Search

An issue for molecular dynamics simulations is that events of interest often involve timescales that are much longer than the simulation time step, which is set by the fastest timescales of the model. Because of this timescale separation, direct simulation of many events is prohibitively computationally costly. This issue can be overcome by aggregating information from many relatively short simulations that sample segments of trajectories involving events of interest. This is the strategy of Markov state models (MSMs) and related approaches, but such methods suffer from approximation error because the variables defining the states generally do not capture the dynamics fully. By contrast, once converged, the weighted ensemble (WE) method aggregates information from trajectory segments so as to yield unbiased estimates of both thermodynamic and kinetic statistics. Unfortunately, errors decay no faster than unbiased simulation in WE as originally formulated and commonly deployed. Here, we introduce a theoretical framework for describing WE that shows that the introduction of an approximate stationary distribution on top of the stratification, as in nonequilibrium umbrella sampling (NEUS), accelerates convergence. Building on ideas from MSMs and related methods, we generalize the NEUS approach in such a way that the approximation error can be reduced systematically. We show that the improved algorithm can decrease the simulation time required to achieve the desired precision by orders of magnitude. [ABSTRACT FROM AUTHOR]

It is shown that the four-component (4C), quasi-four-component (Q4C), and exact two-component (X2C) relativistic Hartree–Fock equations can be implemented in a unified manner by making use of the atomic nature of the small components of molecular 4-spinors. A model density matrix approximation can first be invoked for the small-component charge/current density functions, which gives rise to a static, pre-molecular mean field to be combined with the one-electron term. As a result, only the nonrelativistic-like two-electron term of the 4C/Q4C/X2C Fock matrix needs to be updated during the iterations. A "one-center small-component" approximation can then be invoked in the evaluation of relativistic integrals, that is, all atom-centered small-component basis functions are regarded as extremely localized near the position of the atom to which they belong such that they have vanishing overlaps with all small- or large-component functions centered at other nuclei. Under these approximations, the 4C, Q4C, and X2C mean-field and many-electron Hamiltonians share precisely the same structure and accuracy. Beyond these is the effective quantum electrodynamics Hamiltonian that can be constructed in the same way. Such approximations lead to errors that are orders of magnitude smaller than other sources of errors (e.g., truncation errors in the one- and many-particle bases as well as uncertainties of experimental measurements) and are, hence, safe to use for whatever purposes. The quaternion forms of the 4C, Q4C, and X2C equations are also presented in the most general way, based on which the corresponding Kramers-restricted open-shell variants are formulated for "high-spin" open-shell systems. [ABSTRACT FROM AUTHOR]

Estimating ranks, quantiles, and distributions over streaming data is a central task in data analysis and monitoring. Given a stream of n items from a data universe equipped with a total order, the task is to compute a sketch (data structure) of size polylogarithmic in n. Given the sketch and a query item y, one should be able to approximate its rank in the stream, i.e., the number of stream elements smaller than or equal to y. Most works to date focused on additive εn error approximation, culminating in the KLL sketch that achieved optimal asymptotic behavior. This article investigates multiplicative (1±ε )-error approximations to the rank. Practical motivation for multiplicative error stems from demands to understand the tails of distributions, and hence for sketches to be more accurate near extreme values. The most space-efficient algorithms due to prior work store either O(log(ε²n)/ε²) or O(log³ (εn)/ε ) universe items. We present a randomized sketch storing O(log1.5 (εn)/ε ) items that can (1 ± ε )-approximate the rank of each universe item with high constant probability; this space bound is within an O(√log(εn)) factor of optimal. Our algorithm does not require prior knowledge of the stream length and is fully mergeable, rendering it suitable for parallel and distributed computing environments. [ABSTRACT FROM AUTHOR]

In order to ensure the flight safety of small unmanned aerial vehicles (UAVs), a deep neural network-fused mathematical modeling approach is put up for reliable flight control of small UAVs. First, engine torque, thrust eccentricity and initial stop angle are taken into full consideration. A six-degree-of-freedom nonlinear model is formulated for small UAVs, concerning both ground taxiing and air flight status. Then, the model was linearized using the principle of small disturbances. The linearized model expressions for both ground taxiing and air flight were provided. In addition, radial basis function neural networks are used for online approximation to address the nonlinearity and uncertainty caused by changes in aircraft aerodynamic parameters. At the same time, to compensate for the external disturbance and the approximation error of the neural network, the system robustness is improved by selecting reasonable design parameters. This helps the whole flight control system obtain better tracking control performance. At last, some simulation experiments are carried out to evaluate the performance of the proposed mathematical modeling framework. The simulation results show that the proposal has stronger convergence ability, smaller prediction error, and better performance. Thus, proper proactivity can be acknowledged. [ABSTRACT FROM AUTHOR]

This paper presents a numerical comparison of finite-element methods resulting in local mass conservation at the element level for Poisson's problem, namely the primal hybrid and mixed methods. These formulations result in an indefinite system. Alternative formulations yielding a positive-definite system are obtained after hybridizing each method. The choice of approximation spaces yields methods with enhanced accuracy for the pressure variable, and results in systems with identical size and structure after static condensation. A regular pressure precision mixed formulation is also considered based on the classical RT k space. The simulations are accelerated using Open multi-processing (OMP) and Thread-Building Blocks (TBB) multithreading paradigms alongside either a coloring strategy or atomic operations ensuring a thread-safe execution. An additional parallel strategy is developed using C++ threads, which is based on the producer-consumer paradigm, and uses locks and semaphores as synchronization primitives. Numerical tests show the optimal parallel strategy for these finite-element formulations, and the computational performance of the methods are compared in terms of simulation time and approximation errors. Additional results are developed during the process. Numerical solvers often fail to find an accurate solution to the highly indefinite systems arising from finite-element formulations, and this paper documents a matrix ordering strategy to stabilize the resolution. A procedure to enable static condensation based on the introduction of piecewise constant functions that also fulfills Neumann's compatibility condition, and yet computes an average pressure per element is presented. • Numerical comparison of the mixed, primal hybrid, and other locally conservatives finite element methods. • Introduction of a twice hybridized primal hybrid method, yielding a positive definite matrix after static condensation. • Comparison of multi-threaded strategies and thread-safety mechanisms. • A numerical procedure to stabilize the linear system resolution for indefinite matrices is described. • A general procedure that enables static condensation and fulfills Neumann's compatibility condition. [ABSTRACT FROM AUTHOR]

