Search

In the grey prediction, the nonlinear Grey Bernoulli model NGBM (1, 1) is an important type. The NGBM (1, 1) has good adaptability to data fitting and then small prediction errors, and thus has been applied widely. However, if we improve the modelling method, the prediction precision shall be improved to some extent. The important factors of prediction error are the approximation of background value and the approximation of power exponent. Therefore, the paper tries to combine the optimisation of background value with the optimisation of the power exponent of NGBM (1, 1) model and then improves the model from parameter estimation. The paper gives three methods for the following three cases respectively: the background value in the form of exponential curve, the background value in the form of the polynomial curve and the background value in the form of interpolation function, to combine background value optimisation with power exponent optimisation for parameter optimisation. The final section of the paper builds the NGBM (1, 1) models of China's GDP and energy consumption with three improvement methods. The simulation and prediction results show the three improvement methods all have high precision. The methods given offer good approaches for the in-depth study on nonlinear grey Bernoulli model, enrich the method system of grey modelling and can be applied to the studies on other grey models to promote the study and wide application of the grey model. [ABSTRACT FROM AUTHOR]

Copyright of Basrah Journal of Science / Magallat Al-Barat Li-L-ulum is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) 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 order to improve the effect of Chinese-English machine translation, this paper combines attention mechanism and neural network algorithm and applies it to Chinese-English machine learning translation. Moreover, this paper uses Gaussian distribution instead of chi-square distribution to analyze the approximate error introduced by the Chinese and English speech energy detection method. In addition, this paper studies the overall and specific approximation errors by establishing the normalized mean square error function and the absolute error function, respectively. Finally, this paper proposes a new model for machine translation based on logarithmic position representation and self-attention mechanism. Through the experimental research, it can be seen that the Chinese-English machine translation model integrating attention mechanism and bidirectional neural network proposed in this paper has a good practical translation effect. [ABSTRACT FROM AUTHOR]

In this paper, we propose and solve low phase-rank approximation problems, which serve as a counterpart to the well-known low-rank approximation problem and the Schmidt-Mirsky theorem. It is well known that a nonzero complex number can be specified by its gain and phase, and while it is generally accepted that the gains of a matrix may be defined by its singular values, there is no widely accepted definition for its phases. In this work, we consider sectorial matrices, whose numerical ranges do not contain the origin, and adopt the canonical angles of such matrices as their phases. Similarly to the rank of a matrix being defined as the number of its nonzero singular values, we define the phase-rank of a sectorial matrix as the number of its nonzero phases. While a low-rank approximation problem is associated with the matrix arithmetic mean, it turns out that a natural parallel for the low phase-rank approximation problem is to use the matrix geometric mean to measure the approximation error. Importantly, we derive a majorization inequality between the phases of the geometric mean and the arithmetic mean of the phases, similarly to the Ky-Fan inequality for eigenvalues of Hermitian matrices. A characterization of the solutions to the proposed problem, with the same flavor as the Schmidt-Mirsky theorem, is then obtained in the case where both the objective matrix and the approximant are restricted to be positive-imaginary. In addition, we provide an alternative formulation of the low phase-rank approximation problem using geodesic distances between sectorial matrices. The two formulations give rise to the exact same set of solutions when the involved matrices are additionally assumed to be unitary. [ABSTRACT FROM AUTHOR]

The current study was aimed to assess the reliability and validity of the Urdu version of the Psychosomatic Symptoms Scale (PSSS) in Pakistani patients. The PSSS is time-saving and easy to administer. The field experts drafted the translated version of PSSS. The Urdu version of PSSS, Patient Health Questionnaire- 9 (PHQ-9), and Symptom Checklist-90 (SCL-90) Urdu version were used for assessment. The translation procedure was comprised of three steps, namely forward translation, back translation, and expert panel discussion. A sample of 982 (men = 50.5% and women = 49.5%) was collected with a convenient sample technique from a general hospital and private clinic. The Cronbach's alpha for PSSS was 0.974. The confirmatory factor analysis (CFA) revealed that all the items factor loading of PSSS were more than 0.35, the root mean square error of approximation (RMSEA) was = 0.062, the standardized root mean residual (SRMR) was = 0.043, and the comparative fit index (CFI) was = 0.97 with 90% CI. The results also showed that women (M = 72.08, SD = 6.79) are more likely to have psychosomatic symptoms than men (M = 51.21, SD = 13.36) on P < 0.001. The PSSS Urdu version is proven to be a useful and reliable instrument for screening, monitoring, and assessing Pakistani patients' psychosomatics symptoms. [ABSTRACT FROM AUTHOR]

In this article, an effective hybrid method is proposed to design the subarrays generating multiple functional beams. In the process of synthesis, one of the target patterns is selected as the primary pattern which is generated at the element level. Other secondary patterns are generated at the subarray level by dividing the elements into nonoverlapped subarrays and utilizing attenuators and phase shifters at the subarray outputs. To simplify the complexity of the feed network, the subarray configurations of secondary patterns are the same. The proposed method consists of two steps. In the first step, the subarray configuration of the phased array is optimized by the ${K}$ -means clustering method. In the second step, the convex optimization method is applied to acquiring the complex weighting coefficients of subarrays, which can improve the accuracy of the synthesized secondary patterns. A series of numerical examples demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Satellite-based aerosol retrievals provide global spatially distributed estimates of atmospheric aerosol parameters that are commonly needed in applications such as estimation of atmospherically corrected satellite data products, climate modelling and air quality monitoring. However, a common feature of the conventional satellite aerosol retrievals is that they have reasonably low spatial resolution and poor accuracy caused by uncertainty in auxiliary model parameters, such as fixed aerosol model parameters, and the approximate forward radiative transfer models utilized to keep the computational complexity feasible. As a result, the improvement and reprocessing of the operational satellite data retrieval algorithms would become a tedious and computationally excessive problem. To overcome these problems, we have developed a machine-learning-based post-process correction approach to correct the existing operational satellite aerosol data products. Our approach combines the existing satellite retrieval data and a post-processing step where a machine learning algorithm is utilized to predict the approximation error in the conventional retrieval. With approximation error, we refer to the discrepancy between the true aerosol parameters and the ones retrieved using the satellite data. Our hypothesis is that the prediction of the approximation error with a finite training dataset is a less complex and easier task than the direct, fully learned machine-learning-based prediction in which the aerosol parameters are directly predicted given the satellite observations and measurement geometry. Our approach does not require reprocessing of the satellite retrieval products; it requires only a computationally fast machine-learning-based post-processing step of the existing retrieval product. Our approach is based on neural networks trained based on collocated satellite data and accurate ground-based Aerosol Robotic Network (AERONET) aerosol data. Based on our post-processing approach, we propose a post-process-corrected high-resolution Sentinel-3 Synergy aerosol product, which gives a spectral estimate of the aerosol optical depth at five different wavelengths with a high spatial resolution equivalent to the native resolution of the Sentinel-3 Level-1 data (300 m at nadir). With aerosol data from Sentinel-3A and 3B satellites, we demonstrate that our approach produces high-resolution aerosol data with clearly better accuracy than the operational Sentinel-3 Level-2 Synergy aerosol product, and it also results in slightly better accuracy than the conventional fully learned machine learning approach. We also demonstrate better generalization capabilities of the post-process correction approach over the fully learned approach. [ABSTRACT FROM AUTHOR]

