Approximation Theorem Research Articles

A convolution inequality, yielding a sharper Berry–Esseen theorem for summands Zolotarev-close to normal

The classical Berry–Esseen error bound, for the normal approximation to the law of a sum of independent and identically distributed random variables, is here improved by replacing the standardised third absolute moment with a weak norm distance to normality, using Zolotarev’s ζ \zeta norms. We thus sharpen and simplify two results of Ul’yanov (1976) and of Senatov (1998), each of them previously optimal, in the line of research initiated by Zolotarev (1965) and Paulauskas (1969). Our proof is based on a seemingly incomparable normal approximation theorem of Zolotarev (1986), combined with our main technical result: The Kolmogorov distance (supremum norm of difference of distribution functions) between a convolution of two laws and a convolution of two Lipschitz laws is bounded homogeneously of degree 1 in the pair of the Kantorovich distances (often called Wasserstein distances, the L 1 ^1 norms of differences of distribution functions) of the corresponding factors, and also in the pair of the Lipschitz constants. Side results include a short introduction to ζ \zeta norms on the real line, simpler inequalities for various probability distances, slight improvements of the theorem of Zolotarev (1986) and of a lower bound theorem of Bobkov, Chistyakov and Götze (2012), an application to sampling from finite populations, auxiliary results on rounding and on winsorisation, and computations of a few examples. The introductory section in particular is aimed at analysts in general rather than specialists in probability approximations.

Some approximation theorems in Fourier-Jacobi analysis and Boas type results for Nikol’skii spaces

Some approximation theorems in Fourier-Jacobi analysis and Boas type results for Nikol’skii spaces

Approximation by Schurer Type λ-Bernstein–Bézier Basis Function Enhanced by Shifted Knots Properties

In this article, a novel Schurer form of λ-Bernstein operators augmented by Bézier basis functions is presented by utilizing the features of shifted knots. The shifted knots form of Bernstein operators and the Schurer form of the Bézier basis function are used in this article, then, new operators, the Schurer type λ-Bernstein shifted knots operators are constructed in terms of the Bézier basis function. First, the test functions are calculated and the central moments for these operators are obtained. Then, Korovkin’s type approximation properties are studied by the use of a modulus of continuity of orders one and two. Finally, the convergence theorems for these new operators are obtained by using Peetre’s K-functional and Lipschitz continuous functions. In the end, some direct approximation theorems are also obtained.

Equidistribution of non-uniformly stretching translates of shrinking smooth curves and weighted Dirichlet approximation

AbstractConsider the action of $$a_t=\textrm{diag}(e^{nt},e^{-r_1(t)},\ldots ,e^{-r_n(t)})\in \textrm{SL}(n+1,{\mathbb {R}})$$ a t = diag ( e nt , e - r 1 ( t ) , … , e - r n ( t ) ) ∈ SL ( n + 1 , R ) , where $$r_i(t)\rightarrow \infty $$ r i ( t ) → ∞ for each i, on the space of unimodular lattices in $${\mathbb {R}}^{n+1}$$ R n + 1 . We show that $$a_t$$ a t -translates of segments of size $$e^{-t}$$ e - t about all except countably many points of a nondegenerate smooth horospherical curve get equidistributed in the space as $$t\rightarrow \infty $$ t → ∞ . This result implies that the weighted Dirichlet approximation theorem cannot be improved for almost all points on any nondegenerate $$C^{2n}$$ C 2 n curve in $${\mathbb {R}}^n$$ R n . These results extend the corresponding results for translates of fixed pieces of analytic curves due to Shah (2010) as well as those for uniform translates of shrinking curves due to Shah and Yang (2023), and answer some questions inspired by the work of Davenport and Schmidt (1969) and Kleinbock and Weiss (2008).

Herglotz's representation and Carathéodory's approximation

AbstractHerglotz's representation of holomorphic functions with positive real part and Carathéodory's theorem on approximation by inner functions are two well‐known classical results in the theory of holomorphic functions on the unit disc. We show that they are equivalent. On a multi‐connected domain , a version of Heglotz's representation is known. Carathéodory's approximation was not known. We formulate and prove it and then show that it is equivalent to the known form of Herglotz's representation. Additionally, it also enables us to prove a new Heglotz's representation in the style of Korányi and Pukánszky. Of particular interest is the fact that the scaling technique of the disc is replaced by Carathéodory's approximation theorem while proving this new form of Herglotz's representation. Carathéodory's approximation theorem is also proved for operator‐valued functions on a multi‐connected domain.