For the first time, we propose an efficient difference spectral approximation for Allen–Cahn equation in a circular domain. Firstly, we introduce the polar coordinate transformation and derive the equivalent form of Allen–Cahn equation under this coordinate system, as well as the corresponding essential polar condition. Then, by using first‐order Euler and second‐order backward difference methods in the temporal direction, we deduce the first‐order and second‐order semi‐implicit schemes, based on which the first‐order and second‐order fully discrete schemes are established by employing Legendre‐Fourier spectral approximation in the spatial direction. In addition, the energy stability and error estimations for the two types of numerical schemes are theoretically proved. Finally, we provide some numerical examples, the results of which demonstrate the stability and convergence of the algorithm. [ABSTRACT FROM AUTHOR]

Objective: This study was to evaluate measurement properties of the Chinese version of the Brief Inventory of Perceived Stress (BIPS-C) and confirm possible solutions for measuring the constructs underlying perceived stress. Methods: A total of 1356 community residents enrolled and were randomly split into two halves. The first half was used to explore the underlying constructs of the BIPS-C by exploratory graph analysis (EGA) and the second half was used to compare and confirm the constructs by confirmatory factor analysis (CFA). Results: The EGA identified a one-factor model of the BIPS-C with an accuracy of 99.3%. One-factor, three-factor, second-order, and bifactor models were compared by CFAs. The bifactor model with one general and three specific factors was found to be the most adequate [comparative fit index (CFI) = 0.990; Tucker-Lewis index (TLI) = 0.979; root mean square error of approximation (RMSEA) = 0.058] and was superior to the other models. The related bifactor indices showed a stronger existence of the general factor. The bifactor model of the BIPS-C also showed adequate internal consistency with McDonald's omega and omega subscales ranging from moderate to strong (0.677–0.869). Conclusion: The BIPS-C demonstrates sufficient measurement properties for assessing general perceived stress. [ABSTRACT FROM AUTHOR]

We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs) under model uncertainty. Specifically, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with infinite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a unified framework. We build on the Bayesian approximation error (BAE) approach, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a Gaussian approximation to the posterior at the maximum a posteriori probability point is used to define an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to find optimal designs. We demonstrate the effectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages. [ABSTRACT FROM AUTHOR]

This paper proposes an algorithm to obtain the sampling factors to model any frequency partitioning so that it is realized using low complexity rational decimated non-uniform filter banks (RDNUFBs). The proposed algorithm is employed to approximate the Bark scale to find a rational frequency partitioning that can be realized using RDNUFBs with less approximation error. The proposed Bark frequency partitioning is found to reduce the average deviation in centre frequency and bandwidth by 65.04% and 48.50%, respectively, and root-mean-square bandwidth deviation by 54.46% when compared to those of the existing perceptual wavelet approximation of Bark scale. In the first filter bank design approach discussed in the paper, a near-perfect reconstruction partially cosine modulated filter bank is employed to obtain the improved Bark frequency partitioning. In the second design approach, the hardware complexity of the PCM-based RDNUFB is reduced by deriving the channels with different sampling factors from the same prototype filter by the method of channel merging. There is a considerable reduction of 40.83% in the number of multipliers when merging of partially cosine modulated channels is employed. It is also found that both the proposed approaches can reduce the delay by 95.17% when compared to the existing rational tree approximation methods and hence, are suitable for real-time applications. [ABSTRACT FROM AUTHOR]

In this paper, we propose an a posteriori error estimate of the weak Galerkin finite element method (WG‐FEM) solving the Stokes problems with variable coefficients. Its error estimator, based on the property of Stokes' law conservation, Helmholtz decomposition and bubble functions, yields global upper bound and local lower bound for the approximation error of the WG‐FEM. Error analysis is proved to be valid under the mesh assumptions of the WG‐FEM and the way can be extended to other FEMs with the property of Stokes' law conservation, for example, discontinuous Galerkin (DG) FEMs. Finally, we verify the performance of error estimator by performing a few numerical examples. [ABSTRACT FROM AUTHOR]

In the realm of five-axis Computer Numerical Control (CNC) machining, the challenge of minimizing velocity, acceleration, and jerk fluctuations has prompted the development of various local corner smoothing methods. However, existing techniques often rely on symmetrical spline curves or impose pre-set transition length constraints, limiting their effectiveness in reducing curvature and maintaining velocity at critical corners. To comprehensively address these limitations comprehensively, a novel approach for local smoothing of five-axis linear toolpaths is presented in this paper. The proposed method introduces two asymmetrical B-splines at corners, effectively smoothing both the tool-tip position in the workpiece coordinate system and the tool orientation in the machine coordinate system. To fine-tune transition curve scales and minimize velocity disparities between adjacent corners, an overlap elimination scheme is employed. Furthermore, the two-step strategy emphasizes the synchronization of tool-tip position and tool orientation while considering maximal approximation error. The outcome is a blended five-axis toolpath that significantly reduces curvature extremes in the smoothed path, achieving reductions ranging from 36.28 % to 45.51 %. Additionally, the proposed method streamlines the interpolation process, resulting in time savings ranging from 5.68 % to 8.78 %, all the while adhering to the same geometric and kinematic constraints. [ABSTRACT FROM AUTHOR]

To solve some scheduling problems of batch processes based on timed Petri net models, timed extended reachability graphs (TERGs) and approximated TERGs can be used. Such graphs abstract temporal specifications and represent parts of timed languages. By exploring the feasible trajectories in a TERG, optimal schedules can be obtained with respect to the makespans of batch processes that are modeled by timed Petri nets. Nevertheless, the rapid growth of the number of states in a TERG makes the problem intractable for large systems. In this paper, we improve the existing clustering TERG approach, and we make it suitable for large sized batch processes. We also enlarge a systematic approach to model batch processes with timed Petri nets. Finally, a comprehensive example of scheduling problem is studied for an archetypal chemical production plant in order to illustrate the efficiency of the proposed approach. • The scheduling issue of batch chemical processes is formalized with timed Petri nets. • A graph named ClusTERG approximates the state space through hierarchical clustering. • The approximation error of this ClusTERG is far less than other approximations. • Such an approach is used to obtain near-optimal schedules of a batch process. [ABSTRACT FROM AUTHOR]