In this work, we study several approximation properties of Sz'asz-Mirakjan-Durrmeyer operators with shape parameter λ ∈ [-1, 1]. Firstly, we obtain some preliminaries results such as moments and central moments. Next, we estimate the order of convergence in terms of the usual modulus of continuity, for the functions belong to Lipschitz type class and Peetre's K-functional, respectively. Also, we prove a Korovkin type approximation theorem on weighted spaces and derive a Voronovskaya type asymptotic theorem for these operators. Finally, we show the comparison of the convergence of these newly defined operators to certain functions with some graphics and an error of approximation table. [ABSTRACT FROM AUTHOR]

We present randUBV, a randomized algorithm for matrix sketching based on the block Lanzcos bidiagonalization process. Given a matrix A, it produces a low-rank approximation of the form UBVT, where U and V have orthonormal columns in exact arithmetic and B is block bidiagonal. In finite precision, the columns of both U and V will be close to orthonormal. Our algorithm is closely related to the randQB algorithms of Yu, Gu, and Li [SIAM J. Matrix Anal. Appl., 39 (2018), pp. 1339-1359]. in that the entries of B are incrementally generated and the Frobenius norm approximation error may be efficiently estimated. It is therefore suitable for the fixed-accuracy problem and so is designed to terminate as soon as a user input error tolerance is reached. Numerical experiments suggest that the block Lanczos method is generally competitive with or superior to algorithms that use power iteration, even when A has significant clusters of singular values. [ABSTRACT FROM AUTHOR]

This paper investigates a novel control algorithm to deal with trajectory tracking control problems of robotic manipulators based on adaptive backstepping integral nonsingular terminal sliding mode control (BINTSMC). The proposed approach is developed based on an integration between the integral nonsingular terminal sliding mode control (INTSMC) and stability analysis procedure of backstepping control technique. In addition, the extreme learning machine (ELM) learning algorithm is introduced to approximate lumped uncertain component of the dynamics model. The approximated lumped uncertain terms are transmitted to rebuild the dynamics and provide a compensation feedback for control systems. Moreover, the robust term is utilized to counteract the approximation residual error, wherein the nonconstant ELM approximation error is analyzed. The superior performance of the BINTSMC is validated by comparing conventional NTSMC in the simulations. [ABSTRACT FROM AUTHOR]

This article presents a randomized matrix-free method for approximating the trace of f (A) , where A is a large symmetric matrix and f is a function analytic in a closed interval containing the eigenvalues of A. Our method uses a combination of stochastic trace estimation (i.e., Hutchinson's method), Chebyshev approximation, and multilevel Monte Carlo techniques. We establish general bounds on the approximation error of this method by extending an existing error bound for Hutchinson's method to multilevel trace estimators. Numerical experiments are conducted for common applications such as estimating the log-determinant, nuclear norm, and Estrada index. We find that using multilevel techniques can substantially reduce the variance of existing single-level estimators. [ABSTRACT FROM AUTHOR]

This article analyzes the effect of discretization in the frequency-adaptive flux observer (FAO). Although FAO has structural simplicity and ease of implementation, the discrete-time FAO has phase and gain errors in fundamental flux estimation. As a result, the estimated flux might be erroneous, especially in the high-speed region. For accurate flux estimation, this article analyzes how the voltage synthesis in the pulsewidth-modulation affects the stator flux. Through the analysis, the backward Euler method is selected for the primitive stator flux estimation. To mitigate the FAO's discretization error, the proposed FAO utilizes the integrators constructed at the rotor reference frame (RRF). Since the discretization error comes from the approximation error of the continuous-time integrator at the operating frequency, the proposed FAO conducts the fundamental flux extraction at the RRF, where the operating frequency components are treated as a dc signal. Thanks to the coordinate transformation, the FAO can be discretized with no error while keeping the structure of double integrators. The proposed method is verified through the computer simulation and experimental test. [ABSTRACT FROM AUTHOR]

Modern search services often provide multiple options to rank the search results, e.g., sort "by relevance", "by price" or "by discount" in e-commerce. While the traditional rank by relevance effectively places the relevant results in the top positions of the results list, the rank by attribute could place many marginally relevant results in the head of the results list leading to poor user experience. In the past, this issue has been addressed by investigating the relevance-aware filtering problem, which asks to select the subset of results maximizing the relevance of the attribute-sorted list. Recently, an exact algorithm has been proposed to solve this problem optimally. However, the high computational cost of the algorithm makes it impractical for the Web search scenario, which is characterized by huge lists of results and strict time constraints. For this reason, the problem is often solved using efficient yet inaccurate heuristic algorithms. In this article, we first prove the performance bounds of the existing heuristics. We then propose two efficient and effective algorithms to solve the relevance-aware filtering problem. First, we propose OPT-Filtering, a novel exact algorithm that is faster than the existing state-of-the-art optimal algorithm. Second,we propose an approximate and even more efficient algorithm, ∈-Filtering, which, given an allowed approximation error ∈, finds a (1-∈)--optimal filtering, i.e., the relevance of its solution is at least (1-∈) times the optimum. We conduct a comprehensive evaluation of the two proposed algorithms against state-of-the-art competitors on two real-world public datasets. Experimental results show that OPT-Filtering achieves a significant speedup of up to two orders of magnitude with respect to the existing optimal solution, while ∈-Filtering further improves this result by trading effectiveness for efficiency. In particular, experiments show that ∈-Filtering can achieve quasi-optimal solutions while being faster than all state-of-the-art competitors in most of the tested configurations. [ABSTRACT FROM AUTHOR]

