Approximation Theory Research Articles

Interactive mechanism of bending and torsion and springback prediction method for spatial tubes

Spatial tubes, known for their attributes of lightweight, large section modulus, and high strength, find extensive applications across various industries. Nevertheless, in the course of forming spatial tubes, springback poses a significant challenge in enhancing the forming quality. In this paper, the bound approximations for the combined bending and twisting of circular tubes employing the power-hardening material model are deduced innovatively. Drawing upon the boundary approximation theory, the interactive mechanism of bending and twisting is meticulously examined. The forming principles of the four-axis free-bending (FFB) are expounded. Concurrently, an innovative method for predicting springback, based on the established bound approximations within the framework of total plasticity theory, is introduced, which holds significant importance for addressing combined bending and torsion issues in engineering applications. Noteworthy findings reveal that the interaction between bending and twisting exhibits a predominant sensitivity to the hardening exponent n and the higher n the material has, the less pronounced the interaction will be. The augmentation of torsion contributes to mitigating the springback resulting from bending, and conversely, a similar effect is observed with the influence of bending on torsion. These findings contribute significantly to the understanding of combined bending and twisting problems. The proportionality coefficients of bending and twisting, denoted as λM and λT, are defined and investigated via comprehensive finite element (FE) simulations. Furthermore, this method of springback prediction is corroborated by FE simulations as well as the FFB experiments.

Low-rank solutions to the stochastic Helmholtz equation

In this paper, we consider low-rank approximations for the solutions to the stochastic Helmholtz equation with random coefficients. A Stochastic Galerkin finite element method is used for the discretization of the Helmholtz problem. Existence theory for the low-rank approximation is established when the system matrix is indefinite. The low-rank algorithm does not require the construction of a large system matrix which results in an advantage in terms of CPU time and storage. Numerical results show that, when the operations in a low-rank method are performed efficiently, it is possible to obtain an advantage in terms of storage and CPU time compared to computations in full rank. We also propose a general approach to implement a preconditioner using the low-rank format efficiently.

Approximation of functions from Korobov spaces by shallow neural networks

In this paper, we consider the problem of approximating functions from a Korobov space on [−1,1]d by ReLU shallow neural networks and present a rate O(m−25(1+2d)log⁡m) of uniform approximation by networks of m hidden neurons. This is achieved by combining a novel Fourier analysis approach and a probability argument. We apply our approximation theory to a learning algorithm for regression based on ReLU shallow neural networks and derive learning rates of order O(N−4(d+2)9d+8log⁡N) for the excess generalization error with the sample size N when the regression function lies in the Korobov space.

Effects of errors-in-variables on the internal and external reliability measures

The reliability theory has been an important element of the classical geodetic adjustment theory and methods in the linear Gauss-Markov model. Although errors-in-variables (EIV) models have been intensively investigated, little has been done about reliability theory for EIV models. This paper first investigates the effect of a random coefficient matrix A on the conventional geodetic reliability measures as if the coefficient matrix were deterministic. The effects of such geodetic internal and external reliability measures due to the randomness of the coefficient matrix are worked out, which are shown to depend not only on the noise level of the random elements of A but also on the values of parameters. An alternative, linear approximate reliability theory is accordingly developed for use in EIV models. Both the EIV-affected reliability measures and the corresponding linear approximate measures fully account for the random errors of both the coefficient matrix and the observations, though formulated in a slightly different way. Numerical experiments have been carried to demonstrate the effects of errors-in-variables on reliability measures and compared with the conventional Baarda's reliability measures. The simulations have confirmed our theoretical results that the EIV-reliability measures depend on both the noise level of A and the parameter values. The larger the noise level of A, the larger the EIV-affected internal and external reliability measures; the larger the parameters, the larger the EIV-affected internal and external reliability measures.

Distributed entropy-regularized multi-agent reinforcement learning with policy consensus

Sample efficiency is a limiting factor for existing distributed multi-agent reinforcement learning (MARL) algorithms over networked multi-agent systems. In this paper, the sample efficiency problem is tackled by formally incorporating the entropy regularization into the distributed MARL algorithm design. Firstly, a new entropy-regularized MARL problem is formulated under the model of networked multi-agent Markov decision processes with observation-based policies and homogeneous agents, where the policy parameter sharing among the agents provably preserves the optimality. Secondly, an on-policy distributed actor–critic algorithm is proposed, where each agent shares its parameters of both the critic and actor for consensus update. Then, the convergence analysis of the proposed algorithm is provided based on the stochastic approximation theory under the assumption of linear function approximation of the critic. Furthermore, a practical off-policy version of the proposed algorithm is developed which possesses scalability, data efficiency and learning stability. Finally, the proposed distributed algorithm is compared against the solid baselines including two classic centralized training algorithms in the multi-agent particle environment, whose learning performance is empirically demonstrated through extensive simulation experiments.