Best approximation theorems via relatively u-continuous and Jaggi nonexpansive mappings

In this paper, we introduced a new type of cyclic (noncyclic) mappings, called Jaggi relatively nonexpansive mappings and use to investigate the existence of best proximity points (pairs) in the framework of reflexive (strictly convex) Banach spaces by using a geometric concept of proximal normal structure. We also present a best proximity version of Schauder’s fixed point problem for non-self mappings. Finally, we prove the existence of a common best proximity point for an arbitrary family of cyclic relatively u-continuous mappings which are compact and then we obtain a generalization of Markov-Kakutani’s theorem.

Neural network representation of quantum systems

Abstract It has been proposed that random wide neural networks near Gaussian process are quantum field theories around Gaussian fixed points. In this paper, we provide a novel map with which a wide class of quantum mechanical systems can be cast into the form of a neural network with a statistical summation over network parameters. Our simple idea is to use the universal approximation theorem of neural networks to generate arbitrary paths in the Feynman’s path integral. The map can be applied to interacting quantum systems / field theories, even away from the Gaussian limit. Our findings bring machine learning closer to the quantum world.

Long-term Traffic Flow Prediction using Stochastic Configuration Networks for Smart Cities

Accurate predictions of traffic flow are very meaningful to city managers. With such information, traffic systems can better coordinate traffic signals and reduce congestion. By understanding traffic patterns, navigation systems can provide real-time routing suggestions that avoid traffic jams, save time, and reduce fuel consumption. However, traffic flow will be interfered with by multiple factors such as collection time and place. In this paper, we propose to use stochastic configuration networks (SCNs) to predict the traffic flow. The network is trained through stepwise construction, and the network parameters are effectively optimized based on the approximation theorem and convergence analysis optimization mechanism. The proposed network automatically adjusts its structure according to the complexity of traffic flow to better adapt to the complex non-linearity of traffic flow. We observed that the proposed model achieves better prediction performance overall and greater flexibility in the length of the prediction period compared to the benchmarks using the Guangzhou urban traffic flow dataset. It's worth noting that SCNs consistently outperform other models across different prediction intervals. They yield RMSE improvements of up to 10.73% for 10-minute predictions, 5.02% for 30-minute predictions, and 11.21% for 60-minute predictions compared to the least effective models. The R-value also exhibits steady enhancement, increasing up to 0.78%, 0.65%, and 2.33% for 10-minute, 30-minute, and 60-minute predictions, respectively. These notable advancements, combined with the model's computational efficiency, especially in short-term predictions, underscore the effectiveness and practicality of SCNs in traffic flow prediction tasks.

Interpolation for neural-network operators activated with a generalized logistic-type function

This paper defines a family of neural-network interpolation operators. The first derivative of generalized logistic-type functions is considered as a density function. Using the first-order uniform approximation theorem for continuous functions defined on the finite intervals, the interpolation properties of these operators are presented. A Kantorovich-type variant of the operators Fna,ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$F_{n}^{a,\\varepsilon} $\\end{document} is also introduced. The approximation of Kantorovich-type operators in LP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L_{P}$\\end{document} spaces with 1≤p≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1 \\leq p\\leq \\infty $\\end{document} is studied. Further, different combinations of the parameters of our generalized logistic-type activation function θs,a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\ heta _{s, a}$\\end{document} are examined to see which parameter values might give us a more efficient activation function. By choosing suitable parameters for the operator Fna,ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$F_{n}^{a,\\varepsilon} $\\end{document} and the Kantorovich variant of the operator Fna,ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$F_{n}^{a,\\varepsilon} $\\end{document}, the approximation of various function examples is studied.

Physics-Informed Holomorphic Neural Networks (PIHNNs): Solving 2D linear elasticity problems

We propose physics-informed holomorphic neural networks (PIHNNs) as a method to solve boundary value problems where the solution can be represented via holomorphic functions. Specifically, we consider the case of plane linear elasticity and, by leveraging the Kolosov–Muskhelishvili representation of the solution in terms of holomorphic potentials, we train a complex-valued neural network to fulfill stress and displacement boundary conditions while automatically satisfying the governing equations. This is achieved by designing the network to return only approximations that inherently satisfy the Cauchy-Riemann conditions through specific choices of layers and activation functions. To ensure generality, we provide a universal approximation theorem guaranteeing that, under basic assumptions, the proposed holomorphic neural networks can approximate any holomorphic function. Furthermore, we suggest a new tailored weight initialization technique to mitigate the issue of vanishing/exploding gradients. Compared to the standard PINN approach, noteworthy benefits of the proposed method for the linear elasticity problem include a more efficient training, as evaluations are needed solely on the boundary of the domain, lower memory requirements, due to the reduced number of training points, and C∞ regularity of the learned solution. Several benchmark examples are used to verify the correctness of the obtained PIHNN approximations, the substantial benefits over traditional PINNs, and the possibility to deal with non-trivial, multiply-connected geometries via a domain-decomposition strategy.