This article studies the adaptive finite-time quantized tracking control problem for a class of full state-constrained nonlinear systems with unknown control directions based on a modified fractional-order dynamic surface control (FODSC) technique. First, fractional calculus is introduced to filter design to avoid the issue of the “explosion of complexity” exposed in the traditional backstepping technique. To facilitate the control design, barrier Lyapunov functions and Nussbaum gain technique are utilized to handle full state-constrained problem and the unknown control directions, respectively. In addition, the fuzzy logic systems are employed to approximate the unknown nonlinearity of the system. By integrating with the approximation errors and compensating signals, a composite adaptive quantized controllers is designed to guarantee all the signals of the closed-loop systems are bounded and tracking error converges to an arbitrarily small neighborhood of the zero within a finite time. Finally, a mechanical horizontal platform model and a Brusselator model are carried out to verify the effectiveness of the presented control method. [ABSTRACT FROM AUTHOR]

This article considers the Nussbaum gain adaptive control issue for a type of nonlinear systems, in which some sophisticated and challenging problems, such as periodic disturbances, dead zone output, and unknown control direction are addressed. The Fourier series expansion and radial basis function neural network are incorporated into a function approximator to model time-varying-disturbed function with a known period in nonlinear systems. To deal with the problems of the dead zone output and unknown control direction, the Nussbaum-type function is recommended in the design of the control algorithm. Applying the Lyapunov stability theory and backstepping technique, the proposed control strategy ensures that the tracking error is pulled back to a small neighborhood of origin and all closed-loop signals are bounded. Finally, simulation results are presented to show the availability and validity of the analysis approach. [ABSTRACT FROM AUTHOR]

The $K$ K -means algorithm is one of the most popular clustering methods. However, it is a well-known fact that its performance, in terms of quality of the obtained solution and computational load, highly depends upon its initialization phase. For this reason, different initialization techniques have been developed throughout the years to enable its fast convergence to competitive solutions. In this sense, it is common practice to re-start the $K$ K -means algorithm several times via one of these techniques and keep the solution with the lowest error. Unfortunately, such a choice is still likely to be a poor approximation of the optimal set of centroids. In this article, we introduce a cheap Split-Merge step that can be used to re-start the $K$ K -means algorithm after reaching a fixed point. Under some settings, one can show that this approach reduces the error of the given fixed point without requiring any further iteration of the $K$ K -means algorithm. Moreover, experimental results show that this strategy is able to generate approximations with an associated error that is hard to reach for different multi-start methods, such as multi-start Forgy $K$ K -means, $K$ K -means++ and Hartigan $K$ K -means, while also computing a lower amount of distances than the previous algorithms. [ABSTRACT FROM AUTHOR]

Given an arbitrary function in |$\boldsymbol{H}({\operatorname{div}})$|⁠ , we show that the error attained by the global-best approximation by |$\boldsymbol{H}({\operatorname{div}})$| -conforming piecewise polynomial Raviart–Thomas–Nédélec elements under additional constraints on the divergence and normal flux on the boundary is, up to a generic constant, equivalent to the sum of independent local-best approximation errors over individual mesh elements, without constraints on the divergence or normal fluxes. The generic constant only depends on the shape-regularity of the underlying simplicial mesh, the space dimension, and the polynomial degree of the approximations. The analysis also gives rise to a stable, local, commuting projector in |$\boldsymbol{H}({\operatorname{div}})$|⁠ , delivering an approximation error that is equivalent to the local-best approximation. We next present a variant of the equivalence result, where robustness of the constant with respect to the polynomial degree is attained for unbalanced approximations. These two results together further enable us to derive rates of convergence of global-best approximations that are fully optimal in both the mesh size |$h$| and the polynomial degree |$p$|⁠ , for vector fields that only feature elementwise the minimal necessary Sobolev regularity. We finally show how to apply our findings to derive optimal a priori |$hp$| -error estimates for mixed and least-squares finite element methods applied to a model diffusion problem. [ABSTRACT FROM AUTHOR]

Engaging students in the essential practices of science has been forwarded as an effective means to achieve scientific literacy for all students by promoting the learning of content, procedures, and the epistemology of science concurrently. However, the emphasis on science practices requires teachers not only to develop skills and knowledge specific to the science practices, but also to adopt epistemic orientations congruent with student‐centered, knowledge generation science teaching approaches. This exploratory quantitative study examined the relationships between secondary science teachers' epistemic orientations and their understanding and self‐reported implementation of the science practices described in the US Next Generation Science Standards (NGSS). This study also identified teacher‐level variables that predict teachers' epistemic orientations, their understanding and implementation of the science practices. Data were collected from 128 secondary science teachers in a southeastern state of the US using three instruments: epistemic orientation survey with demographic questions, epistemic nature of science practice survey, and science practice implementation survey. One‐way analysis of variance, correlation analyses, multiple linear regression, and path analyses were used to examine the relationship among these three variables and the influence of teacher‐level variables on each of the three variables. Our results indicated that teachers' participation in NGSS PD programs significantly predicted their self‐reported implementation of science practices. In addition, teachers' lateral entry was the only significant predictor of teachers' epistemic understanding of science practices. Concerning the relationship among the three variables, the path analysis revealed that teachers' epistemic orientations significantly mediated the relationship between their epistemic understanding and implementation of science practices (χ2/df = 1.530, p = 0.216, comparative fit index = 0.980, Tucker–Lewis index = 0.940, root‐mean‐square error of approximation = 0.065). This study highlights a close relationship between teachers' epistemic orientations and enactment of science practices, which suggests prioritizing epistemic orientations in supporting teachers to implement the science‐as‐practice approach. [ABSTRACT FROM AUTHOR]

Frequency response analysis under uncertainty is computationally expensive. Low-rank approximation techniques can significantly reduce the solution times. Thin perforated cylinders, as with all shells, have specific features affecting the approximation error. There exists a rich thickness-dependent boundary layer structure, leading to local features becoming dominant as the thickness tends to zero. Related to boundary layers, there is also a connection between eigenmodes and the perforation patterns. The Krylov subspace approach for proportionally damped systems with uncertain Young's modulus is compared with the full system, and via numerical experiments, it is shown that the relative accuracy of the low-rank approximation of perforated shells measured in energy depends on the dimensionless thickness. In the context of frequency response analysis, it then becomes possible that, at some critical thicknesses, the most energetic response within the observed frequency range is not identified correctly. The reference structure used in the experiments is a trommel screen with a non-regular perforation pattern with two different perforation zones. The low-rank approximation scheme is shown to be feasible in computational asymptotic analysis of trommel designs when the proportional damping model is used. [ABSTRACT FROM AUTHOR]