Joint Service Caching, Resource Allocation and Task Offloading for MEC-Based Networks: A Multi-Layer Optimization Approach

To provide reliable and elastic Multi-access edge computing services, one feasible solution is to federate geographically proximate edge servers to form a logically centralized resource pool. Optimization of such systems, however, becomes challenging. In this paper, we study the problem of maximizing users' QoE in a MEC-based network, through jointly optimizing service caching, resource allocation and task offloading decisions. We formulate a mixed-integer nonlinear programming (MINLP) problem for the task and establish its NP-hardness. To tackle it efficiently, we propose a novel two-stage algorithmic solution based on approximation and decomposition theory. The proposed algorithm achieves high system performance while at the same time, ensures all constraints from different layers are satisfied. Meanwhile, the structure of the algorithm also fits the multi-layer optimizing feature, making it suitable to be implemented at different layers. In addition, we propose a distributed and online version of our mechanism with very limited information exchange between MEC servers, and further demonstrate how the cost of service switches from real MEC systems can be incorporated into our framework. We evaluate our mechanisms through simulations with both synthetic and real-world traces, and results indicate they are effective as compared to representative baseline algorithms.

Propagation of compression solitary waves on tensegrity-like lattices made of truncated octahedrons

This work studies the properties of compression solitary waves propagating through one-dimensional mass–spring lattices, whose unit cells show an interaction law mimicking that of a truncated tensegrity octahedron. Analytic methods based on the Weierstrass criterion for compact solitary waves, and the long wave approximation theory are employed to demonstrate that the analyzed systems support compression solitary waves in a suitable range of wave speeds. These systems exhibit an initially soft response, which progressively turns into an elastically hardening behavior ending with a linear branch in the pre-buckling regime of the units cells. Explicit formulae are given for the lower and upper bounds of the wave speed interval that produce compact solitary waves, together with analytic laws of the shape of the traveling wave. Numerical simulations provide a validation of the presented theory.

On Conic Equations Under Bernstein Operators

One of the most important problems in approximation theory in mathematical analysis is the determination of sequences of polynomials that converge to functions and have the same geometric properties. This type of approximation is called the shape-preserving approximation. These types of problems are usually handled depending on the convexity of the functions, the degree of smoothness depending on the order of differentiability, or whether it satisfies a functional equation. The problem addressed in this paper belongs to the third class. A quadratic bivariate algebraic equation denotes geometrically some well-known shapes such as circles, ellipses, hyperbolas and parabolas. Such equations are known as conic equations. In this study, it is investigated whether conic equations transform into a conic equation under bivariate Bernstein polynomials, and if so, which conic equation it transforms into.

Neural Network Approximation for Pessimistic Offline Reinforcement Learning

Deep reinforcement learning (RL) has shown remarkable success in specific offline decision-making scenarios, yet its theoretical guarantees are still under development. Existing works on offline RL theory primarily emphasize a few trivial settings, such as linear MDP or general function approximation with strong assumptions and independent data, which lack guidance for practical use. The coupling of deep learning and Bellman residuals makes this problem challenging, in addition to the difficulty of data dependence. In this paper, we establish a non-asymptotic estimation error of pessimistic offline RL using general neural network approximation with C-mixing data regarding the structure of networks, the dimension of datasets, and the concentrability of data coverage, under mild assumptions. Our result shows that the estimation error consists of two parts: the first converges to zero at a desired rate on the sample size with partially controllable concentrability, and the second becomes negligible if the residual constraint is tight. This result demonstrates the explicit efficiency of deep adversarial offline RL frameworks. We utilize the empirical process tool for C-mixing sequences and the neural network approximation theory for the Holder class to achieve this. We also develop methods to bound the Bellman estimation error caused by function approximation with empirical Bellman constraint perturbations. Additionally, we present a result that lessens the curse of dimensionality using data with low intrinsic dimensionality and function classes with low complexity. Our estimation provides valuable insights into the development of deep offline RL and guidance for algorithm model design.

Applications of lubrication approximation theory in the analysis of the roll‐coating using a tangent hyperbolic fluid model

AbstractIn this study, the process of roll‐over‐web coating for a tangent hyperbolic fluid is theoretically examined. Fundamental laws of fluid mechanics are used to set up the flow equations and then simplified through lubrication approximation theory (LAT). The governing differential equations are solved with the help of the perturbation method in conjunction with the Runge–Kutta method. The outcomes of the study are presented through the graphs and tables. Study shows that enhancing the Weissenberg number upsurges the velocity and pressure while decreasing the pressure gradient. It is also noted that force shows an increasing trend while power is decreasing for bigger values . Results further depict that for a bigger value of the power law index , pressure gradient, pressure, force, and power decrease, and the coating thickness increases. It is worth mentioning here that for the shear thinning case, the tangent hyperbolic model predicts less pressure in the nip region when compared to the Newtonian model.