Reconfigurable variable stiffness actuator (RVSA) has attracted increasing attention in robotics due to its safety, compliance, and robustness. However, the control of the RVSA is challenging due to nonlinear factors such as high-order nonlinear dynamic, model uncertainties, time-varying model parameters, and disturbances. In this paper, firstly, a lightweight RVSA structure with both passive and active nonlinear variable stiffness characteristic is developed. Secondly, a dynamic surface backstepping control method based on a radial basis neural network and disturbance observer (DSBC-RBFNN-DOB) is proposed to achieve position control of the lightweight RVSA with matched and unmatched uncertainties. To address solve the "complexity explosion" and noise problems in traditional backstepping control, the dynamic surface backstepping control (DSBC) method is used to design the controller. Then, a method based on radial basis neural network (RBFNN) and disturbance observer (DOB) are used to compensate for the matched and unmatched uncertainties in the link and motor. In this method, the matched uncertainties are compensated using RBFNN, and the DOB is integrated to compensate RBFNN approximation errors and unmatched uncertainties. Through Lyapunov stability analysis, the semi-global boundedness of the controller is proven. Finally, the proposed method is simulated and actually implemented, verifying the effectiveness of the method. Simulation and experimental results show that the root mean square error (RMSE) of the proposed method is only 0.97277° and 0.6418°, respectively. Compared with PID, DSBC, and DSBC-RBFNN, the error reduction percentages in simulation (experiment) are 85.6 % (88.9 %), 49.4 % (88.4 %) and 36.1 % (80.0 %) respectively. • A lightweight reconfigurable variable stiffness actuator with smaller mass and more compact structure is proposed.. • A dynamic surface backstepping control method based on neural network and disturbance observer is proposed. • A method of uncertainty and disturbance estimation combining neural network and disturbance observer is proposed. • Experiments and simulations were conducted under the condition of actively changing stiffness. [ABSTRACT FROM AUTHOR]

Negative binomial related distributions have been widely used in practice. The calculation of the corresponding Fisher information matrices involves the expectation of trigamma function values which can only be calculated numerically and approximately. In this paper, we propose a trigamma-free approach to approximate the expectations involving the trigamma function, along with theoretical upper bounds for approximation errors. We show by numerical studies that our approach is highly efficient and much more accurate than previous methods. We also apply our approach to compute the Fisher information matrices of zero-inflated negative binomial (ZINB) and beta negative binomial (ZIBNB) probabilistic models, as well as ZIBNB regression models. [ABSTRACT FROM AUTHOR]

In this work, we introduce a novel stochastic second-order method, within the framework of a non-monotone trust-region approach, for solving the unconstrained, nonlinear, and non-convex optimization problems arising in the training of deep neural networks. The proposed algorithm makes use of subsampling strategies that yield noisy approximations of the finite sum objective function and its gradient. We introduce an adaptive sample size strategy based on inexpensive additional sampling to control the resulting approximation error. Depending on the estimated progress of the algorithm, this can yield sample size scenarios ranging from mini-batch to full sample functions. We provide convergence analysis for all possible scenarios and show that the proposed method achieves almost sure convergence under standard assumptions for the trust-region framework. We report numerical experiments showing that the proposed algorithm outperforms its state-of-the-art counterpart in deep neural network training for image classification and regression tasks while requiring a significantly smaller number of gradient evaluations. [ABSTRACT FROM AUTHOR]

Background: Distributed statistical analyses provide a promising approach for privacy protection when analyzing data distributed over several databases. Instead of directly operating on data, the analyst receives anonymous summary statistics, which are combined into an aggregated result. Further, in discrimination model (prognosis, diagnosis, etc.) development, it is key to evaluate a trained model w.r.t. to its prognostic or predictive performance on new independent data. For binary classification, quantifying discrimination uses the receiver operating characteristics (ROC) and its area under the curve (AUC) as aggregation measure. We are interested to calculate both as well as basic indicators of calibration-in-the-large for a binary classification task using a distributed and privacy-preserving approach. Methods: We employ DataSHIELD as the technology to carry out distributed analyses, and we use a newly developed algorithm to validate the prediction score by conducting distributed and privacy-preserving ROC analysis. Calibration curves are constructed from mean values over sites. The determination of ROC and its AUC is based on a generalized linear model (GLM) approximation of the true ROC curve, the ROC-GLM, as well as on ideas of differential privacy (DP). DP adds noise (quantified by the ℓ 2 sensitivity Δ 2 (f ^) ) to the data and enables a global handling of placement numbers. The impact of DP parameters was studied by simulations. Results: In our simulation scenario, the true and distributed AUC measures differ by Δ AUC < 0.01 depending heavily on the choice of the differential privacy parameters. It is recommended to check the accuracy of the distributed AUC estimator in specific simulation scenarios along with a reasonable choice of DP parameters. Here, the accuracy of the distributed AUC estimator may be impaired by too much artificial noise added from DP. Conclusions: The applicability of our algorithms depends on the ℓ 2 sensitivity Δ 2 (f ^) of the underlying statistical/predictive model. The simulations carried out have shown that the approximation error is acceptable for the majority of simulated cases. For models with high Δ 2 (f ^) , the privacy parameters must be set accordingly higher to ensure sufficient privacy protection, which affects the approximation error. This work shows that complex measures, as the AUC, are applicable for validation in distributed setups while preserving an individual's privacy. [ABSTRACT FROM AUTHOR]

