameters by leveraging calculation results and two groups of experimental data, respectively. The results show that the calibrated values for the key parameters could significantly improve prediction accuracy. Subsequently, the work also delves into the inadequacies of SSG/LRR-ω models when applied to separated flows, revealing an underestimation of Reynolds stress in the separation zone as a primary source of poor predictions. Furthermore, it is considered that using a set of parameter combinations with a larger value of C 3 ω after the separation of the flow will obtain better calculation results in the separation zone. • Uncertainty quantification is performed by a surrogate model constructed by the non-invasive polynomial chaos method. • The Bayesian method is employed to evalute and correct the SSG/LRR-ω model. • The specific reasons for the inaccurate predictions of the model applied to separated flows are analyzed. • The improvement measures of the SSG/LRR-ω model are discussed. [ABSTRACT FROM AUTHOR]

In the context of solving inverse problems for physics applications within a Bayesian framework, we present a new approach, the Markov Chain Generative Adversarial Neural Network (MCGAN), to alleviate the computational costs associated with solving the Bayesian inference problem. GANs pose a very suitable framework to aid in the solution of Bayesian inference problems, as they are designed to generate samples from complicated high-dimensional distributions. By training a GAN to sample from a low-dimensional latent space and then embedding it in a Markov Chain Monte Carlo method, we can highly efficiently sample from the posterior, by replacing both the high-dimensional prior and the expensive forward map. This comes at the cost of a potentially expensive offline stage in which training data must be simulated or gathered and the GAN has to be trained. We prove that the proposed methodology converges to the true posterior in the Wasserstein-1 distance and that sampling from the latent space is equivalent to sampling in the high-dimensional space in a weak sense. The method is showcased in two test cases where we perform both state and parameter estimation simultaneously and it is compared with two conventional approaches, polynomial chaos expansion and ensemble Kalman filter, and a deep learning-based approach, deep Bayesian inversion. The method is shown to be more accurate than alternative approaches while also being computationally faster, in multiple test cases, including the important engineering setting of detecting leaks in pipelines. [ABSTRACT FROM AUTHOR]

This paper focuses on the stochastic response of railway embankments considering the uncertainties in soil cohesion and friction angle. The non-sampling stochastic method concerning generalised polynomial chaos (gPC) expansion was employed for the dynamic numerical simulation. The uncertain parameters, including soil cohesion and friction angle, were defined by the truncated gPC expansions. Furthermore, the system's response, namely, the displacement and acceleration of different embankment sections, was presented by the gPC expansion with unknown deterministic coefficients. The stochastic Galerkin projection was used to calculate a set of deterministic equations. The unknown gPC coefficients of the system's response were determined by a non-intrusive solution as a set of collocation points. In addition, the results of these analyses were compared with classical Monte Carlo simulations. It is essential to note that although only a few collocation points have been used, the results are in good agreement with the MC sampling method. One of the main objectives of this study is to demonstrate the accuracy of the results and the time efficiency of the proposed non-sampling method in quantifying the uncertainty of stochastic systems compared to the sampling procedure (e.g. Monte Carlo simulation). [ABSTRACT FROM AUTHOR]

In light of the current discrepancies between the recent predictions based on QCD factorisation (QCDF) and the experimental data for several non-leptonic colour-allowed two-body B-meson decays, we obtain new determinations of the non-factorisable soft-gluon contribution to the decays B ¯ s 0 → D s + π − and B ¯ 0 → D + K − , using the framework of light-cone sum rule (LCSR), with a suitable three-point correlation function and B-meson light-cone distribution amplitudes. In particular, we discuss the problem associated with a double light-cone (LC) expansion of the correlator, and motivate future determinations of the three-particle B-meson matrix element with the gluon and the spectator quark aligned along different light-cone directions. Performing a LC-local operator product expansion of the correlation function, we find, for both modes considered, the non-factorisable part of the amplitude to be sizeable and positive, however, with very large systematic uncertainties. Furthermore, we also determine for the first time, using LCSR, the factorisable amplitudes at LO-QCD, and thus the corresponding branching fractions. Our predictions are in agreement with the experimental data and consistent with the results based on QCDF, although again within very large uncertainties. In this respect, we provide a rich outlook for future improvements and investigations. [ABSTRACT FROM AUTHOR]