This paper investigates the use of approximate fixed-width and static segment multipliers in the design of Delayed Least Mean Square (DLMS) adaptive filters based on the sign–magnitude representation for the error signal. The fixed-width approximation discards part of the partial product matrix and introduces a compensation function for minimizing the approximation error, whereas the static segmented multipliers reduce the bit-width of the multiplicands at runtime. The use of sign–magnitude representation for the error signal reduces the switching activity in the filter learning section, minimizing power dissipation. Simulation results reveal that, by properly sizing the two approaches, a steady state mean square error practically unchanged with respect to the exact DLMS can be achieved. The hardware syntheses in a 28 nm CMOS technology reveal that the static segmented multipliers perform better in the learning section of the filter and are the most efficient approach to reduce area occupation, while the fixed-width multipliers offer the best performances in the finite impulse response section and provide the lowest power dissipation. The investigated adaptive filters overcome the state-of-the-art, exhibiting an area and power reduction with respect to the standard implementation up to −18.0% and −64.1%, respectively, while preserving the learning capabilities. [ABSTRACT FROM AUTHOR]

In this article we consider the extension of the (L)SIAC-MRA enhancement procedure to nonuniform meshes. We demonstrate that error reduction can be obtained on perturbed quadrilateral and Delaunay meshes, and investigate the effect of limited resolution and its impact on the procedure for various function types. We show that utilizing mesh-based localized kernel scalings, which were shown to reduce approximation errors for LSIAC filters, improve the performance of the LSIAC-MRA enhancement procedure. Lastly, we demonstrate the usefulness of enhanced approximations generated by (L)SIAC-MRA in mesh adaptivity applications, and show that SIAC reconstruction can be used in identification of regions of high error in steady-state DG approximations. [ABSTRACT FROM AUTHOR]

Deep learning algorithms have been widely used to solve linear Kolmogorov partial differential equations (PDEs) in high dimensions, where the loss function is defined as a mathematical expectation. We propose to use the randomized quasi-Monte Carlo (RQMC) method instead of the Monte Carlo (MC) method for computing the loss function. In theory, we decompose the error from empirical risk minimization (ERM) into the generalization error and the approximation error. Notably, the approximation error is independent of the sampling methods. We prove that the convergence order of the mean generalization error for the RQMC method is O (n − 1 + ϵ) for arbitrarily small ϵ > 0 , while for the MC method it is O (n − 1 / 2 + ϵ) for arbitrarily small ϵ > 0. Consequently, we find that the overall error for the RQMC method is asymptotically smaller than that for the MC method as n increases. Our numerical experiments show that the algorithm based on the RQMC method consistently achieves smaller relative L 2 error than that based on the MC method. [ABSTRACT FROM AUTHOR]

DARTS has emerged as a popular method for neural architecture search (NAS) owing to its efficiency and simplicity. It employs gradient-based bi-level optimization to iteratively optimize the upper-level architecture parameters and lower-level super-network weights. The key challenge in DARTS is the accurate estimation of gradients for two-level object functions, leading to significant errors in gradient approximation. To address this issue, we propose a new approach, MR-DARTS, that incorporates a momentum term and a recursive scheme to improve gradient estimation. Specifically, we leverage historical information by using a running average of past observed gradients to enhance the quality of current gradient estimation in both upper-level and lower-level functions. Our theoretical analysis shows that the variance of our estimated gradient decreases with each iteration. By utilizing momentum and a recursive scheme, MR-DARTS effectively controls the error in stochastic gradient updates that result from inaccurate gradient estimation. Furthermore, we utilize the Neumann series approximation and Hessian Vector Product scheme to reduce computational requirements and memory usage. We evaluate our proposed method on several benchmarks and demonstrate its effectiveness through comprehensive experiments. • We pioneer a momentum recursive gradient estimator for differentiable NAS, improving gradient quality in upper-level and lower-level functions. • We leverage the historical gradient to enhance the quality of the current gradient estimation. • We provide a theoretical variance bound for the estimated hypergradients. • Extensive experiments demonstrate that MR-DARTS outperforms other methods. [ABSTRACT FROM AUTHOR]

In this article, we explore the further study of filter based schemes along with parameter choice rules and then apply to pseudo-differential operator equations and the analytic continuation problem. As we know that the stability of the regularization schemes depends on the regularization parameter and here we derive an a posteriori parameter choice rule which is best among all other parameter choice rules; and this parameter choice rule minimizes an upper bound of the approximation error ‖ x α , δ − x † ‖. Numerical experiments are also provided to validate the proposed theory. • The parameter choice rule considered here is the best among all existing parameter choice rules. • We studied filter-based regularization schemes along with the parameter choice rule. • The parameter choice rule provides optimal rate of convergence for filter-based schemes. • We studied the parameter choice rule in the context of two practical examples. [ABSTRACT FROM AUTHOR]

This paper presents a theoretical analysis of the value iteration Q-learning with non-discounted costs. The analysis focuses on two main aspects: the convergence of the iterative Q-function and the stability of the system under the final iterative control policy. Unlike previous theoretical results on Q-learning, our analysis takes into account the effect of approximation errors, leading to a more comprehensive investigation. We first discuss the effect of approximation errors on the iterative Q-function update. Then, considering the presence of approximation errors in each iteration, we analyze the convergence of the iterative Q-function. Furthermore, we establish a sufficient condition, also accounting for the approximation errors, to ensure the stability of the system under the final iterative control policy. Finally, two simulation cases are conducted to validate the presented convergence and stability results. [ABSTRACT FROM AUTHOR]