The primary focus of this research paper is to explore the realm of dynamic learning in sampled‐data strict‐feedback nonlinear systems (SFNSs) by leveraging the capabilities of radial basis function (RBF) neural networks (NNs) under the framework of adaptive control. First, the exact discrete‐time model of the continuous‐time system is expressed as an Euler strict‐feedback model with a sampling approximation error. We provide the consistency condition that establishes the relationship between the exact model and the Euler model with meticulous detail. Meanwhile, a novel lemma is derived to show the stability condition of a digital first‐order filter. To address the non‐causality issues of SFNSs with sampling approximation error and the input data dimension explosion of NNs, the auxiliary digital first‐order filter and backstepping technology are combined to propose an adaptive neural dynamic surface control (ANDSC) scheme. Such a scheme avoids the n$$ n $$‐step time delays associated with the existing NN updating laws derived by the common n$$ n $$‐step predictor technology. A rigorous recursion method is employed to provide a comprehensive verification of the stability, guaranteeing its overall performance and dependability. Following that, the NN weight error systems are systematically decomposed into a sequence of linear time‐varying subsystems, allowing for a more detailed analysis and understanding. In order to ensure the recurrent nature of the input variables, a recursive design is employed, thereby satisfying the partial persistent excitation condition specifically designed for the RBF NNs. Meanwhile, it can verify that the NN estimated weights converge to their ideal values. Compared with the common n$$ n $$‐step predictor technology, there is no need to redesign the learning rules due to the designed NN weight updating laws without time delays. Subsequently, after capturing and storing the convergence weights, a novel neural learning dynamic surface control (NLDSC) scheme is specifically formulated by leveraging the acquired knowledge. The introduced methodology reduces computational complexity and facilitates practical implementation. Finally, empirical evidence obtained from simulation experiments validates the efficacy and viability of the proposed methodology. [ABSTRACT FROM AUTHOR]

In this article, the adaptive event-triggered fault-tolerant control (FTC) issue of uncertain nonlinear systems suffering from sensor and actuator faults as well as external disturbances is studied. A disturbance observer (DO) is constructed to compensate for external disturbances and approximation errors generated by neural networks (NNs). Then, switching event-triggered control and command filtering techniques are introduced to balance the communication frequency and tracking performance of the controlled systems while avoiding “explosion of complexity” issue caused by iterative derivations of virtual controllers. Furthermore, compensation signals are designed to eliminate filtered errors. Finally, it is proven that the developed FTC scheme can esure that all signals are bounded and free from Zeno phenomenon. Simulation results of a spring damping mechanical system verifies the merits of the designed control algorithm. [ABSTRACT FROM AUTHOR]

Background: Single-nucleotide polymorphism (SNP) effects can be backsolved from ssGBLUP genomic estimated breeding values (GEBV) and used for genome-wide association studies (ssGWAS). However, obtaining p-values for those SNP effects relies on the inversion of dense matrices, which poses computational limitations in large genotyped populations. In this study, we present a method to approximate SNP p-values for ssGWAS with many genotyped animals. This method relies on the combination of a sparse approximation of the inverse of the genomic relationship matrix ( G A P Y - 1 ) built with the algorithm for proven and young (APY ) and an approximation of the prediction error variance of SNP effects which does not require the inversion of the left-hand side (LHS) of the mixed model equations. To test the proposed p-value computing method, we used a reduced genotyped population of 50K genotyped animals and compared the approximated SNP p-values with benchmark p-values obtained with the direct inverse of LHS built with an exact genomic relationship matrix ( G - 1 ) . Then, we applied the proposed approximation method to obtain SNP p-values for a larger genotyped population composed of 450K genotyped animals. Results: The same genomic regions on chromosomes 7 and 20 were identified across all p-value computing methods when using 50K genotyped animals. In terms of computational requirements, obtaining p-values with the proposed approximation reduced the wall-clock time by 38 times and the memory requirement by ten times compared to using the exact inversion of the LHS. When the approximation was applied to a population of 450K genotyped animals, two new significant regions on chromosomes 6 and 14 were uncovered, indicating an increase in GWAS detection power when including more genotypes in the analyses. The process of obtaining p-values with the approximation and 450K genotyped individuals took 24.5 wall-clock hours and 87.66GB of memory, which is expected to increase linearly with the addition of noncore genotyped individuals. Conclusions: With the proposed method, obtaining p-values for SNP effects in ssGWAS is computationally feasible in large genotyped populations. The computational cost of obtaining p-values in ssGWAS may no longer be a limitation in extensive populations with many genotyped animals. [ABSTRACT FROM AUTHOR]

Forecast error growth as a function of lead time of atmospheric phenomena is caused by initial and model errors. When studying the initial error growth, it may turn out that small-scale phenomena, which contribute little to the forecast product, significantly affect the ability to predict this product. The question under investigation is whether omitting these atmospheric phenomena will improve the predictability of the resulting value. The topic is studied in the extended Lorenz (2005) system. This system shows that omitting small spatiotemporal scales that significantly affect prediction ability will reduce predictability more than modeling it. In other words, a system with model error (omitting phenomena) will not improve predictability. A hypothesis explaining and describing this behavior is developed, with the difference between systems (model error) produced at each time step seen as the error of the initial conditions. The resulting model error is then defined as the sum of the increments of the time evolution of the initial conditions so defined. The hypothesis is compared to the fit parameters that define the model error in certain approximations of the average forecast error growth. Parameters are interpreted in this context, and the approximations are used to estimate the errors described in the hypothesis. A method is proposed to distinguish increments of prediction error growth from small-spatiotemporal-scale phenomena and model errors. Results are presented for the error growth of the ECMWF system, where a 40 % reduction in model error between 1987 and 2011 is calculated based on the developed hypothesis, while over the same period the instability (error growth rate) of the system with respect to initial condition errors has grown. [ABSTRACT FROM AUTHOR]

The relaxation spectrum is a fundamental viscoelastic characteristic from which other material functions used to describe the rheological properties of polymers can be determined. The spectrum is recovered from relaxation stress or oscillatory shear data. Since the problem of the relaxation spectrum identification is ill-posed, in the known methods, different mechanisms are built-in to obtain a smooth enough and noise-robust relaxation spectrum model. The regularization of the original problem and/or limit of the set of admissible solutions are the most commonly used remedies. Here, the problem of determining an optimally smoothed continuous relaxation time spectrum is directly stated and solved for the first time, assuming that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. The relaxation time spectrum model that reproduces the relaxation modulus measurements and is the best smoothed in the class of continuous square-integrable functions is sought. Based on the Hilbert projection theorem, the best-smoothed relaxation spectrum model is found to be described by a finite sum of specific exponential–hyperbolic basis functions. For noise-corrupted measurements, a quadratic with respect to the Lagrange multipliers term is introduced into the Lagrangian functional of the model's best smoothing problem. As a result, a small model error of the relaxation modulus model is obtained, which increases the model's robustness. The necessary and sufficient optimality conditions are derived whose unique solution yields a direct analytical formula of the best-smoothed relaxation spectrum model. The related model of the relaxation modulus is given. A computational identification algorithm using the singular value decomposition is presented, which can be easily implemented in any computing environment. The approximation error, model smoothness, noise robustness, and identifiability of the polymer real spectrum are studied analytically. It is demonstrated by numerical studies that the algorithm proposed can be successfully applied for the identification of one- and two-mode Gaussian-like relaxation spectra. The applicability of this approach to determining the Baumgaertel, Schausberger, and Winter spectrum is also examined, and it is shown that it is well approximated for higher frequencies and, in particular, in the neighborhood of the local maximum. However, the comparison of the asymptotic properties of the best-smoothed spectrum model and the BSW model a priori excludes a good approximation of the spectrum in the close neighborhood of zero-relaxation time. [ABSTRACT FROM AUTHOR]