• The 6.polynomial chaos expansion is extended to high dimensional scenarios by a novel modeling method. • A data driven method is implemented for computing the orthogonal polynomial bases within PCE layer. • High dimensional uncertainty propagation problems involving correlated input variables can be solved. • Several numerical examples are analyzed for validating the effectiveness of the proposed method. In this paper, deep kernel polynomial chaos expansion (DKPCE) is proposed as a surrogate model for high dimensional uncertainty propagation. Firstly, deep neural network (DNN) and polynomial chaos expansion (PCE) are connected to create a novel network model, the input dimensionality of PCE layer can thus be controlled by restricting the number of neurons in the feature layer. Then, the back-propagation algorithm is employed for computing all the parameters of DKPCE, the dimension reduction and modeling process of DKPCE are thus executed simultaneously. During the modeling process, a data driven method is first implemented for computing the orthogonal polynomial bases within the PCE layer in the forward propagation step, and the partial derivatives for the coefficients of orthogonal polynomial bases are computed first in the back-propagation step. After constructing DKPCE, the coefficients of PCE layer can be utilized to compute the statistical characteristics of system response. Finally, several numerical examples are utilized for validating the effectiveness of DKPCE. [ABSTRACT FROM AUTHOR]

Multiclass probability estimation is the problem of estimating conditional probabilities of a data point belonging to a class given its covariate information. It has broad applications in statistical analysis and data science. Recently a class of weighted Support Vector Machines (wSVMs) has been developed to estimate class probabilities through ensemble learning for K-class problems (Wu et al., 2010; Wang et al., 2019), where K is the number of classes. The estimators are robust and achieve high accuracy for probability estimation, but their learning is implemented through pairwise coupling, which demands polynomial time in K. In this paper, we propose two new learning schemes, the baseline learning and the One-vs-All (OVA) learning, to further improve wSVMs in terms of computational efficiency and estimation accuracy. In particular, the baseline learning has optimal computational complexity in the sense that it is linear in K. Though not the most efficient in computation, the OVA is found to have the best estimation accuracy among all the procedures under comparison. The resulting estimators are distribution-free and shown to be consistent. We further conduct extensive numerical experiments to demonstrate their finite sample performance. [ABSTRACT FROM AUTHOR]

The focus of this work is on the application of classical Model Order Reduction techniques, such as Active Subspaces and Proper Orthogonal Decomposition, to Deep Neural Networks. We propose a generic methodology to reduce the number of layers in a pre-trained network by combining the aforementioned techniques for dimensionality reduction with input-output mappings, such as Polynomial Chaos Expansion and Feedforward Neural Networks. The motivation behind compressing the architecture of an existing Convolutional Neural Network arises from its usage in embedded systems with specific storage constraints. The conducted numerical tests demonstrate that the resulting reduced networks can achieve a level of accuracy comparable to the original Convolutional Neural Network being examined, while also saving memory allocation. Our primary emphasis lies in the field of image recognition, where we tested our methodology using VGG-16 and ResNet-110 architectures against three different datasets: CIFAR-10, CIFAR-100, and a custom dataset. [ABSTRACT FROM AUTHOR]

With the gradual popularization of high-power electric vehicle wireless power transfer (EV-WPT) applications, the safety issue of human exposure to electromagnetic fields leaked from EV-WPT devices has received considerable attention. In particular, careful attention should be devoted to human protection from electromagnetic field issues among people with medical implants. Considering the electromagnetic coupling between a human aortic valve metal stent (AVS) and the leakage field, this study establishes a numerical simulation model of the electromagnetic exposure of a human implanted with AVS to the leakage electromagnetic field of EV-WPT on the basis of human medical ethics. Given the existence of many uncertainties in actual WPT charging, which may cause damage to a human heart implanted with AVS, an orthogonal matching pursuit sparse generalized polynomial chaos expansion (OMP-sgPCE) method is developed to conduct an uncertainty quantification of the maximum induced electric field intensity (Emax) of a human heart implanted with AVS. Results indicate that the induced Emax obtained by this method can exceed the ICNIRP guideline limit and may seriously endanger human heart safety. This study also adopts the Sobol method to obtain the degree of influence of the coil group's spatial location parameters and the AVS geometric parameters on the induced Emax, thereby providing a reasonable theoretical basis and scientific guidance for the optimal design of EV-WPT devices and AVS. [ABSTRACT FROM AUTHOR]

