Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Accuracy, limits, and approximation01:28

Accuracy, limits, and approximation

Accuracy, limits, and approximations are common in many fields, especially in engineering calculations. These concepts are imperative for ensuring that a given value is as close as possible to its true value.
Accuracy is defined as the closeness of the measured value to the true or actual value. In engineering mechanics, repeated measurements are taken during theoretical or experimental analyses to ensure that the result is precise and accurate.
The accuracy of any solution is based on the...
Linearization and Approximation01:26

Linearization and Approximation

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
Approximate Integration01:24

Approximate Integration

In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...
Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
Application of Linearization and Approximation01:29

Application of Linearization and Approximation

A drone flying through complex terrain often relies on more than one sensing method to estimate small changes in altitude. Along with direct measurements, air pressure provides a useful indirect indicator of vertical movement. Atmospheric pressure decreases as altitude increases, and this relationship is commonly described using an exponential model. Although accurate, converting pressure measurements into altitude values requires calculations that are too complex to perform repeatedly during...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

GATE: Adaptive learning with working memory by information gating in multi-lamellar hippocampal formation.

PLoS computational biology·2026
Same author

ED-SAM: Sharpness-aware minimization with energy-adjusted perturbations and direction-corrected updates.

Neural networks : the official journal of the International Neural Network Society·2026
Same author

Privacy-preserving Online Federated Learning for Massive Infinite Streams.

IEEE transactions on pattern analysis and machine intelligence·2026
Same author

Enantioselective synthesis of chiral 1,2-oxazinane spiro-oxindoles <i>via</i> carbene-catalyzed [3 + 3] annulation of isatin-derived nitrones with enals.

Organic & biomolecular chemistry·2026
Same author

A prediction model of gastric cancer based on M2-like tumor-associated macrophage infiltration verified by immunohistochemistry.

Translational cancer research·2026
Same author

DAC-MR: Data Augmentation Consistency Based Meta-Regularization for Meta-Learning.

IEEE transactions on pattern analysis and machine intelligence·2026
Same journal

Q-learning based asynchronous Boolean control networks stabilization with data loss.

Neural networks : the official journal of the International Neural Network Society·2026
Same journal

New results on prescribed-time synchronization of complex networks via intermittent control.

Neural networks : the official journal of the International Neural Network Society·2026
Same journal

Variance-constrained multi-view ensemble broad network for imbalanced data.

Neural networks : the official journal of the International Neural Network Society·2026
Same journal

Dynamic analysis and reliable mechanical optimization application of ring HNN effected with a memristive neuron.

Neural networks : the official journal of the International Neural Network Society·2026
Same journal

DAFF-Net: A detection and search method for small-scale low surface brightness galaxies.

Neural networks : the official journal of the International Neural Network Society·2026
Same journal

Quasi-synchronization for complex networks with hybrid pinning intermittent control.

Neural networks : the official journal of the International Neural Network Society·2026
See all related articles

Related Experiment Videos

Essential rate for approximation by spherical neural networks.

Shaobo Lin1, Feilong Cao, Zongben Xu

  • 1Institute for Information and System Sciences, Xi'an Jiaotong University, Xi'an 710049, Shannxi Province, PR China. ssxvkihc@yahoo.com.cn

Neural Networks : the Official Journal of the International Neural Network Society
|May 31, 2011
PubMed
Summary
This summary is machine-generated.

This study shows the optimal approximation rate for neural networks on a sphere. Spherical neural networks achieve an essential approximation rate of n to the power of (-2r/d-1).

Related Experiment Videos

Area of Science:

  • Machine Learning
  • Numerical Analysis
  • Functional Analysis

Background:

  • Feed-forward neural networks are powerful tools for function approximation.
  • Approximation rates quantify how well a function class can be represented by neural networks.
  • The unit sphere presents unique challenges for approximation due to its geometry.

Purpose of the Study:

  • To determine the optimal approximation rate of single hidden layer feed-forward neural networks on the unit sphere.
  • To analyze the asymptotic behavior of neural network approximation for Sobolev classes on spherical domains.
  • To establish the essential approximation rate achievable by spherical neural networks.

Main Methods:

  • Utilizing theoretical analysis of neural network approximation capabilities.
  • Considering feed-forward neural networks with sigmoidal activation functions.
  • Analyzing the deviation between Sobolev classes W²(2r)(S(d)) and neural network function classes Φ(n)(ϕ) on the unit sphere S(d).

Main Results:

  • Proving the existence of a neural network with n neurons and a specific activation function.
  • Demonstrating that the deviation asymptotically behaves as n(-2r/d-1).
  • Establishing n(-2r/d-1) as the essential approximation rate for spherical neural networks.

Conclusions:

  • The essential approximation rate for neural networks on the unit sphere is determined by the network size (n), the smoothness of the function (r), and the dimension of the sphere (d).
  • This finding provides a precise understanding of the efficiency of neural networks in approximating functions on spherical manifolds.
  • The results are crucial for designing efficient neural network architectures for spherical data analysis.