• Unique Augmentation Algorithm: Combines variational autoencoders with metric learning for time series. • Normalization of Irregularities: Normalizes heteroscedastic and non-stationary data effectively. • Advanced VAE Integration: Integrates GRU into VAE, enhanced by metric learning for better representation. • Improved Data Interpretability: Maintains temporal structure through multi-seasonal decomposition. • Superior Performance: Shows enhanced accuracy and fidelity in data-limited scenarios. In the field of time series classification, deep learning techniques have shown remarkable performance; however, their effectiveness is often compromised when confronted with challenges of insufficient data and class imbalance. To address this challenge, we propose an interpretable time series data augmentation algorithm integrating variational autoencoders (VAE) and metric learning. The core contribution of this algorithm is manifested in three aspects: First, it eliminates the heteroscedasticity and non-stationarity of the data, ensuring that the data satisfies the hypothesis of normal distribution in the potential space of the encoder, and effectively avoids the approximation error of the real data distribution; Secondly, the algorithm constructs a discriminant VAE potential space, suitable for data augmentation, with metric learning, ensuring that the hidden variable distribution accurately reflects the characteristics of the original data. Finally, this paper explores the multi-seasonal decomposition algorithm of time series to seamlessly integrate the structural features of the original time series in the generated data, thereby enhancing the interpretability of data generation. Through experimental verification on four multivariate time series data sets, including the electrical energy data set, the results demonstrate that the proposed algorithm outperforms existing methods in fidelity and prediction performance, exhibiting high stability and generalization ability, particularly in cases of limited data volume. The introduction of this algorithm not only contributes to enhancing the overall performance of time series classification models but also substantially reduces the cost of data collection and labeling, thereby demonstrating its significant value in practical applications. [ABSTRACT FROM AUTHOR]

High-dimensional dynamical systems often require computationally intensive physics-based simulations, making full physical space data assimilation impractical. Latent data assimilation methods perform assimilation in reduced-order latent space for efficiency but struggle with complex, nonlinear state-observation mappings. Recent solutions like Generalized Latent Data Assimilation (GLA) and Latent Space Data Assimilation (LSDA) address heterogeneous latent spaces by incorporating surrogate mapping functions but introduce computational costs and uncertainties. Furthermore, current algorithms that integrate data assimilation and deep learning still face limitations when it comes to handling non-explicit mapping functions. To address these challenges, this paper introduces a novel deep-learning-based data assimilation scheme, named Multi-domain Encoder–Decoder Latent Data Assimilation (MEDLA), capable of handling diverse data sources by sharing a common latent space. The proposed approach significantly reduces the computational burden since the complex mapping functions are mimicked by the multi-domain encoder–decoder neural network. It also enhances assimilation accuracy by minimizing interpolation and approximation errors. Extensive numerical experiments from three different test cases assess MEDLA's performance in high dimensional dynamical systems, benchmarking it against state-of-the-art latent data assimilation methods. The numerical results consistently underscore MEDLA's superiority in managing multi-scale observational data and tackling intricate, non-explicit mapping functions. [ABSTRACT FROM AUTHOR]

It is shown that weighted linear combinations of the generalised- α method and certain related higher order schemes allow for the formulation of unconditionally stable single step time integration methods with improved second order accuracy and more targeted high-frequency damping. It is also shown that, if the user controlled high frequency damping parameter ρ ∞ is set to zero, the new schemes can be expressed as linear multistep backward difference formulae and, in a particular case, recover Park's method. The performance of the proposed methods is illustrated in terms of mathematical analysis and a number of linear and nonlinear numerical examples including finite element based solutions of the incompressible Navier-Stokes equations. • New single step, second order, unconditionally stable implicit time integration schemes with controlled numerical damping. • New second order backward difference formulae. • Smaller approximation errors than generalised- α method and standard second order backward difference formula. • Trivial computer implementation and negligible additional computational cost. [ABSTRACT FROM AUTHOR]

In this paper, we propose an augmented subspace based adaptive proper orthogonal decomposition (POD) method for solving the time dependent partial differential equations. We use the difference between the approximation obtained in the augmented subspace and that obtained in the original POD subspace to construct an error indicator, by which we obtain a general framework for augmented subspace based adaptive POD method. We then provide two strategies to construct the augmented subspaces, the residual type augmented subspace and the coarse-grid approximation type augmented subspace. We apply our new methods to two typical 3D advection-diffusion equations with the advection being the Kolmogorov flow and the ABC flow. Numerical results show that both the residual type augmented subspace based adaptive POD method and the coarse-grid approximation type augmented subspace based adaptive POD method are more efficient than the existing adaptive POD methods, especially for the advection dominated models. • Propose an augmented subspace approximation based error indicator for the POD approximation of the time dependent PDEs. • Obtain a general framework of augmented subspace based adaptive POD method for solving the time dependent PDEs. • Provide two strategies to obtain the augmented subspaces, the residual type and the coarse-grid approximation type. • Numerical experiments show that the new method is more efficient than the existing adaptive POD methods. [ABSTRACT FROM AUTHOR]

Solving high dimensional partial differential equations (PDEs) has historically posed a considerable challenge when utilizing conventional numerical methods, such as those involving domain meshes. Recent advancements in the field have seen the emergence of neural PDE solvers, leveraging deep networks to effectively tackle high dimensional PDE problems. This study introduces Inf-SupNet, a model-based unsupervised learning approach designed to acquire solutions for a specific category of elliptic PDEs. The fundamental concept behind Inf-SupNet involves incorporating the inf-sup formulation of the underlying PDE into the loss function. The analysis reveals that the global solution error can be bounded by the sum of three distinct errors: the numerical integration error, the duality gap of the loss function (training error), and the neural network approximation error for functions within Sobolev spaces. To validate the efficacy of the proposed method, numerical experiments conducted in high dimensions demonstrate its stability and accuracy across various boundary conditions, as well as for both semi-linear and nonlinear PDEs. • Introduce a novel InfSupNet for learning PDE solutions. • Establish the equivalence between the PDE and the inf-sup formulation. • Decompose the total error and estimate the error bounds. • Demonstrate the effectiveness of our method in comprehensive experiments. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the position value as a centrality measure to evaluate the relevance of the edges and players in a network, with the additional advantage that this value integrates the degree measure of each player in it. In fact, in the real world, it is particularly important to consider the natural influence of connections of a player in a network. Its applications were very limited in real-world situations due to the high computational complexity of exactly obtaining this value. With the aim of solving this problem we provide a method, based on sampling theory, to estimate the position value, which is analyzed in terms of the theoretical properties of the resulting estimator. Moreover, we establish specific statistical results for bounding the absolute error in this approximation. It is important to emphasize that this approach allows for obtaining rankings not only of the nodes but also of the edges of the network. To illustrate the advantages and interest of the proposed methodology, as well as the variety of problems that can be analyzed in this framework, we applied it in three very different settings, the suburban train network of Madrid in the year 2000, the Spanish national team in a match against Portugal, and the Zerkani network responsible for the terrorist attacks of Paris (2015) and Brussels (2016). • The position value gives a measure of the relevance of the links and players in a network. • We rank the members of networks using the position value. • Sampling techniques are used to solve the computational drawbacks in these settings. • As an application, we rank the Spanish national team and the Zerkani network. [ABSTRACT FROM AUTHOR]