First-order localization and quantum phase transition induced by quasicrystal imaginary domain

The non-Hermitian extension of quasicrystals provides a highly tunable system for exploring novel material phases. While extended-localized phase transitions have been observed in one dimension, quantum phase transitions in higher dimensions and various system sizes remain unexplored. Here, we show the discovery of a new critical phase and first-order structural transition induced by imaginary zeros in the two-dimensional Haldane model with a quasicrystal potential on the upper boundary. Initially, we illustrate a phase diagram that evolves with the amplitude and phase of the quasicrystal potential, which is divided into three distinct phases by two critical boundaries: phase (I) with extended wave functions, parity-time-restore phase (II) with localized wave functions, and a critical phase (III) with multifunctional wave functions. To characterize the wave functions in these distinct phases, we introduce a low-energy approximation theory and an effective two-chain model. Additionally, we uncover a first-order quantum phase transition induced by quasicrystal domains. As we increase the size of the potential boundary, we observe the partitioning of the critical phase into regions in proportion to the growing number of quasicrystal domains. Importantly, these observations are consistent with ground state fidelity and energy gap calculations. Our research advances the understanding of phase diagrams associated with high-dimensional quasicrystal potentials and marks the first discovery of a first-order quantum phase transition induced by quasicrystal domains in non-Hermitian systems. Published by the American Physical Society 2024

Comparing machine learning potentials for water: Kernel-based regression and Behler-Parrinello neural networks.

In this paper, we investigate the performance of different machine learning potentials (MLPs) in predicting key thermodynamic properties of water using RPBE + D3. Specifically, we scrutinize kernel-based regression and high-dimensional neural networks trained on a highly accurate dataset consisting of about 1500 structures, as well as a smaller dataset, about half the size, obtained using only on-the-fly learning. This study reveals that despite minor differences between the MLPs, their agreement on observables such as the diffusion constant and pair-correlation functions is excellent, especially for the large training dataset. Variations in the predicted density isobars, albeit somewhat larger, are also acceptable, particularly given the errors inherent to approximate density functional theory. Overall, this study emphasizes the relevance of the database over the fitting method. Finally, this study underscores the limitations of root mean square errors and the need for comprehensive testing, advocating the use of multiple MLPs for enhanced certainty, particularly when simulating complex thermodynamic properties that may not be fully captured by simpler tests.

Tuning Second Chern Number in a Four-Dimensional Topological Insulator by High-Frequency Time-Periodic Driving

Floquet engineering has attracted considerable attention as a promising approach for tuning topological phase transitions. We investigate the effects of high-frequency time-periodic driving in a four-dimensional (4D) topological insulator, focusing on topological phase transitions at the off-resonant quasienergy gap. The 4D topological insulator hosts gapless three-dimensional boundary states, characterized by the second Chern number C 2. We demonstrate that the second Chern number of 4D topological insulators can be modulated by tuning the amplitude of time-periodic driving. This includes transitions from a topological phase with C 2 = ±3 to another topological phase with C 2 = ±1, or to a topological phase with an even second Chern number C 2 = ±2, which is absent in the 4D static system. Finally, the approximation theory in the high-frequency limit further confirms the numerical conclusions.

Learning scattering waves via coupling physics-informed neural networks and their convergence analysis

Time harmonic acoustic wave scattering problems modeled by the unbounded Helmholtz equations are widely used in medical and military fields. A fast and flexible solver for the Helmholtz equation is expected in engineering applications. Fortunately, data-driven neural network algorithms show extraordinary promise in scientific computing and artificial intelligence. We solve the Helmholtz equation by decomposing the equation into a coupling equation system that explicitly includes the real and imaginary parts of the solution. The real and imaginary parts are approximated by the different neural networks, respectively. The unbounded domain is reduced to a problem on a bounded domain via the Dirichlet-to-Neumann operator. Thus, a coupling physics-informed neural network (cPINN) is constructed. On the other hand, in exploring the convergence analysis for PINNs, we obtain the convergence results for cPINNs through the universal approximation and Sobolev space theory instead of using the neural tangent kernel theory. We prove that the cPINNs converge to the solution of the Helmholtz equation in H01 space as the number of neurons approaches infinity. Finally, we propose some numerical results.

