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Related Concept Videos

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...
Lagrange Multipliers: Two Constraints01:28

Lagrange Multipliers: Two Constraints

The method of Lagrange multipliers with two constraints is used to optimize a function subject to two independent constraints. In many applications, the objective function represents a quantity to be maximized or minimized, such as cost, area, distance, or energy. The two constraints represent requirements that the solution must satisfy, such as fixed volume, limited resources, or prescribed dimensions.For a function of three variables, each constraint forms a surface in three-dimensional space.
Multi-input and Multi-variable systems01:22

Multi-input and Multi-variable systems

Cruise control systems in cars are designed as multi-input systems to maintain a driver's desired speed while compensating for external disturbances such as changes in terrain. The block diagram for a cruise control system typically includes two main inputs: the desired speed set by the driver and any external disturbances, such as the incline of the road. By adjusting the engine throttle, the system maintains the vehicle's speed as close to the desired value as possible.
In the absence of...
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.
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Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

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Multivariable Functions and Higher Derivatives01:30

Multivariable Functions and Higher Derivatives

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Related Experiment Videos

A constructive method for multivariate function approximation by multilayer perceptrons.

S Geva1, J Sitte

  • 1Fac. of Inf. Technol., Queensland Univ. of Technol., Brisbane, Qld.

IEEE Transactions on Neural Networks
|January 1, 1992
PubMed
Summary

This study demonstrates constructing a two-hidden-layer perceptron for multivariate function approximation. This neural network approach offers a practical method for function approximation using local functions, similar to Gaussian potential functions.

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Area of Science:

  • Artificial Intelligence
  • Machine Learning
  • Computational Neuroscience

Background:

  • Mathematical theorems confirm the capability of feedforward multilayered neural networks with sigmoidal neurons to approximate continuous multivariate functions.
  • Existing theorems lack practical guidance for determining network parameters.

Purpose of the Study:

  • To present a constructive method for designing a perceptron capable of multivariate function approximation.
  • To demonstrate an alternative approach to function approximation using neural networks.

Main Methods:

  • Construction of a feedforward neural network with two hidden layers.
  • Utilizing neurons with sigmoidal transfer functions.
  • Employing a linear combination of local functions for approximation.

Main Results:

  • A specific network architecture is proposed for practical multivariate function approximation.
  • The constructed perceptron achieves function approximation capabilities comparable to networks using Gaussian potential functions.
  • The method relies on the combination of local basis functions.

Conclusions:

  • Provides a practical construction for neural network-based multivariate function approximation.
  • Offers an alternative to existing methods by using a linear combination of local functions.
  • Highlights the potential of sigmoidal neural networks for complex function approximation tasks.