Many real multi-objective optimization problems with 20-50 decision variables often have only a small number of function evaluations available, because of their heavy time/money burden. Therefore, surrogate models are often utilized as alternatives for expensive function evaluations. However, the approximation error of the surrogate model is inevitable compared to the real function evaluation. The approximation error has a similar impact on the algorithm as noise, i.e., different optimization stages suffer from various impacts. Therefore, the current optimization stage can be indirectly detected via measuring the impact of the noise formulated by the approximation error. In addition, the rising dimension of the search space leads to an increase in the approximation errors of the surrogate models, which poses a huge challenge for existing surrogate-assisted multi-objective evolutionary algorithms. In this work, we propose a stage-adaptive surrogate-assisted multi-objective evolutionary algorithm to solve the medium-scale optimization problems. In the proposed algorithm, the ensemble model consisting of the latest and historical models is used as the surrogate model, on the basis of which a set of potential candidates can be discovered. Then, a stage-adaptive infill sampling strategy selects the most suitable sampling strategy by analyzing the demand of the current optimization stage on convergence, diversity, model accuracy to sample from the candidates. As for the current optimization stage, it is detected by a noise impact indicator, where the approximation errors of surrogate models are formulated as noise. The experimental results on a series of medium-scale expensive test problems demonstrate the superiority of the proposed algorithm over six state-of-the-art compared algorithms. [ABSTRACT FROM AUTHOR]

The core-set approach is a discretization method for Markov state models of complex molecular dynamics. Core sets are disjoint metastable regions in the conformational space, which need to be known prior to the construction of the core-set model. We propose to use density-based cluster algorithms to identify the cores. We compare three different density-based cluster algorithms: the CNN, the DBSCAN, and the Jarvis-Patrick algorithm. While the core-set models based on the CNN and DBSCAN clustering are well-converged, constructing core-set models based on the Jarvis-Patrick clustering cannot be recommended. In a well-converged core-set model, the number of core sets is up to an order of magnitude smaller than the number of states in a conventional Markov state model with comparable approximation error. Moreover, using the density-based clustering one can extend the core-set method to systems which are not strongly metastable. This is important for the practical application of the core-set method because most biologically interesting systems are only marginally metastable. The key point is to perform a hierarchical density-based clustering while monitoring the structure of the metric matrix which appears in the core-set method. We test this approach on a molecular-dynamics simulation of a highly flexible 14-residue peptide. The resulting core-set models have a high spatial resolution and can distinguish between conformationally similar yet chemically different structures, such as register-shifted hairpin structures. [ABSTRACT FROM AUTHOR]

The adoption of energy systems powered by renewable sources requires substantial economic investments. Hence, selecting system components of an appropriate size becomes a critical step, which is significantly influenced by their distinct characteristics. Furthermore, the availability of renewable energy varies over time, and estimating this availability introduces considerable uncertainty. In this paper, we present a technique for the optimal design of hybrid energy systems that accounts for the uncertainty associated with resource estimation. Our method is based on stochastic programming theory and employs a surrogate model to estimate battery lifespan using a feedforward neural network (FFNN). The optimization analysis for system design was conducted using a genetic algorithm (GA) and the poplar optimization algorithm (POA). We assessed the effectiveness of the proposed technique through a hypothetical case study. The introduction of a surrogate model, based on an FFNN, resulted in an approximation error of 9.6 % for cost estimation and 20.6 % for battery lifespan estimation. The probabilistic design indicates an energy system cost that is 25.7 % higher than that obtained using a deterministic approach. Both the GA and POA achieved solutions that likely represent the global optimum. [ABSTRACT FROM AUTHOR]

A neural network control strategy based on the immune algorithm is proposed to address the problem of chattering and trajectory tracking performance degradation of multi-joint manipulators due to unknown backlash-like hysteresis and system uncertainty. The controller is designed based on backstepping sliding mode technology and is suitable for Euler-Lagrange nonlinear systems. Firstly, to deal with the uncertainty problem of the manipulator model, we consider using radial basis function neural networks (RBFNN) to approximate, and on this basis, we add immune algorithms to optimize the RBFNN parameters to improve the approximation ability. Secondly, to compensate for the unknown backlash-like hysteresis error and RBFNN approximation error, an adaptive reaching law is designed to improve the control accuracy and reduce chattering to a certain extent. Finally, taking a 3-DOF manipulator as the research object, based on comparative simulation results, the feasibility and superiority of this control method can be proven. [ABSTRACT FROM AUTHOR]

A numerical method for the fast evaluation of ratcheting in solid-contaminated contact between rollers was developed. The physical problem was simplified with a plane-strain half-infinite body, with a distribution of pressure and tangential stress on the contact surface obtained by considering the presence of evenly spaced solid contaminant particles between the contacting bodies. Such a distribution was obtained by an analytical method, which considered the material of the contacting bodies as linear elastic. The stresses under the contact surfaces were calculated using the Boussinesq-Cerruti model. Subsequently, the plastic strain accumulated in the roller after each application of the contact load was calculated by means of a non-linear kinematic-hardening constitutive law, also considering the effect of wear in removing material layers from the surface. First, the effects of the introduced approximations were evaluated. In particular, the error in calculating the plastic strain using contact load distributions from a theory based on linear elastic behavior was evaluated by comparison with the results of finite element models. Subsequently, the method was used to assess the experimental results of previously published studies on sand-contaminated contact between rail and wheel materials. Despite the approximations and indetermination of some model constants, the method was proven to be able to qualitatively explain many of the damage mechanisms related to the interaction of the contacting bodies with the contaminant particles. • A numerical method for assessment of solid-contaminated contact was developed. • A contact pressure distribution with solid particle contamination was assumed. • The cyclic plastic strain was calculated by a non-linear kinematic-hardening model. • The error due to the approximations was evaluated. • This is a valuable tool for comparative design for ratcheting and wear damage. [ABSTRACT FROM AUTHOR]