A Policy Gradient Algorithm for the Risk-Sensitive Exponential Cost MDP

We study the risk-sensitive exponential cost Markov decision process (MDP) formulation and develop a trajectory-based gradient algorithm to find the stationary point of the cost associated with a set of parameterized policies. We derive a formula that can be used to compute the policy gradient from (state, action, cost) information collected from sample paths of the MDP for each fixed parameterized policy. Unlike the traditional average cost problem, standard stochastic approximation theory cannot be used to exploit this formula. To address the issue, we introduce a truncated and smooth version of the risk-sensitive cost and show that this new cost criterion can be used to approximate the risk-sensitive cost and its gradient uniformly under some mild assumptions. We then develop a trajectory-based gradient algorithm to minimize the smooth truncated estimation of the risk-sensitive cost and derive conditions under which a sequence of truncations can be used to solve the original, untruncated cost problem. Funding: This work was supported by the Office of Naval Research Global [Grant N0001419-1-2566], the Division of Computer and Network Systems [Grant 21-06801], the Army Research Office [Grant W911NF-19-1-0379], and the Division of Computing and Communication Foundations [Grants 17-04970 and 19-34986].

Radiative Neutron Capture Rate of 11\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{11}$$\\end{document}B(n,γ)12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n,\\gamma )^{12}$$\\end{document}B Reaction from the Coulomb Dissociation of 12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{12}$$\\end{document}B

We calculate the 11\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{11}$$\\end{document}B(n,γ)12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n,\\gamma )^{12}$$\\end{document}B reaction rate which is an important constituent in nucleosynthesis networks and is contributed by resonant as well as non-resonant capture. For the resonant rate, we use the narrow resonance approximation whereas the non-resonant contribution is calculated with the Coulomb dissociation method for which we use finite-range distorted wave Born approximation theory. We then compare our calculated rate of 11\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{11}$$\\end{document}B(n,γ)12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n,\\gamma )^{12}$$\\end{document}B reaction with those reported earlier and with other charged particle reactions on 11\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{11}$$\\end{document}B.

First-principles investigation of the structure and thermodynamic properties of titanium hydrides

The formation of titanium hydrides is closely related to the hydrogen embrittlement in titanium and its alloy. In this work, a comprehensive study on the stable and metastable structures of TiHx (x=0.5, 1, 1.5 and 2) has been performed by means of first-principles calculations in combination with the evolutionary algorithm. The structural features, dynamical stability and elastic properties of these searched titanium hydrides have been systematically investigated. The influence of temperature on the structure stability and phase transition of the Ti2H, TiH, Ti2H3 and TiH2 phases is considered using the quasi-harmonic approximation theory. Our calculations predict that Ti2H, Ti2H3 and TiH2 undergo temperature-driven phase transitions. The reason that the theoretical calculation overestimates the critical temperature of the tetragonal to cubic phase transition in TiH2 within the quasi-harmonic approximations approach is also explored and discussed. The thermodynamic stability of these TiHx (x=0.5, 1, 1.5 and 2) structures is further analyzed over the temperature range of 0–1100 K. TiH2 is the only stable structure in the Ti–H system and Ti2H, TiH, and Ti2H3 are thermodynamically metastable within 900 K.

New Approximate Symmetry Theorems and Comparisons With Exact Symmetries

Three new approximate symmetry theories are proposed. The approximate symmetries are contrasted with each other and with the exact symmetries. The theories are applied to nonlinear ordinary differential equations for which exact solutions are available. It is shown that from the symmetries, approximate solutions as well as exact solutions in some restricted cases can be retrievable. Depending on the specific approximate theory and the equations considered, the approximate symmetries may expand the Lie Algebra of the exact symmetries, may be a perturbed form of the exact symmetries or may be a subalgebra of the exact symmetries. Exact and approximate solutions are retrieved using the symmetries.

Approximate theory of armour perforation by ductile hole expansion

The present study suggests a complete analytical model for armour perforation by ductile hole enlargement process. The new approximate model is applicable for entry, tunnelling and exit phases of a rigid nose-pointed projectile with an arbitrary nose-shape. The theory is based on dividing the target into infinitesimal thin layers, in which a cylindrical hole expansion is assumed. The elastoplastic work that is consumed by the target for infinitesimal projectile advancement is based on the specific cavitation energy and leads to reduction of the projectile kinetic energy. Several equivalent formulae for ballistic limit velocity, and a formula for the ballistic limit thickness are found to be independent of projectile nose-shape and nose length. The approximate model is compared with numerical simulations for AA6061-T651 targets that are impacted by two different ogive nose-shape projectiles at several striking velocities. A small nose-shape effect on the ballistic limit velocity is observed in the numerical simulations and is attributed mainly to the target axial effect and the bulge formation at the target rear surface, which are not considered in the approximate model.

A new meshless method of solving the distributed-order time-fractional mobile-immobile equations

A new method is introduced by combining neural network’s huge expressive power with technologies of the minimum residual approximate solutions (MRASs). In spatial direction, a single hidden layer neural network makes the construction of bases functions easier. The cost of calculations becomes acceptable in high dimensions space using the technologies of the MRASs. And this paper establish approximation and convergence order theories based on neural network.