Background: DASS-21 is the common and widely used tool for assessing depression, anxiety and stress. However, its validity and Reliability in Ethiopian Amharic language is not assessed. Objective: To translate the DASS-21 and assess its validity and reliability among Ethiopian Defense University college of health science students in Bishoftu, Ethiopia. Method: An institutional based cross-sectional study was conducted using a self-administered questionnaire. A total of 435 students from different departments in accordance with their proportional size were participated in this study. As to the sampling technique, the study units were selected from each department and year of study using simple random sampling proportional to size. Confirmatory factor analyses were employed to assess the factor structure and construct validity of Amharic version of the DASS-21. Cronbach alpha coefficient and corrected item total correlation was calculated to assess the internal consistency of Amharic version of DASS-21. Result: Among 435 undergraduate students who participated in the study, 246(56.6%) were the age of 18–25 year and majority 347(79.8%) were males. Regarding their year of study; 200 (46.0%) were first year students. Confirmatory factor analysis indicates a good model fit of the three correlated factors (Comparative fit index (CFI) = 0.92 with root mean square error of approximation (RMSEA) = 0.059[0.052–0.066] and standardize root mean residual SRMR = 0.045). The internal consistency of overall DASS-21 and each sub scale were in acceptable range (0.91, 0.82, 0.77 and 0.77) respectively. Conclusion: Amharic version of DASS-21 was found to be a valid and reliable instrument to measure the mental problem especially, Depression, Anxiety and Stress among university students. [ABSTRACT FROM AUTHOR]

In this work, the capability to generate the negative group delay (NGD) phenomenon at those frequencies higher than a certain value, that is, the NGD high‐pass (HP) filtering function, of the bilinear double‐order transfer function has been demonstrated. Based on the Type‐I bilinear double‐order filter circuit, our theory has been verified by strong agreements between the formulae and their proof‐of‐concept (POC) circuit‐based simulation results. Both formula‐based and POC circuit‐based simulations give the minimum group delay of −5 ms yet the cut‐off frequency of 10 and 9 rad/s, respectively. Such slight deviation is caused by the approximation error of the bilinear double‐order impedance. By employing two orders, the bilinear double‐order transfer function has been found to be the basis transfer function for the NGD filtering function with the highest degree of freedom. Based on our design equation for the NGD filtering function, it can be seen that the distances between zero and pole and the characteristic frequency of the bilinear double‐order transfer function are governed by such characteristic frequency itself, the cut‐off frequency of the NGD filtering function, and the fractional order of the Laplacian operator. In addition, the effects of the fractional order of the Laplacian operator, the fractional order of the transfer function, and the ratio of the abovementioned distances to the characteristics of the newly found double‐order NGD filtering function have been studied in detail. [ABSTRACT FROM AUTHOR]

In this paper, we determine the error of approximation of functions, conjugate to the functions of two variables ( 2 ⁢ π 2\pi -periodic in both variables) belonging to the Lipschitz class Lip ( ( ξ 1 , ξ 2 ) ; r \operatorname{Lip}((\xi_{1},\xi_{2});r ) ( r ≥ 1 r\geq 1 ), through the double Nörlund means of its conjugate double Fourier series. Additionally, in the form of corollaries, we present some particular cases of the theorem proved here. [ABSTRACT FROM AUTHOR]

We give investigations on a kernel function approximation problem arising from learning theory and show the convergence rate from the view of classical Fourier analysis. First, we provide the general definition for a modulus of smoothness and a K -functional, and show that they are equivalent. In particular, we give explicit representation for some moduli of smoothness. Second, we establish some Jackson-type inequalities for the approximation error associated with some non-radial kernels. Also we apply these results to some concrete classical kernel function spaces and give Jackson-type inequalities for some concrete RKHS approximation problems. Finally, we apply these discussions to learning theory and describe the learning rates with the moduli of smoothness. The tools we used are Fourier analysis and the semigroup operator. The results show that the Jackson-type inequalities of approximation by some radial kernel functions on compact set with nonempty interiors cannot be expressed with the classical moduli of smoothness. [ABSTRACT FROM AUTHOR]

A novel cascade visual control scheme is proposed to tailor for electrically driven robotic manipulators that operate under kinematic and dynamic uncertainties, utilizing an uncalibrated stationary camera. The proposed control approach incorporates adaptive weight radial basis function neural networks (RBFNNs) to learn the behaviors of the uncertain dynamics of the robot and the joint actuators. The controllers are designed to nullify the approximation error and mitigate unknown disturbances through an integrated robust adaptive mechanism. A major advantage of the proposed approach is that prior knowledge of the dynamics of the robotic manipulator and its actuator is no longer required. The controller autonomously assimilates the robot and actuator dynamics online, thereby obviating the need for fussy regression matrix derivation and advance dynamic measurement to establish the adaptive dynamic parameter update algorithm. The proposed scheme ensures closed-loop system stability, bounded system states, and the convergence of tracking errors to zero. Simulation results, employing a PUMA manipulator as a testbed, substantiate the viability of the proposed control policy. [ABSTRACT FROM AUTHOR]