Inverse Theorem for Approximation on Subsets of a Domain with Cusps

Inverse Theorem for Approximation on Subsets of a Domain with Cusps

Assuring optimality in surrogate‐based optimization: A novel theorem and its practical implementation in pressure swing adsorption optimization

AbstractSurrogate‐based optimization has gained significant traction in several engineering fields, given its ability to handle complex systems without significant computational burdens. However, surrogate models are approximations and may have limitations, including the possibility of artificial minima. The main contribution of this work is the derivation of a robustness test that guarantees the optimality of surrogate‐based optimization. The derivation of this metric is based on the universal approximation theorem. The full framework proposed in this work is also composed by a sampling sizing methodology to randomly select samples within a feasible operating region (FOR) resulting from the optimization population, reducing the computational cost of the analysis and avoiding biases in the robustness calculation. The applicability and importance of this methodology are demonstrated through a case study of a complex chemical process—a pressure swing adsorption (PSA) unit—which presents a high computational cost to solve optimization problems. The results highlight the need and importance of evaluating the optimality of surrogate‐based optimization schemes.

Douglas-Rudin approximation theorem for operator-valued functions on the unit ball of [formula omitted]

Douglas and Rudin proved that any unimodular function on the unit circle T can be uniformly approximated by quotients of inner functions. We extend this result to the operator-valued unimodular functions defined on the boundary of the open unit ball of Cd. Our proof technique combines the spectral theorem for unitary operators with the Douglas-Rudin theorem in the scalar case to bootstrap the result to the operator-valued case. This yields a new proof and a significant generalization of Barclay's result (2009) [4] on the approximation of matrix-valued unimodular functions on T.

New Kantorovich-type Szász–Mirakjan Operators

In this paper, we present a Kantorovich-type Szász–Mirakjan operators. Initially, we establish the recurrence relationship for the moments of these operators and provide the central moments up to the fourth degree. Subsequently, we analyze the local approximation properties of these operators using Peetre’s K-function. We investigate the rate of convergence, by utilizing the ordinary modulus of continuity and Lipschitz-type maximal functions. Additionally, we prove weighted approximation theorems and Voronoskaja-type theorems specific to these new operators. Following this, we introduce bivariate extension of these operators and investigate some approximation properties. Lastly, we include several numerical illustrative examples.

Prediction of Wind Turbine Gearbox Oil Temperature Based on Stochastic Differential Equation Modeling

Aiming at the problem of high failure rate and inconvenient maintenance of wind turbine gearboxes, this paper establishes a stochastic differential equation model that can be used to fit the change of gearbox oil temperature and adopts an iterative computational method and Markov-based modified optimization to fit the prediction sequence in order to realize the accurate prediction of gearbox oil temperature. The model divides the oil temperature change of the gearbox into two parts, internal aging and external random perturbation, adopts the approximation theorem to establish the internal aging model, and uses Brownian motion to simulate the external random perturbation. The model parameters were calculated by the Newton–Raphson iterative method based on the gearbox oil temperature monitoring data. Iterative calculations and Markov-based corrections were performed on the model prediction data. The gearbox oil temperature variations were simulated in MATLAB, and the fitting and testing errors were calculated before and after the iterations. By comparing the fitting and testing errors with the ordinary differential equations and the stochastic differential equations before iteration, the iterated model can better reflect the gear oil temperature trend and predict the oil temperature at a specific time. The accuracy of the iterated model in terms of fitting and prediction is important for the development of preventive maintenance.

Direct and inverse approximation theorems for functions defined in Damek–Ricci spaces

UDC 517.5 We introduce the notion of k th modulus of smoothness and establish the direct and inverse theorems in terms of the quantities E s ( f ) and the moduli of smoothness generated by the spherical mean operator defined on the L 2 -space for the Damek–Ricci spaces. These theorems are analogous to the well-known theorems of Jackson and Bernstein. We also consider some problems related to the constructive characteristics of functional classes defined by the majorants of the moduli of smoothness of their elements.