The assessment of the future thermodynamics performance of a retrofitted heat and power production unit is prone to many uncertainties due to the large number of parameters involved in the modeling of all its components. To carry out uncertainty quantification analysis, alternatives to the traditional Monte Carlo method must be used due to the large stochastic dimension of the problem. In this paper, sparse polynomial chaos expansion (SPCE) is applied to the retrofit of a large coal-fired power plant into a biomass-fired combined heat and power unit to quantify the main drivers and the overall uncertainty on the plant's performance. The thermodynamic model encompasses over 180 components and 1500 parameters. A methodology combining the use of SPCE and expert judgment is proposed to narrow down the sources of uncertainty and deliver reliable probability distributions for the main key performance indicators (KPIs). The impact of the uncertainties on each input parameter vary with the considered KPI and its assessment through the computation of Sobol' indices. For both coal and biomass operations, the most impactful input parameters are the composition of the fuel and its heating value. The uncertainty on the performance and steam quality parameters is not much affected by the retrofit. Key furnace parameters exhibit a skewed probability distribution with large uncertainties, which is a strong attention point in terms of boiler operation and maintenance. [ABSTRACT FROM AUTHOR]

Polynomial chaos expansion (PCE) is widely adopted in geotechnical engineering as a surrogate model for probabilistic analysis. However, the traditional low-order PCE may be unfeasible for unsaturated transient-state models due to the high nonlinearity. In this study, a temporal-spatial surrogate model of adaptive sparse polynomial chaos expansions (AS-PCE) is established based on hyperbolic truncation with stepwise regression as surrogate models to improve computational efficiency. The uncertainty of pore water pressure of an unsaturated slope under transient-state rainfall infiltration considering hydraulic spatial variability is studied. The saturated coefficient of permeability k s is chosen to be spatial variability to account for the soil hydraulic uncertainty. The effects of location and time and the performances of AS-PCE are investigated. As rainfall goes on, the range of the pore pressure head becomes larger and the spatial variability of k s has little influence in the unsaturated zone with high matric suction. The pore pressure head under the water table suffers more uncertainty than it in the unsaturated zone. The R 2 in the high matric suction zone has a trend of rising first and then falling. Except for the high matric suction zone, the R 2 rise over time and they are almost 1 at the end of the time. It can be concluded that the AS-PCE performs better for low matric suction and positive pore pressure head and the fitting effect gradually increases as the rainfall progresses. The quartiles and at least up to second statistical moments can be characterized by the AS-PCE for transient infiltration in unsaturated soil slopes under rainfall. [ABSTRACT FROM AUTHOR]

To improve the generality and computational efficiency of robust trajectory optimization technology under multi parameter uncertainty, a robust trajectory optimization method based on stochastic response surface method and polynomial chaos is designed. Firstly, non-intrusive polynomial chaos is combined with stochastic response surface, and a Latin hypercube sampling strategy is introduced to establish a multi-dimensional uncertainty quantification model, which realizes the effective and accurate transformation of the uncertain optimization problem. Secondly, combining the model with the convex optimization method, a trajectory optimization solution strategy based on the sequential convex optimization algorithm is designed to realize the rapid solution of the high-dimensional deterministic optimization problem. Finally, a hypersonic glide trajectory optimization problem considering multi parameter uncertainty such as initial state and process parameters is simulated and compared with the existing methods. The results show that compared with the traditional deterministic trajectory optimization algorithm, the proposed method can obtain the hypersonic glide trajectory with both reliability and robustness. Compared with the existing robust trajectory optimization methods, the proposed method has considerable accuracy and significant computational efficiency, which can ensure the efficiency and reliability of the robust trajectory optimization trajectory generation when the dimension of random variables increases. [ABSTRACT FROM AUTHOR]

Robust optimization design (ROD) is playing an increasingly significant role in aerodynamic shape optimization and aircraft design. However, an efficient ROD framework that couples uncertainty quantification (UQ) and a powerful optimization algorithm for three-dimensional configurations is lacking. In addition, it is very important to reveal the maintenance mechanism of aerodynamic robustness from the design viewpoint. This paper first combines gradient-based optimization using the discrete adjoint-based approach with the polynomial chaos expansion (PCE) method to establish the ROD framework. A flying-wing configuration is optimized using deterministic optimization and ROD methods, respectively. The uncertainty parameters are Mach and the angle of attack. The ROD framework with the mean as an objective achieves better robustness with a lower mean (6.7% reduction) and standard derivation (Std, 18.92% reduction) compared to deterministic results. Moreover, we only sacrifice a minor amount of the aerodynamic performance (an increment of 0.64 counts in the drag coefficient). In comparison, the ROD with Std as an objective obtains a very different result, achieving the lowest Std and largest mean The far-field drag decomposition method is applied to compute the statistical moment variation of drag components and reveal how the ROD framework adjusts the drag component to realize better aerodynamic robustness. The ROD with the mean as the objective decreases the statistical moment of each drag component to improve aerodynamic robustness. In contrast, the ROD with Std as an objective reduces Std significantly by maintaining the inverse correlation relationship between the induced drag and viscous drag with an uncertainty parameter, respectively. The established ROD framework can be applied to future engineering applications that consider uncertainties. The unveiled mechanism for maintaining aerodynamic robustness will help designers understand ROD results more deeply, enabling them to reasonably construct ROD optimization problems. [ABSTRACT FROM AUTHOR]

