scholarly journals Neural network approximation

Acta Numerica ◽  
2021 ◽  
Vol 30 ◽  
pp. 327-444
Author(s):  
Ronald DeVore ◽  
Boris Hanin ◽  
Guergana Petrova
Keyword(s):  
Autonomous Navigation ◽  
Numerical Approximation ◽  
Convex Polytopes ◽  
Rate Distortion ◽  
Rational Functions ◽  
Linear Functions ◽  
Computational Time ◽  
Approximation Properties ◽  
Space Filling ◽  
The Stability

Neural networks (NNs) are the method of choice for building learning algorithms. They are now being investigated for other numerical tasks such as solving high-dimensional partial differential equations. Their popularity stems from their empirical success on several challenging learning problems (computer chess/Go, autonomous navigation, face recognition). However, most scholars agree that a convincing theoretical explanation for this success is still lacking. Since these applications revolve around approximating an unknown function from data observations, part of the answer must involve the ability of NNs to produce accurate approximations.This article surveys the known approximation properties of the outputs of NNs with the aim of uncovering the properties that are not present in the more traditional methods of approximation used in numerical analysis, such as approximations using polynomials, wavelets, rational functions and splines. Comparisons are made with traditional approximation methods from the viewpoint of rate distortion, i.e. error versus the number of parameters used to create the approximant. Another major component in the analysis of numerical approximation is the computational time needed to construct the approximation, and this in turn is intimately connected with the stability of the approximation algorithm. So the stability of numerical approximation using NNs is a large part of the analysis put forward.The survey, for the most part, is concerned with NNs using the popular ReLU activation function. In this case the outputs of the NNs are piecewise linear functions on rather complicated partitions of the domain of f into cells that are convex polytopes. When the architecture of the NN is fixed and the parameters are allowed to vary, the set of output functions of the NN is a parametrized nonlinear manifold. It is shown that this manifold has certain space-filling properties leading to an increased ability to approximate (better rate distortion) but at the expense of numerical stability. The space filling creates the challenge to the numerical method of finding best or good parameter choices when trying to approximate.

Symmetric Multistep Methods Revisited: II. Numerical Experiments

1999 ◽  
Vol 173 ◽  
pp. 309-314 ◽  
Author(s):  
T. Fukushima
Keyword(s):  
Multistep Methods ◽  
Computational Time ◽  
Integration Time ◽  
Implicit Methods ◽  
Symmetric Methods ◽  
Celestial Bodies ◽  
The Stability ◽  
Predictor Corrector ◽  
Symmetric Multistep Methods ◽  
Integration Errors

AbstractBy using the stability condition and general formulas developed by Fukushima (1998 = Paper I) we discovered that, just as in the case of the explicit symmetric multistep methods (Quinlan and Tremaine, 1990), when integrating orbital motions of celestial bodies, the implicit symmetric multistep methods used in the predictor-corrector manner lead to integration errors in position which grow linearly with the integration time if the stepsizes adopted are sufficiently small and if the number of corrections is sufficiently large, say two or three. We confirmed also that the symmetric methods (explicit or implicit) would produce the stepsize-dependent instabilities/resonances, which was discovered by A. Toomre in 1991 and confirmed by G.D. Quinlan for some high order explicit methods. Although the implicit methods require twice or more computational time for the same stepsize than the explicit symmetric ones do, they seem to be preferable since they reduce these undesirable features significantly.

AN ADVANCED ADAPTIVE FINITE ELEMENT CODE APPLIED FOR COATING-SUBSTRATE SIMULATION

2011 ◽  
Vol 03 (01n02) ◽  
pp. 91-107 ◽  
Author(s):  
JÜRGEN LEOPOLD ◽  
KATRIN HELLER ◽  
ARNDT MEYER ◽  
REINER WOHLGEMUTH
Keyword(s):  
Finite Element ◽  
Fe Simulation ◽  
Scale Up ◽  
Computational Time ◽  
Adaptive Finite Element ◽  
Workpiece Surface ◽  
Coated Tools ◽  
Geometrical Simulation ◽  
The Stability ◽  
And Storage

The stability of coating-substrate systems influences the chip formation and the surface integrity of the new generated workpiece surface, too. Using finite element (FE) simulation, deformations, strains and stresses in coated tools, caused by external and internal loads, can be computed on a microscopic scale. Since both, the whole macroscopic tool (in mm-scale) and the microscopic coating layers (in μm-scale up to nm-scale) must be included in the same geometrical simulation model, graded high-resolution FE meshes must be used. Nevertheless, the number of nodes in the 3D computational FE grid reaches some millions, leading to large computational time and storage requirements. For this reason, an advanced adaptive finite element (AAFEM) software has been developed and used for the simulation.

