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neural network approximation of Refinable Functions

IEEE TRANSACTIONS ON INFORMATION THEORY 2023年第1期69卷 482-495页

作者： Daubechies, Ingrid De Vore, Ronald Dym, Nadav Faigenbaum-Golovin, Shira Kovalsky, Shahar Z. Lin, Kung-Chin Park, Josiah Petrova, Guergana Sober, Barak Duke Univ Dept Math Durham NC 27708 USA Texas A&M Univ Dept Math College Stn TX 77843 USA Technion Israel Inst Technol Fac Math IL-3200003 Haifa Israel Duke Univ Rhodes Informat Initiat Durham NC 27708 USA Univ North Carolina Chapel Hill Dept Math Chapel Hill NC 27599 USA Hebrew Univ Jerusalem Dept Stat & Data Sci & Digital Humanities IL-9190501 Jerusalem Israel

In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there exists a variety of results which show that a wide range of functions can be approximated with sometimes surprising accuracy by these outputs. For example, it is known that the set of functions that can be approximated with exponential accuracy (in terms of the number of parameters used) includes, on one hand, very smooth functions such as polynomials and analytic functions and, on the other hand, very rough functions such as the Weierstrass function, which is nowhere differentiable. In this paper, we add to the latter class of rough functions by showing that it also includes refinable functions. Namely, we show that refinable functions are approximated by the outputs of deep ReLU neural networks with a fixed width and increasing depth with accuracy exponential in terms of their number of parameters. Our results apply to functions used in the standard construction of wavelets as well as to functions constructed via subdivision algorithms in Computer Aided Geometric Design.

关键词： Artificial neural networks Standards Deep learning approximation algorithms Upper bound Stress Terminology neural networks neural network approximation refinable functions exponential accuracy cascade algorithm

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neural network approximation-based backstepping sliding mode control for spacecraft with input saturation and dynamics uncertainty

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ACTA ASTRONAUTICA 2022年 191卷 1-10页

作者： Liu, Erjiang Yan, Ye Yang, Yueneng Natl Univ Def Technol Coll Aerosp Sci & Engn Changsha 410073 Peoples R China Chinese Acad Mil Sci Natl Innovat Inst Def Technol Beijing 100091 Peoples R China Natl Univ Def Technol Coll Intelligence Sci Changsha 410073 Peoples R China

This paper proposed a neural network approximation-based backstepping sliding mode control approach (NN-BSMC) to address the problem of attitude tracking control for spacecraft in the presence of inertial uncertainties, external disturbances and input saturation. First, the attitude dynamics model of the spacecraft is presented, and the error dynamics model with input saturation is derived. Second, a control scheme using sliding mode control (SMC) and backstepping technique is proposed to guarantee the robustness against the dynamics uncertainty and deal with the effect of input saturation. Under this framework, a neural network (NN) approximator is developed to online estimate the dynamics uncertainty of the spacecraft. Adaptive laws are designed to update the NN weight and estimate the unknown bound of approximation error. The control law does not require the prior knowledge about the bounds on the uncertainty and can provide complete compensation for the uncertainty. Moreover, a Lyapunov-based approach is employed to prove the global stability of the closed-loop system and the asymptotical convergence of the attitude tracking errors. Finally, numerical simulations are carried out to demonstrate the effectiveness and robustness of the proposed controllers. Contrasting simulation results indicate that the neural network backstepping sliding mode controller reduces the chattering effectively and has better performance against the sliding mode controller.

关键词： neural network approximation Backstepping control Input saturation Attitude tracking Sliding mode control

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neural network approximation of Helicopter Turboshaft Engine Parameters for Improved Efficiency

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ENERGIES 2024年第9期17卷 2233页

作者： Vladov, Serhii Yakovliev, Ruslan Bulakh, Maryna Vysotska, Victoria Kharkiv Natl Univ Internal Affairs Kremenchuk Flight Coll 17-6 Peremohy St UA-39605 Kremenchuk Ukraine Rzeszow Univ Technol Fac Mech & Technol PL-37450 Stalowa Wola Poland Lviv Polytech Natl Univ Informat Syst & Networks Dept 12 Bandera St UA-79013 Lvov Ukraine Osnabruck Univ Inst Comp Sci 1 Friedrich Janssen St D-49076 Osnabruck Germany