Regarding the nonconvexity of the optimal power flow (OPF) problem, power industry typically employs convex approximation approaches to satisfy the computational demand. However, the accuracy performances of the convex OPF models are hard to be predicted in actual operating power systems. To control the maximum effect of the approximation error on power dispatch, it is important to solve and compare the worst-case approximation errors of these OPF models (i.e., the error bounds). In this paper, the error bound is represented by the maximum gap of the optimal solutions of the nonconvex and convex OPF models (i.e., the optimality gap) which is further equivalent to be the Hausdorff distance of the feasible regions of the two OPF models. Subsequently, an optimization model is proposed to generally solve the Hausdorff distance as well as the error bound. Case studies show that the proposed error bound is tight and effective when the power system is not seriously congested. The relaxation rate of the error bounds is less than 5% when congestion rate is less than 10%. It provides theoretical guidance for selecting the convex OPF model under the condition without historical operating data. • The error bound of convex OPF models is generally modeled based on Fenchel duality. • An approach for solving error bounds is proposed from Hausdorff distance calculation. • The accuracy and application of the error bound measurement are numerically presented. [ABSTRACT FROM AUTHOR]

The expressive power of deep neural networks is manifested by their remarkable ability to approximate multivariate functions in a way that appears to overcome the curse of dimensionality. This ability is exemplified by their success in solving high-dimensional problems where traditional numerical solvers fail due to their limitations in accurately representing high-dimensional structures. To provide a theoretical framework for explaining this phenomenon, we analyze the approximation of Hölder functions defined on a d -dimensional smooth manifold M embedded in R D , with d ≪ D , using deep neural networks. We prove that the uniform convergence estimates of the approximation and generalization errors by deep neural networks with ReLU activation functions do not depend on the ambient dimension D of the function but only on its lower manifold dimension d , in a precise sense. Our result improves existing results from the literature where approximation and generalization errors were shown to depend weakly on D. • Deep ReLU neural networks exhibit remarkable approximation properties for multivariate functions. • We prove that the approximation error of manifold-embedded data depends only on the manifold dimension. • Argument relies on preservation of pairwise ambient distances on the manifold. • Our result improves existing results where the approximation error depends weakly on the ambient dimension. • Our result implies an improved estimate of the generalization error. [ABSTRACT FROM AUTHOR]

Invertible neural networks (INNs) represent an important class of deep neural network architectures that have been widely used in applications. The universal approximation properties of INNs have been established recently. However, the approximation rate of INNs is largely missing. In this work, we provide an analysis of the capacity of a class of coupling-based INNs to approximate bi-Lipschitz continuous mappings on a compact domain, and the result shows that it can well approximate both forward and inverse maps simultaneously. Furthermore, we develop an approach for approximating bi-Lipschitz maps on infinite-dimensional spaces that simultaneously approximate the forward and inverse maps, by combining model reduction with principal component analysis and INNs for approximating the reduced map, and we analyze the overall approximation error of the approach. Preliminary numerical results show the feasibility of the approach for approximating the solution operator for parameterized second-order elliptic problems. [ABSTRACT FROM AUTHOR]

High-order mesh optimization has many goals, such as improving smoothness, reducing approximation error, and improving mesh quality. The previous methods do not optimize these objectives together, resulting in suboptimal results. To this end, we propose a multi-objective optimization method for high-order meshes. Central to our algorithm is using the multi-objective genetic algorithm (MOGA) to adapt to the multiple optimization objectives. Specifically, we optimize each control point one by one, where the MOGA is applied. We demonstrate the feasibility and effectiveness of our method over various models. Compared to other state-of-the-art methods, our method achieves a favorable trade-off between multiple objectives. • We propose a multi-objective optimization method for high-order tetrahedral meshes. • Employ the multi-objective genetic algorithm to relocate control points to achieve a favorable trade-off among objectives. • Select the best individual via the weighted TOPSIS method, with weights determined by combining multi-objective problem. [ABSTRACT FROM AUTHOR]

The classical Peano–Sard theorem is a very useful result in approximation theory, bounding the errors of approximations that are exact on sets of polynomials. A fractional version was developed by Diethelm for fractional derivatives of Riemann–Liouville type, which we here extend to fractional derivatives of Caputo type. We indicate some applications to quadrature and interpolation formulae. These results will be useful in the approximate solution of fractional differential equations involving Caputo-type operators, which are often said to be more natural for applications. [ABSTRACT FROM AUTHOR]

In this paper the interpolating rational functions introduced by Floater and Hormann are generalized leading to a whole new family of rational functions depending on γ , an additional positive integer parameter. For γ = 1 , the original Floater–Hormann interpolants are obtained. When γ > 1 we prove that the new rational functions share a lot of the nice properties of the original Floater–Hormann functions. Indeed, for any configuration of nodes in a compact interval, they have no real poles, interpolate the given data, preserve the polynomials up to a certain fixed degree, and have a barycentric-type representation. Moreover, we estimate the associated Lebesgue constants in terms of the minimum (h ∗) and maximum (h) distance between two consecutive nodes. It turns out that, in contrast to the original Floater–Hormann interpolants, for all γ > 1 we get uniformly bounded Lebesgue constants in the case of equidistant and quasi-equidistant nodes configurations (i.e., when h ∼ h ∗ ). For such configurations, as the number of nodes tends to infinity, we prove that the new interpolants (γ > 1) uniformly converge to the interpolated function f , for any continuous function f and all γ > 1. The same is not ensured by the original FH interpolants (γ = 1). Moreover, we provide uniform and pointwise estimates of the approximation error for functions having different degrees of smoothness. Numerical experiments illustrate the theoretical results and show a better error profile for less smooth functions compared to the original Floater–Hormann interpolants. [ABSTRACT FROM AUTHOR]

It is increasingly realized that taking stochastic effects into account is important in order to study biological cells. However, the corresponding mathematical formulation, the chemical master equation (CME), suffers from the curse of dimensionality and thus solving it directly is not feasible for most realistic problems. In this paper we propose a dynamical low-rank algorithm for the CME that reduces the dimensionality of the problem by dividing the reaction network into partitions. Only reactions that cross partitions are subject to an approximation error (everything else is computed exactly). This approach, compared to the commonly used stochastic simulation algorithm (SSA, a Monte Carlo method), has the advantage that it is completely noise-free. This is particularly important if one is interested in resolving the tails of the probability distribution. We show that in some cases (e.g. for the lambda phage) the proposed method can drastically reduce memory consumption and run time and provide better accuracy than SSA. • We propose a low-rank algorithm that does not rely on single orbital basis functions. • The proposed approach is better suited for biological/chemical reaction networks. • We demonstrate better accuracy and improved efficiency compared to SSA for some problems. • The proposed complexity reduction algorithm is noise free (in contrast to SSA). • We demonstrate that our algorithm accurately resolves the tails of the distribution (in contrast to SSA). [ABSTRACT FROM AUTHOR]