scholarly journals Autonomous system to control a mobile robot

2020 ◽  
Vol 9 (4) ◽  
pp. 1711-1717
Author(s):  
Ayman Abu Baker ◽  
Yazeed Yasin Ghadi
Keyword(s):  
Mobile Robot ◽  
Obstacle Avoidance ◽  
Autonomous Navigation ◽  
Fuzzy Controller ◽  
Computational Time ◽  
Ongoing Effort ◽  
Unstructured Environment ◽  
Adaptive Inference ◽  
Real Time Applications ◽  
Linguistic Labels

This paper presents an ongoing effort to control a mobile robot in unstructured environment. Obstacle avoidance is an important task in the field of robotics, since the goal of autonomous robot is to reach the destination without collision. Several algorithms have been proposed for obstacle avoidance, having drawbacks and benefits. In this paper, the fuzzy controller is used to tackle the problem of mobile robot autonomous navigation in unstructured environment. The objective is to make the robot move along a collision free trajectory until it reaches its target. The proposed approach uses the fuzzified, adaptive inference engine and defuzzification engine. Also number of linguistic labels is optimized for the input of the mobile robot in order to reduce computational time for real-time applications. The proposed fuzzy controller is evaluated subjectively and objectively with other approaches and also the processing time is taken in consideration.

scholarly journals Predictive modelling to support sensitivity analysis for robust design in aerospace engineering

2020 ◽  
Vol 61 (5) ◽  
pp. 2177-2192 ◽  
Author(s):  
Siva Krishna Dasari ◽  
Abbas Cheddad ◽  
Petter Andersson
Keyword(s):  
Sensitivity Analysis ◽  
Response Surface ◽  
Robust Design ◽  
Aerospace Engineering ◽  
Multivariate Adaptive Regression Splines ◽  
Linear Functions ◽  
Computational Time ◽  
Surface Modelling ◽  
Non Linear ◽  
Response Surface Modelling

AbstractThe design of aircraft engines involves computationally expensive engineering simulations. One way to solve this problem is the use of response surface models to approximate the high-fidelity time-consuming simulations while reducing computational time. For a robust design, sensitivity analysis based on these models allows for the efficient study of uncertain variables’ effect on system performance. The aim of this study is to support sensitivity analysis for a robust design in aerospace engineering. For this, an approach is presented in which random forests (RF) and multivariate adaptive regression splines (MARS) are explored to handle linear and non-linear response types for response surface modelling. Quantitative experiments are conducted to evaluate the predictive performance of these methods with Turbine Rear Structure (a component of aircraft) case study datasets for response surface modelling. Furthermore, to test these models’ applicability to perform sensitivity analysis, experiments are conducted using mathematical test problems (linear and non-linear functions) and their results are presented. From the experimental investigations, it appears that RF fits better on non-linear functions compared with MARS, whereas MARS fits well on linear functions.

scholarly journals Stability Analysis for Autonomous Dynamical Switched Systems through Nonconventional Lyapunov Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
V. Nosov ◽  
J. A. Meda-Campaña ◽  
J. C. Gomez-Mancilla ◽  
J. O. Escobedo-Alva ◽  
R. G. Hernández-García
Keyword(s):  
Stability Analysis ◽  
Finite Number ◽  
Switched Systems ◽  
Lyapunov Functions ◽  
Switched System ◽  
Linear Functions ◽  
Asymptotically Stable ◽  
Multiple Lyapunov Functions ◽  
Stability Theorems ◽  
The Stability

The stability of autonomous dynamical switched systems is analyzed by means of multiple Lyapunov functions. The stability theorems given in this paper have finite number of conditions to check. It is shown that linear functions can be used as Lyapunov functions. An example of an exponentially asymptotically stable switched system formed by four unstable systems is also given.