The work is devoted to the development of a method for neural network approximation of helicopter turboshaft engine parameters, which is the basis for researching engine energy characteristics to improve efficiency, reliability, and flight safety. It is proposed to use a three-layer direct propagation neural network with linear neurons in the output layer for training in which the scale conjugate gradient algorithm is modified by introducing a moment coefficient into the analytical expression. This modification helps in calculating new model parameters to avoid falling into a local minimum. The dependence of the energy released during helicopter turboshaft engine compressor rotation on the gas-generator rotor r.p.m. was obtained. This enables the determination of the optimal gas-generator rotor r.p.m. region for a specific type of helicopter turboshaft engine. The optimal ratio of energy consumption and compressor operating efficiency is achieved, thereby ensuring helicopter turboshaft engines' optimal performance and reliability. Experimental data support the high efficiency of using a three-layer feed-forward neural network with linear neurons in the output layer, trained using a modified scale conjugate gradient algorithm, for approximating parameters of helicopter turboshaft engines compared to the analogues. Specifically, this method better predicts the relations between the energy release during compressor rotation and gas-generator rotor r.p.m. The efficiency coefficient of the proposed method was 0.994, which exceeded that of the closest analogue (0.914) by 1.09 times.

关键词： neural network approximation helicopter turboshaft engines energy power efficiency training gas-generator rotor r.p.m. scaled conjugate gradient algorithm

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Convergence rates for random feature neural network approximation in molecular dynamics

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BIT NUMERICAL MATHEMATICS 2025年第1期65卷 1-40页

作者： Huang, Xin Plechac, Petr Sandberg, Mattias Szepessy, Anders Kungl Tekn Hogskolan Inst Matemat S-10044 Stockholm Sweden Univ Delaware Dept Math Sci Newark DE 19716 USA

Random feature neural network approximations of the potential in Hamiltonian systems yield approximations of molecular dynamics correlation observables that have the expected error OK-1+J-1212\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}\left( \left( K<^>{-1}+J<^>{-\fracapproximationapproximation}\right) <^>{\fracapproximationapproximation}\right) $$\end{document}, for networks with K nodes using J data points, provided the Hessians of the potential and the observables are bounded. The loss function is based on the least squares error of the potential and regularizations, with the data points sampled from the Gibbs density. The proof uses a new derivation of the generalization error for random feature networks that does not apply the Rademacher or related complexities.

关键词： Random Fourier feature representation Generalization error estimate neural network approximation Canonical molecular dynamics Correlation observable

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A novel preview control for MLD models and its neural network approximation for real-time implementation: Application to semi-active vibration control of a vehicle suspension

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IET CONTROL THEORY AND APPLICATIONS 2023年第4期17卷 433-445页

作者： Sato, Kaoru Hiramoto, Kazuhiko Niigata Univ Grad Sch Sci & Technol Dept Adv Mat Sci & Technol Niigata Japan Niigata Univ Fac Engn Mech Engn Program Niigata Japan

Advances in image processing technology have made it possible to measure the surface shape of the road ahead while driving. A new semi-active suspension control method considering the forward road surface shape is proposed. A vehicle model equipped with a semi-active suspension can be expressed as an mixed logical dynamical model. When the shape of the road ahead can be measured, the information on future disturbances is available before the vehicle undergoes. In this paper, the finite time optimization problem for the mixed logical dynamical model is formulated to consider the future disturbances as a mixed integer quadratic programming problem in the same way as the conventional control problem without future disturbance. However, the mixed integer quadratic programming problem is hard to obtain the control action within the control cycle period required in the real-time vibration control with general computers for vehicles. In this paper, the reduction of the computational load is achieved by constructing an approximation function of the designed controller. A neural network is adopted for the approximation. The performance evaluation of the proposed method is evaluated by simulations. In the simulation study, the proposed method achieves better ride comfort with the equivalent suspension stroke compared to the traditional methods.