On a nonlinear general eigenvalue problem in Musielak–Orlicz spaces

Abstract In this paper, we concern the existence result of the following general eigenvalue problem: { 𝒜 ⁢ ( u ) = λ ⁢ ℬ ⁢ ( u ) in ⁢ Ω , D α ⁢ ( u ) = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}\mathcal{A}(u)&\displaystyle={\lambda}% \mathcal{B}(u)&&\displaystyle\phantom{}\text{in }{\Omega},\\ \displaystyle D^{\alpha}(u)&\displaystyle=0&&\displaystyle\phantom{}\text{on }% {\partial\Omega},\end{aligned}\right. in an arbitrary Musielak–Orlicz spaces, where 𝒜 {\mathcal{A}} and ℬ {\mathcal{B}} are quasilinear operators in divergence form of order 2 ⁢ n {2n} and 2 ⁢ ( n - 1 ) {2(n-1)} , respectively. The main assumptions in this case are that 𝒜 {\mathcal{A}} and ℬ {\mathcal{B}} are potential operators with 𝒜 {\mathcal{A}} being elliptic and monotone. In this study, we intentionally avoid imposing constraints on the growth of a generalized N-function, including the Δ 2 {\Delta_{2}} -condition for both the generalized N-function and its conjugate. Consequently, this necessitates the formulation of the approximation theorem and the extensive utilization of modular convergence concepts.

The research on Adaptive Fuzzy Tracking Supervisory Control in the control system of average coolant temperature of lead–bismuth-cooled reactor under multiple operating conditions in the marine environments

Lead–bismuth-cooled reactor nuclear power equipment is severely affected by marine environments, causing operational attitude changes such as heeling and rolling. Conventional controllers cannot track set points rapidly. Therefore, this paper proposes an Adaptive Fuzzy Tracking Supervisory Control method. Firstly, the reactor mathematical model based on fuzzy basis functions is obtained through fuzzy mathematics. Secondly, a coolant average temperature tracking controller is designed on Lyapunov stability theory. To ensure the system returns to stable domain, supervisory compensatory control algorithm is developed. Next, the parameters of fuzzy model are adjusted using Fuzzy universal approximation theorem. By introducing discrepancies between actual system and fuzzy model outputs, parameter adaptive law is designed through Lyapunov theorem. This enables real-time parameter adjustment for fuzzy model and fuzzy tracking supervisory controller, enhancing load tracking performance of coolant’s average temperature. Finally, simulation experiments demonstrate that adaptive fuzzy tracking supervisory controller has stronger adaptability to multiple operating conditions under marine environments.

Interplay between depth and width for interpolation in neural ODEs

Neural ordinary differential equations have emerged as a natural tool for supervised learning from a control perspective, yet a complete understanding of the role played by their architecture remains elusive. In this work, we examine the interplay between the width p and the number of transitions between layers L (corresponding to a depth of L+1). Specifically, we construct explicit controls interpolating either a finite dataset D, comprising N pairs of points in Rd, or two probability measures within a Wasserstein error margin ɛ>0. Our findings reveal a balancing trade-off between p and L, with L scaling as 1+O(N/p) for data interpolation, and as 1+Op−1+(1+p)−1ɛ−d for measures.In the high-dimensional and wide setting where d,p>N, our result can be refined to achieve L=0. This naturally raises the problem of data interpolation in the autonomous regime, characterized by L=0. We adopt two alternative approaches: either controlling in a probabilistic sense, or by relaxing the target condition. In the first case, when p=N we develop an inductive control strategy based on a separability assumption whose probability increases with d. In the second one, we establish an explicit error decay rate with respect to p which results from applying a universal approximation theorem to a custom-built Lipschitz vector field interpolating D.

Universal approximation theorem for vector- and hypercomplex-valued neural networks

The universal approximation theorem states that a neural network with one hidden layer can approximate continuous functions on compact sets with any desired precision. This theorem supports using neural networks for various applications, including regression and classification tasks. Furthermore, it is valid for real-valued neural networks and some hypercomplex-valued neural networks such as complex-, quaternion-, tessarine-, and Clifford-valued neural networks. However, hypercomplex-valued neural networks are a type of vector-valued neural network defined on an algebra with additional algebraic or geometric properties. This paper extends the universal approximation theorem for a wide range of vector-valued neural networks, including hypercomplex-valued models as particular instances. Precisely, we introduce the concept of non-degenerate algebra and state the universal approximation theorem for neural networks defined on such algebras.