Preventing unauthorized access to sensitive data has always been one of the main concerns in the field of information security. Accordingly, various solutions have been proposed to meet this requirement, among which encryption can be considered as one of the first and most effective solutions. The continuous increase in the computational power of computers and the rapid development of artificial intelligence techniques have made many previous encryption solutions not secure enough to protect data. Therefore, there is always a need to provide new and more efficient strategies for encrypting information. In this article, a two-way approach for information encryption based on chaos theory is presented. To this end, a new chaos model is first proposed. This model, in addition to having a larger key space and high sensitivity to slight key changes, can demonstrate a higher level of chaotic behavior compared to previous models. In the proposed method, first, the input is converted to a vector of bytes and first diffusion is applied on it. Then, the permutation order of chaotic sequence is used for diffusing bytes of data. In the next step, the chaotic sequence is used for applying second diffusion on confused data. Finally, to further reduce the data correlation, an iterative reversible rule-based model is used to apply final diffusion on data. The performance of the proposed method in encrypting image, text, and audio data was evaluated. The analysis of the test results showed that the proposed encryption strategy can demonstrate a pattern close to a random state by reducing data correlation at least 28.57% compared to previous works. Also, the data encrypted by proposed method, show at least 14.15% and 1.79% increment in terms of MSE and BER, respectively. In addition, key sensitivity of 10−28 and average entropy of 7.9993 in the proposed model, indicate its high resistance to brute-force, statistical, plaintext and differential attacks. [ABSTRACT FROM AUTHOR]

The present study devotes to introducing and analyzing the use of the adaptive Fourier decomposition (AFD)‐type methods in the area of stochastic processes and random fields. We involve two types of algorithms, namely, stochastic AFD (SAFD) and stochastic pre‐orthogonal AFD (SPOAFD), for, respectively, the Hardy space format and non‐Hardy space ones, as may be regarded. We provide both their theoretical results and practical algorithms and compare them with the well adopted and, in fact, dominating Karhunen–Loève (KL)‐type expansions. The AFD methods involve a finite or infinite sequence of optimally chosen parameters; they, in contrast, do not rely on and hence not have to compute the eigenpairs of the second type Fredholm integral equation with the covariance function as the kernel. Apart from such computational conveniences, they with the same convergence rate have flexibility of choosing best suitable dictionaries for doing the specific task in the practice. We include a number of experiments showing that in terms of effectiveness, the AFD methods give better approximations before all the positive eigenvalues running out in the case the integral operator being of a finite rank or before the KL iteration step becomes excessively large: The AFD methods normally give better approximations from the very beginning of the iterations. [ABSTRACT FROM AUTHOR]

The paper presents a mechanical-based framework for the evaluation of local-scale seismic fragility curves. The approach is oriented to a seismic vulnerability assessment of unreinforced masonry buildings and makes use of basic exposure data easily obtained from survey or available in existing database. An efficient finite element model and static nonlinear analyses are employed to assess the structural behaviour. The mechanical-based fragility curves are evaluated using Monte Carlo simulations that allow to account for the uncertainties propagation. The proposed approach is tested on a case-study regarding the city centre of Cosenza, in southern Italy, using exposure information available from CARTIS database. [ABSTRACT FROM AUTHOR]

In this article, q -Chlodowsky operators are utilized for synchronizing a chaotic master-slave system. The universal approximation property enables q -Chlodowsky operators to approximate uncertainties, which consist of disturbances and unmodeled dynamics. It is verified that uniformly ultimately bounded stability of the tracking/approximation errors can be assured if the q -Chlodowsky operators are utilized as regressors and the polynomial coefficients are tuned by the adaptive laws calculated in the stability analysis. Moreover, it has been assumed that the rate of synchronization error is not at hand and a filter-based strategy has been adopted instead of designing an observer. The Duffing-Holmes oscillator is studied and the results are also compared with three powerful approximation-based and robust control methods. Unlike neural network/fuzzy estimators in which system states are necessary to define the regressor vector, the proposed uncertainty estimator does not require them in the regressor vector. Furthermore, in comparison with radial basis functions neural networks (RBFNNs), we are involved in fewer adaptive coefficients in the regressor vector of the proposed uncertainty estimator. As a result, the tedious and time-consuming procedure of trial and error for tuning the uncertainty estimator is eliminated. [ABSTRACT FROM AUTHOR]