Collocation Method for the Modeling of Membrane Gas Permeation Systems

2015 ◽  
Vol 16 (3-4) ◽  
pp. 141-149
Author(s):  
Anna Feichtinger ◽  
Aleksander Makaruk ◽  
Ewa Weinmüller ◽  
Anton Friedl ◽  
Michael Harasek
Keyword(s):  
Collocation Method ◽  
Numerical Approximation ◽  
Numerical Approach ◽  
Adaptation Strategy ◽  
Gas Permeation ◽  
Computational Time ◽  
Efficient Computation ◽  
Grid Adaptation ◽  
Crucial Feature ◽  
Comparison Of The Results

AbstractIn this work, we describe a numerical method which enables an efficient computation of membrane gas permeation processes that involve multiple membrane stages and multiple gas components. The utilized numerical approach is a collocation method equipped with a grid adaptation strategy based on a dependable error estimate of the numerical approximation. The comparison of the results provided by the collocation method with those calculated from an experimentally validated finite difference method has demonstrated that the accuracy of both numerical approximations is practically the same. However, the current procedure is characterized by a much better computational efficiency that allows to considerably reduce the computational time. This is a crucial feature when combining computation of membrane permeation processes with optimization algorithms. In such a setting the computation of the permeation process is frequently repeated and naturally, results in long computational times when the efficiency is not adequately improved.

scholarly journals HILBERT C*-BIMODULES OF FINITE INDEX AND APPROXIMATION PROPERTIES OF C*-ALGEBRAS

2017 ◽  
Vol 60 (2) ◽  
pp. 321-331
Author(s):  
MARZIEH FOROUGH ◽  
MASSOUD AMINI
Keyword(s):  
Finite Index ◽  
Morita Equivalence ◽  
Discrete Groups ◽  
Crossed Products ◽  
Approximation Properties ◽  
C Algebras ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation ◽  
The Stability

AbstractLet A and B be arbitrary C*-algebras, we prove that the existence of a Hilbert A–B-bimodule of finite index ensures that the WEP, QWEP, and LLP along with other finite-dimensional approximation properties such as CBAP and (S)OAP are shared by A and B. For this, we first study the stability of the WEP, QWEP, and LLP under Morita equivalence of C*-algebras. We present examples of Hilbert A–B-bimodules, which are not of finite index, while such properties are shared between A and B. To this end, we study twisted crossed products by amenable discrete groups.

STABILITY OF NUMERICAL DISCRETIZATIONS IN MODELLING VIBRATIONAL CHARACTERISTICS OF PIEZOELECTRIC CYLINDRICAL SHELLS

2002 ◽  
Vol 02 (02) ◽  
pp. 241-264
Author(s):  
R. V. N. MELNIK ◽  
K. N. MELNIK
Keyword(s):  
Conventional Treatment ◽  
Numerical Approximation ◽  
Piezoelectric Materials ◽  
Analytical Techniques ◽  
Stability Conditions ◽  
Cylindrical Shape ◽  
Vibrational Characteristics ◽  
Electromechanical Energy ◽  
The Stability ◽  
Piezoelectric Structures

Many problems in applications of piezoelectric materials are essentially time-dependent, and a conventional treatment of such problems with analytical or semi-analytical techniques based on the analysis of harmonic oscillations become inadequate in those cases where a complete dynamic picture of electromechanical energy transfer is required. For such situations we have developed an efficient explicit numerical methodology allowing us to compute dynamic electromechanical characteristics of piezoelectric structures and devices under various loading conditions. In this paper we demonstrate that the stability conditions for our numerical approximation can be obtained from a discrete conservation law, and can be cast in a form similar to that of the classical CFL condition. However, in our case the velocities of wave propagations, participating in the formulation of the stability conditions, are clearly dependent on the pattern of electromectromechanical coupling. Our discussion in this paper, including computational examples, is centred around finite piezoelectric shells of cylindrical shape.

On Dynamic Tooth Load and Stability of a Spur-Gear System Using the State-Space Approach

1985 ◽  
Vol 107 (1) ◽  
pp. 54-60 ◽  
Author(s):  
A. S. Kumar ◽  
T. S. Sankar ◽  
M. O. M. Osman
Keyword(s):  
State Space ◽  
Dynamic Load ◽  
Initial Conditions ◽  
Gear Tooth ◽  
Spur Gear ◽  
The State ◽  
Gear System ◽  
Computational Time ◽  
Steady State Condition ◽  
The Stability

In this study, a new approach using the state-space method is presented for the dynamic load analysis of spur gear systems. This approach gives the dynamic load on gear tooth in mesh as well as information on the stability of the gear system. Also a procedure is given for the selection of proper initial conditions that enable the steady-state condition to be reached faster, conditions that result in considerable savings in computational time. The variations in the dynamic load with respect to changes in contact position, operating speed, backlash, damping, and stiffness are also investigated. In addition, the stability of the gear system is studied, using the Floquet theory and the well-known stability conditions of difference systems.

Sign in / Sign up

Export Citation Format

Share Document