关键词： MLD models Optimisation techniques finite time optimization problem forward road surface shape semiactive suspension control method novel preview control future disturbance vehicle model equivalent suspension stroke image processing technology approximation function Control technology and theory (production) semiactive vibration control quadratic programming Mechanical components Optimal control Vibrations and shock waves (mechanical engineering) control system synthesis vehicle dynamics real-time implementation mixed integer quadratic programming problem control cycle period real-time vibration control Road-traffic system control integer programming control action optimal control mixed logical dynamical model vibration control vehicle suspension conventional control problem Control system analysis and synthesis methods predictive control Mechanical variables control neural network approximation Vehicle mechanics road vehicles suspensions (mechanical components)

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Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions

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IMA JOURNAL OF NUMERICAL ANALYSIS 2022年第3期42卷 2055-2082页

作者： Grohs, Philipp Herrmann, Lukas Univ Vienna Fac Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria Austrian Acad Sci Johann Radon Inst Computat & Appl Math Altenbergerstr 69 A-4040 Linz Austria

In recent work it has been established that deep neural networks (DNNs) are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whole Euclidean domain. On the other hand, most problems in engineering and in the sciences are formulated on finite domains and subjected to boundary conditions. The present paper considers an important such model problem, namely the Poisson equation on a domain D. Rd subject to Dirichlet boundary conditions. It is shown that DNNs are capable of representing solutions of that problem without incurring the curse of dimension. The proofs are based on a probabilistic representation of the solution to the Poisson equation as well as a suitable sampling method.

关键词： high-dimensional approximation neural network approximation Monte Carlo methods

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Hierarchical fusion network for periocular and iris by neural network approximation and sparse autoencoder

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MACHINE VISION AND APPLICATIONS 2020年第1期32卷 p1-10页

作者： Algashaam, Faisal Kien Nguyen Banks, Jasmine Chandran, Vinod Tuan-Anh Do Alkanhal, Mohamed Queensland Univ Technol Brisbane Qld Australia King Abdulaziz City Sci & Technol Riyadh Saudi Arabia Hanoi Univ Sci & Technol Hanoi Vietnam

The eye region is one of the most attractive sources for identification and verification due to the representative availability of such biometric modalities as periocular and iris. Many score-level fusion approaches have been proposed to combine these two modalities targeting to improve the robustness. The score-level approaches can be grouped into three categories: transformation-based, classification-based and density-based. Each category has its own benefits, if combined can lead to a robust fusion mechanism. In this paper, we propose a hierarchical fusion network to fuse multiple fusion approaches from transformation-based and classification-based categories into a unified framework for classification. The proposed hierarchical approach relies on the universal approximation theorem for neural networks to approximate each fusion approach as one child neural network and then ensemble them into a unified parent network. This mechanism takes advantage of both categories to improve the fusion performance, illustrated by an improved equal error rate of the multimodal biometric system. We subsequently force the parent network to learn the representation and interaction strategy between the child networks from the training data through a sparse autoencoder layer, leading to further improvements. Experiments on two public datasets (MBGC version 2 and CASIA-Iris-Thousand) and our own dataset validate the effectiveness of the proposed hierarchical fusion approach for periocular and iris modalities.

关键词： Periocular recognition Iris recognition Hierarchical multimodal fusion neural network approximation Sparse autoencoder

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A New L1 neural network Adaptive Wind Turbine Pitch Control Strategy

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JOURNAL OF CIRCUITS SYSTEMS AND COMPUTERS 2025年第1期34卷

作者： Hu, Hui Peng, Long Li, Xiaoqin Chen, Ying Yan, Jiande Guo, Peng Cheng, Bozhi Hunan Inst Engn Coll Elect & Informat Engn Xiangtan Hunan Peoples R China Xiangtan Tract Locomot Factory Co Ltd Xiangtan Hunan Peoples R China

This paper proposes a novel L1 neural network (NN) adaptive control strategy to address the challenges posed by the non-affine nonlinear characteristics of wind turbines operating at high wind speed. These challenges complicate the design of control inputs and the issues of ensuring system transient performance due to unmodeled dynamics and external disturbances in variable-pitch control. This strategy comprises an estimator, control laws, and adaptive laws. The L1 adaptive controller ensures rapid adaptation and robustness of the system, and the NN is utilized to approximate the ideal implicit controller. This approach guarantees the uniform boundedness of system transient responses without the need for wind speed measurements. Finally, the new strategy proposed in this work is compared with traditional NN adaptive and PID control methods through simulation experiments, demonstrating enhanced robustness and superior transient performance.