Artificial Intelligence and Machine learning have been widely used in various fields of mathematical computing, physical modeling, computational science, communication science, and stochastic analysis. Approaches based on Deep Artificial Neural Networks (DANN) are very popular in our days. Depending on the learning task, the exact form of DANNs is determined via their multi-layer architecture, activation functions and the so-called loss function. However, for a majority of deep learning approaches based on DANNs, the kernel structure of neural signal processing remains the same, where the node response is encoded as a linear superposition of neural activity, while the non-linearity is triggered by the activation functions. In the current paper, we suggest to analyze the neural signal processing in DANNs from the point of view of homogeneous chaos theory as known from polynomial chaos expansion (PCE). From the PCE perspective, the (linear) response on each node of a DANN could be seen as a 1st degree multi-variate polynomial of single neurons from the previous layer, i.e. linear weighted sum of monomials. From this point of view, the conventional DANN structure relies implicitly (but erroneously) on a Gaussian distribution of neural signals. Additionally, this view revels that by design DANNs do not necessarily fulfill any orthogonality or orthonormality condition for a majority of data-driven applications. Therefore, the prevailing handling of neural signals in DANNs could lead to redundant representation as any neural signal could contain some partial information from other neural signals. To tackle that challenge, we suggest to employ the data-driven generalization of PCE theory known as arbitrary polynomial chaos (aPC) to construct a corresponding multi-variate orthonormal representations on each node of a DANN. Doing so, we generalize the conventional structure of DANNs to Deep arbitrary polynomial chaos neural networks (DaPC NN). They decompose the neural signals that travel through the multi-layer structure by an adaptive construction of data-driven multi-variate orthonormal bases for each layer. Moreover, the introduced DaPC NN provides an opportunity to go beyond the linear weighted superposition of single neurons on each node. Inheriting fundamentals of PCE theory, the DaPC NN offers an additional possibility to account for high-order neural effects reflecting simultaneous interaction in multi-layer networks. Introducing the high-order weighted superposition on each node of the network mitigates the necessity to introduce non-linearity via activation functions and, hence, reduces the room for potential subjectivity in the modeling procedure. Although the current DaPC NN framework has no theoretical restrictions on the use of activation functions. The current paper also summarizes relevant properties of DaPC NNs inherited from aPC as analytical expressions for statistical quantities and sensitivity indexes on each node. We also offer an analytical form of partial derivatives that could be used in various training algorithms. Technically, DaPC NNs require similar training procedures as conventional DANNs, and all trained weights determine automatically the corresponding multi-variate data-driven orthonormal bases for all layers of DaPC NN. The paper makes use of three test cases to illustrate the performance of DaPC NN, comparing it with the performance of the conventional DANN and also with plain aPC expansion. Evidence of convergence over the training data size against validation data sets demonstrates that the DaPC NN outperforms the conventional DANN systematically. Overall, the suggested re-formulation of the kernel network structure in terms of homogeneous chaos theory is not limited to any particular architecture or any particular definition of the loss function. The DaPC NN Matlab Toolbox is available online and users are invited to adopt it for own needs. • Kernel structure reformulation of deep artificial neural networks in terms of homogeneous chaos theory. • Response on each node of the deep network represented through polynomial chaos expansion. • Orthonormal decomposition mitigates the non-optimal representation while processing the neural signal. • Accounting for high-order effects reflecting simultaneous neural impacts. • Analytical estimation of mean and variance of neural signal on each node is provided. [ABSTRACT FROM AUTHOR]

Uncertainty quantification (UQ) is deemed critical in steel reheating simulations due to the significant input uncertainties arising when defining steel surface properties and atmospheric furnace conditions. In order to conduct UQ, the study utilizes polynomial chaos expansion, which has been found to significantly curtail the computational effort needed to obtain reliable convergent statistics for the model of interest. Results from a comprehensive UQ analysis of a walking‐beam reheat furnace simulated using Tata Steel's reheat furnace control model, online slab temperature calculation, are presented. Slab temperature evolution and oxide scale growth are chosen as the study's QoIs. The analysis reveals that at the earlier stages of reheating, the majority of the output variance in slab temperature can be traced back exclusively to the emissivity of the slab surface, and the majority of the output variance in oxide scale growth is traced back to the combination of slab's surface emissivity and the initial scale thickness found on steel products prior to reheating. However, as the steel product advances toward the furnace's discharge end, inputs related to oxide scale growth become increasingly important, ultimately becoming the most influential input parameters, although the dynamics of this transition differ between the QoIs. [ABSTRACT FROM AUTHOR]