In this article, a pseudo-model-based iterative learning control (ILC) is exploited for multi-phase batch processes which can be described as a nonlinear switched system with unknown functions and identical states in different phases. The nonlinear switched system is converted into a linear model whose system parameter matrix is approximated by minimizing the discrepancy from the real system output increment to the approximated system output increment. A data-driven ILC is constructed in an interactive form with system parameter matrix approximate algorithm. Meanwhile, the signs of the diagonal elements of system lower triangular parameter matrix are introduced into the construction of control input law. Theoretical analysis shows that the pseudo-model-based ILC (PM-ILC) concept can be extended to multi-phase batch processes with non-identical states in different phases. Furthermore, the approximation error of the system parameters matrix is bounded and the proposed PM-ILC is robust if the parameter is appropriately chosen. Simulation results illustrate the effectiveness and practicability of the proposed PM-ILC. [ABSTRACT FROM AUTHOR]

We study the approximation of integrals of the form ∫ D f (x ⊤ A) d μ (x) , where the measure μ is of product form and A is a matrix, by quasi-Monte Carlo (QMC) rules N − 1 ∑ k = 0 N − 1 f (x k ⊤ A). We are interested in cases where the main computational cost in the approximation arises from calculating the products x k ⊤ A. We design QMC rules for which the computation of x k ⊤ A , k = 0 , 1 , ... , N − 1 , can be done in a fast way, and for which the approximation error of the QMC rule is similar to the standard QMC error. We do not require that the matrix A has any particular structure. Problems of this form arise in some important applications in statistics and uncertainty quantification. For instance, this approach can be used when approximating the expected value of some function with a multivariate normal random variable with some given covariance matrix, or when approximating the expected value of the solution of a PDE with random coefficients. The speed-up of the computation time of our approach is sometimes better and sometimes worse than the fast QMC matrix–vector product from [Josef Dick, Frances Y. Kuo, Quoc T. Le Gia, and Christoph Schwab, Fast QMC Matrix-Vector Multiplication, SIAM J. Sci. Comput. 37 (2015), no. 3, A1436–A1450]. As in that paper, our approach applies to lattice point sets and polynomial lattice point sets, but also applies to digital nets (we are currently not aware of any approach which allows one to apply the fast QMC matrix–vector paper from the aforementioned paper of Dick, Kuo, Le Gia, and Schwab to digital nets). The method in this paper does not make use of the fast Fourier transform, instead we use repeated values in the quadrature points to derive a significant reduction in the computation time. Such a situation naturally arises from the reduced CBC construction of lattice rules and polynomial lattice rules. The reduced CBC construction has been shown to reduce the computation time for the CBC construction. Here we show that it can additionally be used to also reduce the computation time of the underlying QMC rule. One advantage of the present approach is that it can be combined with random (digital) shifts, whereas this does not apply to the fast QMC matrix–vector product from the earlier paper of Dick, Kuo, Le Gia, and Schwab. [ABSTRACT FROM AUTHOR]

• A simple implementation of aperture modulation quantitative differential phase contrast (amqDPC) imaging and the phase reconstruction (PR) process were proposed. • Complete theoretical model was established, and the result of simulations showed that the partial spectrum center pass (PSCP) demonstrated higher PR accuracy than the full spectral center pass (FSCP). • The PR process, which could eliminate streak background error and compensate weak phase approximation error, was introduced in detail. • Experiments confirmed that the PSCP amqDPC imaging and the PR process could achieve an isotropic lateral resolution of 0.87 μm with a 20 × 0.4NA objective and higher axial accuracy. • This method can simply and steadily implement amqDPC imaging by only 2 axisymmetric images in the phase range of [0, π], which could provide another efficient method for QPI of transparent samples and biological tissues and cells. As one of the quantitative phase imaging methods, aperture modulation quantitative differential phase contrast (amqDPC) imaging has isotropic lateral resolution, being widely used in many fields such as biological cell morphology detection. However, the devices required to implement it are poorly controlled and very expensive. In addition, due to the weak phase approximation, there is a large axial phase error, and the full spectrum center pass during aperture modulation will cause distortion. Therefore, a method that enables simple and low-cost implementation of amqDPC, high axial phase accuracy, and isotopic lateral resolution is reported. Through simulations and experimental verifications, axial accuracy is greatly improved in the phase range of [0, π], and an isotropic lateral resolution of 870 nm can be achieved. In addition, experiments on transparent glass circular step sample and mouse kidney tissue slice also prove the practicability of this method. According to this study, researchers can accurately and simply measure the axial height of a slightly thick sample, which is achieved by capturing two intensity images to reconstruct a quantitative phase. [ABSTRACT FROM AUTHOR]

Despite the widespread success of deep learning in various applications, neural network theory has been lagging behind. The choice of the activation function plays a critical role in the expressivity of a neural network but for reasons that are not yet fully understood. While the rectified linear unit (ReLU) is currently one of the most popular activation functions, ReLU squared has only recently been empirically shown to be pivotal in producing consistently superior results for state-of-the-art deep learning tasks (So et al., 2021). To analyze the expressivity of neural networks with ReLU powers, we employ the novel framework of Gribonval et al. (2022) based on the classical concept of approximation spaces. We consider the class of functions for which the approximation error decays at a sufficiently fast rate as network complexity, measured by the number of weights, increases. We show that when approximating sufficiently smooth functions that cannot be represented by sufficiently low-degree polynomials, networks with ReLU powers need less depth than those with ReLU. Moreover, if they have the same depth, networks with ReLU powers can have potentially faster approximation rates. Lastly, our computational experiments on approximating the Rastrigin and Ackley functions with deep neural networks showed that ReLU squared and ReLU cubed networks consistently outperform ReLU networks. • ReLU powers need less depth than ReLU in approximating sufficiently smooth functions • For the same depth, ReLU powers can potentially learn faster than those with ReLU • ReLU powers outperform ReLU in approximating Rastrigin and Ackley functions [ABSTRACT FROM AUTHOR]