关键词： Wind turbine variable-pitch control non-affine nonlinear systems neural network approximation L1 adaptive control

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Mean-Field Multiagent Reinforcement Learning: A Decentralized network Approach

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MATHEMATICS OF OPERATIONS RESEARCH 2025年第1期50卷 1-781 C2页

作者： Gu, Haotian Guo, Xin Wei, Xiaoli Xu, Renyuan Univ Calif Berkeley Dept Math Berkeley CA 94720 USA Univ Calif Berkeley Dept Ind Engn & Operat Res Berkeley CA 94720 USA Tsinghua Shenzhen Int Grad Sch Shenzhen 518071 Peoples R China Univ Southern Calif Ind & Syst Engn Los Angeles CA 90089 USA

One of the challenges for multiagent reinforcement learning (MARL) is designing efficient learning algorithms for a large system in which each agent has only limited or partial information of the entire system. Whereas exciting progress has been made to analyze decentralized MARL with the network of agents for social networks and team video games, little is known theoretically for decentralized MARL with the network of states for modeling self -driving vehicles, ride -sharing, and data and traffic routing. This paper proposes a framework of localized training and decentralized execution to study MARL with the network of states. Localized training means that agents only need to collect local information in their neighboring states during the training phase;decentralized execution implies that agents can execute afterward the learned decentralized policies, which depend only on agents' current states. The theoretical analysis consists of three key components: the first is the reformulation of the MARL system as a networked Markov decision process with teams of agents, enabling updating the associated team Q -function in a localized fashion;the second is the Bellman equation for the value function and the appropriate Q -function on the probability measure space;and the third is the exponential decay property of the team Q -function, facilitating its approximation with efficient sample efficiency and controllable error. The theoretical analysis paves the way for a new algorithm LTDE-neural-AC, in which the actor-critic approach with overparameterized neural networks is proposed. The convergence and sample complexity are established and shown to be scalable with respect to the sizes of both agents and states. To the best of our knowledge, this is the first neural network-based MARL algorithm with network structure and provable convergence guarantee.

关键词： multiagent reinforcement learning mean-field neural network approximation

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Adaptive neural Preassigned-Time Control for Macro-Micro Composite Positioning Stage with Displacement Constraints

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IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS 2024年第2期20卷 1103-1112页

作者： Chen, Xiangyong Liu, Huawei Wen, Guanghui Liu, Yang Cao, Jinde Qiu, Jianlong Linyi Univ Sch Automat & Elect Engn Key Lab Complex Syst & Intelligent Comp Univ Shan Linyi 276005 Shandong Peoples R China Southeast Univ Sch Math Nanjing 210096 Peoples R China Southeast Univ Res Ctr Complex Syst & Network Sci Nanjing 210096 Peoples R China Qingdao Univ Sci & Technol Sch Automat Sci & Technol Qingdao 266100 Peoples R China

This paper considers the rapid vibration reduction problem of macro-micro composite positioning stage (MMCPS) using an adaptive neural preassigned-time control strategy. Based on Newton's second law, the MMCPS is modeled as an inter-connected system with unknown perturbations, and for the first time, the vibration reduction problem of MMCPS is transformed into a displacement constraint problem. Through adaptive neural network approximation and backstepping control, a preassigned-time controller with a novel performance function-related term is developed, which not only significantly improves the positioning accuracy and reduces the vibration amplitude but also ensures that the displacements of the voice coil motor axis and the stage are constrained to a predefined region in a finite time. Another distinguished feature of the proposed controller lies in the fact that the settling time of the displacement signals can be set as an arbitrary positive value. Moreover, all signals of the closed-loop system are proved to be semi-globally uniformly ultimately bounded. Finally, the feasibility of the designed control strategy is demonstrated via a simulation experiment.

关键词： Rapid vibration reduction macro-micro composite positioning stage (MMCPS) adaptive neural preassigned-time control displacement constraints neural network approximation

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