Velocity response spectrum (SV) plays a critical role in the seismic design of structures equipped with velocity-dependent dampers. Often, the SV is approximated using the pseudo-velocity response spectrum (PSV), because of the lack of information on the SV in seismic codes. As indicated in the existing literature, the error of this approximation is significant for short and long periods, and therefore, researchers have started to develop PSV to SV conversion models for obtaining SV more accurately. Recent studies discovered that the relationship between the PSV and SV is affected not only by structural parameters, but also by magnitude, distance, and site class. However, models for converting PSV to SV including magnitude, distance, and site class as input parameters have not been developed. To this end, a PSV to SV conversion model including magnitude, distance, and site class is proposed in this study based on a large number of real ground motion records (16,660 horizontal acceleration time histories) selected from the Japan Strong Motion Network. Furthermore, since the magnitude and distance are not specified in the seismic design, a response-spectrum-shape factor, s, is discussed to reflect the influences of magnitude and distance. Accordingly, an SV/PSV model incorporating s is established. The proposed models show better accuracy than the existing models for cases with different magnitudes, distances, and site classes. [ABSTRACT FROM AUTHOR]

Particle-based Variational Inference (ParVI) methods have been widely adopted in deep Bayesian inference tasks such as Bayesian neural networks or Gaussian Processes, owing to their efficiency in generating high-quality samples given the score of the target distribution. Typically, ParVI methods evolve a weighted-particle system by approximating the first-order Wasserstein gradient flow to reduce the dissimilarity between the particle system's empirical distribution and the target distribution. Recent advancements in ParVI have explored sophisticated gradient flows to obtain refined particle systems with either accelerated position updates or dynamic weight adjustments. In this paper, we introduce the semi-Hamiltonian gradient flow on a novel Information–Fisher–Rao space, known as the SHIFR flow, and propose the first ParVI framework that possesses both accelerated position update and dynamical weight adjustment simultaneously, named the General Accelerated Dynamic-Weight Particle-based Variational Inference (GAD-PVI) framework. GAD-PVI is compatible with different dissimilarities between the empirical distribution and the target distribution, as well as different approximation approaches to gradient flow. Moreover, when the appropriate dissimilarity is selected, GAD-PVI is also suitable for obtaining high-quality samples even when analytical scores cannot be obtained. Experiments conducted under both the score-based tasks and sample-based tasks demonstrate the faster convergence and reduced approximation error of GAD-PVI methods over the state-of-the-art. [ABSTRACT FROM AUTHOR]

We introduce new finite-dimensional spaces specifically designed to approximate the solutions to high-frequency Helmholtz problems with smooth variable coefficients in dimension d. These discretization spaces are spanned by Gaussian coherent states, that have the key property to be localised in phase space. We carefully select the Gaussian coherent states spanning the approximation space by exploiting the (known) micro-localisation properties of the solution. For a large class of source terms (including plane-wave scattering problems), this choice leads to discrete spaces that provide a uniform approximation error for all wavenumber k with a number of degrees of freedom scaling as k d - 1 / 2 , which we rigorously establish. In comparison, for discretization spaces based on (piecewise) polynomials, the number of degrees of freedom has to scale at least as k d to achieve the same property. These theoretical results are illustrated by one-dimensional numerical examples, where the proposed discretization spaces are coupled with a least-squares variational formulation. [ABSTRACT FROM AUTHOR]

Monte Carlo integration is a commonly used technique to compute intractable integrals and is typically thought to perform poorly for very high-dimensional integrals. To show that this is not always the case, we examine Monte Carlo integration using techniques from the high-dimensional statistics literature by allowing the dimension of the integral to increase. In doing so, we derive nonasymptotic bounds for the relative and absolute error of the approximation for some general classes of functions through concentration inequalities. We provide concrete examples in which the magnitude of the number of points sampled needed to guarantee a consistent estimate varies between polynomial to exponential, and show that in theory arbitrarily fast or slow rates are possible. This demonstrates that the behavior of Monte Carlo integration in high dimensions is not uniform. Through our methods we also obtain nonasymptotic confidence intervals which are valid regardless of the number of points sampled. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Central China Normal University is the property of Huazhong Normal University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this article we introduce an algorithm that approximates the nondominated sets of multiobjective mixed-integer convex optimization problems. The algorithm constructs an inner and outer approximation of the front exploiting the convexity of the patches for problems with an arbitrary number of criteria. In the algorithm, the problem is decomposed into patches, which are multiobjective convex problems, by fixing the integer assignments. The patch problems are solved using (simplicial) Sandwiching. We identify parts of patches that are dominated by other patches and ensure that these patch parts are not refined further. We prove that the algorithm converges and show a bound on the reduction of the approximation error in the course of the algorithm. We illustrate the behaviour of our algorithm using some numerical examples and compare its performance to an algorithm from literature. [ABSTRACT FROM AUTHOR]

The conventional theory of elastic-wave propagation is based on classical elastodynamics, assuming linear small deformations of particles. However, recent observations of seismic rotation have revealed significant disparities between actual rotational motions induced by earthquakes in focal areas and near fields compared to theoretical calculations and simulations. Considering the nonlinearity may be the main cause of the discrepancies and based on classical elastodynamic principle, we derive seismic elastic-wave equations with Green strain tensor without the linear small deformation assumption, a different way from using complex nonlinear constitutive relation and try to interpret the mechanism of seismic rotation. By simulating and analyzing translational and rotational components subjected to the three basic and typical vibrating sources, namely, isotropic (ISO), double couple (DC), and compensated linear vector dipole (CLVD), represented by moment tensors, we investigate the wavefield differences between elastic-wave equations based on linear and nonlinear geometric relations and quantify the differences in homogeneous elastic full-space model. Subsequently, we simulate two observed six-component Taiwan earthquakes and compare their differences caused by nonlinear simulations. The results indicate that linear approximation errors are more pronounced in seismic ISO and CLVD sources. And the nonlinearity of small deformation has a more pronounced effect on rotational motions deduced by strong earthquakes. Also, the nonlinear mechanics of seismic rotation can attribute to the complex propagation paths and source mechanisms simultaneously. [ABSTRACT FROM AUTHOR]

The tracking control of a non-linear fractional-order system (FOS) is highly tedious since cumulative approximation errors are encountered during the control process. Hence, to cope with that problem, several controllers have been adopted previously, such as backstepping controllers, sliding mode controllers, dynamic surface controllers, etc. These controllers have successfully reduced the approximation errors in the integer order non-linear system. However, the performance of those effective controllers in fractional-order non-linear systems is evaluated in this paper. The controllers such as the backstepping controller, sliding mode controller, dynamic surface controller and fuzzy dynamic surface controller are examined on FOS to verify the stability of control systems. As a result, the backstepping controller has performed better than other controllers in the case of actuator errors but still has the problem of the increased number of derivative terms and greater complexity in repeated calculations. The tracking error of the backstepping controller has been reduced to 0.0358 with 1.58 s of response time. [ABSTRACT FROM AUTHOR]

