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

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 Approximations01:23

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...
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...
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...
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

Adrenocorticotropic hormone-secreting pheochromocytomas: the exception to the rule.

Surgery·1995
Same author

The mouse obese gene. Genomic organization, promoter activity, and activation by CCAAT/enhancer-binding protein alpha.

The Journal of biological chemistry·1995
Same author

Circadian variation in coronary tone in patients with stable angina. Protective role of the endothelium.

Circulation·1995
Same author

Purification of a heteromeric CCAAT binding protein from Neurospora crassa.

Molecular & general genetics : MGG·1995
Same author

Requirement for neuregulin receptor erbB2 in neural and cardiac development.

Nature·1995
Same author

Suppression of retinal neovascularization in vivo by inhibition of vascular endothelial growth factor (VEGF) using soluble VEGF-receptor chimeric proteins.

Proceedings of the National Academy of Sciences of the United States of America·1995

Related Experiment Videos

Approximation capability in C(R (n)) by multilayer feedforward networks and related problems.

T Chen1, H Chen, R W Liu

  • 1Dept. of Math., Fudan Univ., Shanghai.

IEEE Transactions on Neural Networks
|January 1, 1995
PubMed
Summary

Neural networks can approximate functions using sigmoidal activation. Boundedness of the sigmoidal function is crucial for this capability, simplifying n-dimensional approximation to one dimension.

Related Experiment Videos

Area of Science:

  • Computational mathematics
  • Neural network theory
  • Function approximation

Background:

  • Three-layered neural networks are widely used for function approximation.
  • Sigmoidal functions are common activation functions in neural networks.
  • Understanding the theoretical capabilities of neural networks is essential for their application.

Purpose of the Study:

  • To investigate the function approximation capabilities of three-layered neural networks with sigmoidal activation in C(R^n).
  • To determine the essential properties of sigmoidal functions for successful approximation.
  • To simplify the proof of approximation for n-dimensional cases.

Main Methods:

  • Theoretical analysis of function approximation by neural networks.
  • Focus on the role of boundedness, continuity, and monotonicity of sigmoidal functions.
  • Reduction of the n-dimensional approximation problem to a one-dimensional case.

Main Results:

  • The boundedness condition of the sigmoidal function is essential for approximation in C(R^n).
  • Continuity or monotonicity alone is insufficient for the approximation capability.
  • The n-dimensional approximation problem can be effectively reduced to a one-dimensional case.

Conclusions:

  • Boundedness is a critical property for sigmoidal activation functions in neural network function approximation.
  • The theoretical framework for n-dimensional approximation can be simplified by focusing on the one-dimensional case.
  • The study provides insights into the fundamental requirements for neural network approximation capabilities.