The demand for thermal power generation from natural gas has increased globally due to its cleaner burning properties compared to other fossil fuels. Optimizing the gas flow through the network to meet this demand is challenging due to the nonconvex Weymouth equation constraining gas flow and nodal pressures in pipelines. Traditional methods for addressing this nonconvexity lead to significant approximation errors or high operational costs. This study poses the Weymouth constraint as a Mathematical Programming with Complementarity Constraints (MPCC) for an optimal gas flow problem. The complementarity constraints reformulate the discontinuous sign function using binary-behaving continuous variables. This MPCC-based approach avoids solving mixed-integer programming problems while enhancing the accuracy of conventional linear and second-order approximations. Testing the approach on various interconnected systems, including Colombia's national gas transportation grid, demonstrated significant reductions in Weymouth approximation errors, thereby supporting effective optimization for interconnected networks. [ABSTRACT FROM AUTHOR]

One‐hidden‐layer feedforward neural networks are described as functions having many real‐valued parameters. Approximation properties of neural networks are established (universal approximation property), and the approximation error is related to the number of parameters in the network. The essentially optimal order of approximation error bounds was already derived in 1996. We focused on the numerical experiment that indicates the neural networks whose parameters contain stochastic perturbations gain better performance than ordinary neural networks and explored the approximation property of neural networks with stochastic perturbations. In this paper, we derived the quantitative order of variance of stochastic perturbations to achieve the essentially optimal approximation order and verified the justifiability of our theory by numerical experiments. [ABSTRACT FROM AUTHOR]

Directional interpolation is a fast and efficient compression technique for high‐frequency Helmholtz boundary integral equations, but requires a very large amount of storage in its original form. Algebraic recompression can significantly reduce the storage requirements and speed up the solution process accordingly. During the recompression process, weight matrices are required to correctly measure the influence of different basis vectors on the final result, and for highly accurate approximations, these weight matrices require more storage than the final compressed matrix. We present a compression method for the weight matrices and demonstrate that it introduces only a controllable error to the overall approximation. Numerical experiments show that the new method leads to a significant reduction in storage requirements. [ABSTRACT FROM AUTHOR]

Let X,Y$X, Y$ be two Banach spaces, F:X→Y$F:X\rightarrow Y$ be an ε$\varepsilon$‐norm‐additive map for some ε⩾0$\varepsilon \geqslant 0$, that is, |∥F(x)+F(y)∥−∥x+y∥|⩽ε,for allx,y∈X.$$\begin{equation*} \big\vert \Vert F(x)+F(y)\Vert -\Vert x+y\Vert \big\vert \leqslant \varepsilon, {\text{\,for all\,}} x,y\in X. \end{equation*}$$In this paper, we prove that if F$F$ is surjective with F(0)=0$F(0)=0$, then there exists a linear surjective isometry U:X→Y$U:X\rightarrow Y$ such that ∥F(x)−U(x)∥⩽32ε,for allx∈X.$$\begin{equation*} \Vert F(x)-U(x)\Vert \leqslant \frac{3}{2}\varepsilon, {\text{\,for all\,}} x\in X. \end{equation*}$$The estimate 32ε$\frac{3}{2}\varepsilon$ is sharp. We also approximate standard surjective ε$\varepsilon$‐norm‐additive maps between the positive cones of continuous function spaces by linear isometries within a sharp approximation error 32ε$\frac{3}{2}\varepsilon$. [ABSTRACT FROM AUTHOR]

The predefined performance tracking control issue of a vertical take-off and landing (VTOL) aircraft system is discussed in this work. Employing the presented control performance function, the constrained error signal of the aircraft is switched into an equivalent unconstrained signal. Moreover, fuzzy logic systems are introduced to address unknown nonlinear dynamics problems in aircraft system. Furthermore, in order to address the approximation errors and the external disturbances of aircraft system, parameter-adaptive update laws are proposed in the analysis process. Subsequently, adaptive fuzzy sliding-mode control laws are designed by combining fuzzy control method with sliding-mode control technique. By applying the redesigned sliding-mode functions and the Lyapunov stability method, the output of the VTOL aircraft system can track the desired trajectory, and the tracking errors are constrained in a predefined range. Finally, the availability of the proposed control laws is testified through simulation examples. [ABSTRACT FROM AUTHOR]

To strike a balance between modeling accuracy and computational efficiency for simulations of ultrasound waves in soft tissues, we derive a pseudodifferential factorization of the wave operator with fractional attenuation. This factorization allows us to approximately solve the Helmholtz equation via one-way (transmission) or two-way (transmission and reflection) sweeping schemes tailored to high-frequency wave fields. We provide explicitly the three highest order terms of the pseudodifferential expansion to incorporate the well-known square-root first order symbol for wave propagation, the zeroth order symbol for amplitude modulation due to changes in wave speed and damping, and the next symbol to model fractional attenuation. We also propose wide-angle Padé approximations for the pseudodifferential operators corresponding to these three highest order symbols. Our analysis provides insights regarding the role played by the frequency and the Padé approximations in the estimation of error bounds. We also provide a proof-of-concept numerical implementation of the proposed method and test the error estimates numerically. [ABSTRACT FROM AUTHOR]

Building on the successes of local kernel methods for approximating the solutions to partial differential equations (PDEs) and the evaluation of definite integrals (quadrature/cubature), a local estimate of the error in such approximations is developed. This estimate is useful for determining locations in the solution domain where increased node density (equivalently, reduction in the spacing between nodes) can decrease the error in the solution. An adaptive procedure for adding nodes to the domain for both the approximation of derivatives and the approximate evaluation of definite integrals is described. This method efficiently computes the error estimate at a set of prescribed points and adds new nodes for approximation where the error is too large. Computational experiments demonstrate close agreement between the error estimate and actual absolute error in the approximation. Such methods are necessary or desirable when approximating solutions to PDEs (or in the case of quadrature/cubature), where the initial data and subsequent solution (or integrand) exhibit localized features that require significant refinement to resolve and where uniform increases in the density of nodes across the entire computational domain is not possible or too burdensome. [ABSTRACT FROM AUTHOR]

In this work is considered an elliptic problem, referred to as the Ventcel problem, involving a second-order term on the domain boundary (the Laplace--Beltrami operator). A variational formulation of the Ventcel problem is studied, leading to a finite element discretization. The focus is on the construction of high-order curved meshes for the discretization of the physical domain and on the definition of the lift operator, which is aimed at transforming a function defined on the mesh domain into a function defined on the physical one. This lift is defined in such a way as to satisfy adapted properties on the boundary relative to the trace operator. The Ventcel problem approximation is investigated both in terms of geometrical error and of finite element approximation error. Error estimates are obtained both in terms of the mesh order r > 1 and to the finite element degree k > 1, whereas such estimates usually have been considered in the isoparametric case so far, involving a single parameter k = r. The numerical experiments we led in both 2 and 3 dimensions allow us to validate the results obtained and proved on the a priori error estimates depending on the 2 parameters k and r. A numerical comparison is made between the errors using the former lift definition and the lift defined in this work establishing an improvement in the convergence rate of the error in the latter case. [ABSTRACT FROM AUTHOR]

In approximation of functions based on point values, least-squares methods provide more stability than interpolation, at the expense of increasing the sampling budget. We show that near-optimal approximation error can nevertheless be achieved, in an expected L2 sense, as soon as the sample size m is larger than the dimension n of the approximation space by a constant multiplicative ratio. On the other hand, for m = n, we obtain an interpolation strategy with a stability factor of order n. The proposed sampling algorithms are greedy procedures based on [Batson, Spielman, and Srivastava, Twice-Ramanujan sparsifiers, in Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, 2009, pp. 255--262] and [Lee and Sun, SIAM J. Comput., 47 (2018), pp. 2315--2336], with polynomial computational complexity. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fast and accurate method for approximating the Minkowski sum of two rotational ellipsoids with a superellipsoid. The Minkowski sum is used in a variety of applications such as robot motion planning and particle flow simulation requiring collision detection. Many of them are computed based on Minkowski sum, whose accuracy and processing time depend on how the mesh is created. We approximate Minkowski sum with a superellipsoid function. The superellipsoid has various shapes with the value of the exponents, and the computation of the parameters including the exponents is an algebraic computation with the time complexity O (1) . As a result, approximation error is small and computation is fast in all combinations of rotational ellipsoids. In addition, collision is quickly detected by solving the inequality of superellipsoid. [ABSTRACT FROM AUTHOR]

The exponential function maps the imaginary axis to the unit circle and for many applications this unitarity property is also desirable from its approximations. We show that this property is conserved not only by the |$(k,k)$| -rational interpolant of the exponential on the imaginary axis, but also by |$(k,k)$| -rational barycentric approximants that minimize a linearized approximation error. These results are a consequence of certain properties of singular vectors of Loewner-type matrices associated to linearized approximation errors. Prominent representatives of this class are rational approximants computed by the adaptive Antoulas–Anderson (AAA) method and the AAA–Lawson method. Our results also lead to a modified procedure with improved numerical stability of the unitarity property and reduced computational cost. [ABSTRACT FROM AUTHOR]

Fully coupled McKean–Vlasov forward-backward stochastic differential equations (MV-FBSDEs) arise naturally from large population optimization problems. Judging the quality of given numerical solutions for MV-FBSDEs, which usually require Picard iterations and approximations of nested conditional expectations, is typically difficult. This paper proposes an a posteriori error estimator to quantify the |$L^2$| -approximation error of an arbitrarily generated approximation on a time grid. We establish that the error estimator is equivalent to the global approximation error between the given numerical solution and the solution of a forward Euler discretized MV-FBSDE. A crucial and challenging step in the analysis is the proof of stability of this Euler approximation to the MV-FBSDE, which is of independent interest. We further demonstrate that, for sufficiently fine time grids, the accuracy of numerical solutions for solving the continuous MV-FBSDE can also be measured by the error estimator. The error estimates justify the use of residual-based algorithms for solving MV-FBSDEs. Numerical experiments for MV-FBSDEs arising from mean field control and games confirm the effectiveness and practical applicability of the error estimator. [ABSTRACT FROM AUTHOR]

We consider approximating the solution of the Helmholtz exterior Dirichlet problem for a nontrapping obstacle, with boundary data coming from plane-wave incidence, by the solution of the corresponding boundary value problem where the exterior domain is truncated and a local absorbing boundary condition coming from a Padé approximation (of arbitrary order) of the Dirichlet-to-Neumann map is imposed on the artificial boundary (recall that the simplest such boundary condition is the impedance boundary condition). We prove upper- and lower-bounds on the relative error incurred by this approximation, both in the whole domain and in a fixed neighbourhood of the obstacle (i.e. away from the artificial boundary). Our bounds are valid for arbitrarily-high frequency, with the artificial boundary fixed, and show that the relative error is bounded away from zero, independent of the frequency, and regardless of the geometry of the artificial boundary. [ABSTRACT FROM AUTHOR]

We propose a simulation-based technique for the parameterization and the stability analysis of parametric Ordinary Differential Equations. This technique is an adaptation of Statistical Model Checking, often used to verify the validity of biological models, to the setting of Ordinary Differential Equations systems. The aim of our technique is to estimate the probability of satisfying a given property under the variability of the parameter or initial condition of the ODE, with any metrics of choice. To do so, we discretize the values space and use statistical model checking to evaluate each individual value w.r.t. provided data. Contrary to other existing methods, we provide statistical guarantees regarding our results that take into account the unavoidable approximation errors introduced through the numerical integration of the ODE system performed while simulating. In order to show the potential of our technique, we present its application to two case studies taken from the literature, one relative to the growth of a jellyfish population, and the other concerning a well-known oscillator model. [ABSTRACT FROM AUTHOR]

In the present article, we extend the results of the neural network interpolation operators activated by smooth ramp functions proposed by Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016). We give different results from Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016) we discuss the high-order approximation result using the smoothness of φ and a related Voronovskaya-type asymptotic expansion for the error of approximation. In addition, we showcase the related fractional estimates result and the fractional Voronovskaya type asymptotic expansion. We investigate the approximation degree for the iterated and complex extensions of the aforementioned operators. Finally, we provide numerical examples and graphs to effectively illustrate and validate our results. [ABSTRACT FROM AUTHOR]