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Related Concept Videos

Accuracy, limits, and approximation01:28

Accuracy, limits, and approximation

1.0K
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...
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Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

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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....
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Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

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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,...
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Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

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Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
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Transformers with Off-Nominal Turns Ratios01:25

Transformers with Off-Nominal Turns Ratios

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In scenarios involving parallel transformers with disparate ratings, developing per-unit models requires accommodating off-nominal turns ratios. This situation arises when the selected base voltages are not proportional to the transformer’s voltage ratings. Consider a transformer where the rated voltages are related by the term a. If the chosen voltage bases satisfy a relationship involving term b, term c is defined as the ratio of these bases. This ratio is then substituted into the...
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Convolution Properties II01:17

Convolution Properties II

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The important convolution properties include width, area, differentiation, and integration properties.
The width property indicates that if the durations of input signals are T1 and T2, then the width of the output response equals the sum of both durations, irrespective of the shapes of the two functions. For instance, convolving two rectangular pulses with durations of 2 seconds and 1 second results in a function with a width of 3 seconds.
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Related Experiment Video

Updated: Dec 20, 2025

Deep Neural Networks for Image-Based Dietary Assessment
13:19

Deep Neural Networks for Image-Based Dietary Assessment

Published on: March 13, 2021

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Approximation rates for neural networks with general activation functions.

Jonathan W Siegel1, Jinchao Xu1

  • 1Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA.

Neural Networks : the Official Journal of the International Neural Network Society
|May 30, 2020
PubMed
Summary

This study advances neural network approximation theory, proving new dimension-independent rates for general activation functions. These findings enhance understanding of neural network expressivity and approximation capabilities.

Keywords:
Approximation theoryNeural networksStratified sampling

Related Experiment Videos

Last Updated: Dec 20, 2025

Deep Neural Networks for Image-Based Dietary Assessment
13:19

Deep Neural Networks for Image-Based Dietary Assessment

Published on: March 13, 2021

9.8K

Area of Science:

  • Computational mathematics
  • Machine learning theory
  • Neural network analysis

Background:

  • Neural networks are powerful function approximators.
  • Understanding their approximation rates is crucial for theoretical guarantees.
  • Previous work focused on specific activation functions.

Purpose of the Study:

  • To establish new approximation rates for neural networks with general activation functions.
  • To extend dimension-independent approximation results to non-sigmoidal activation functions.
  • To investigate the impact of activation function properties on approximation rates.

Main Methods:

  • Analysis of two-layer neural networks.
  • Derivation of approximation bounds.
  • Extension of existing theoretical frameworks.
  • Application of stratified sampling techniques.

Main Results:

  • Dimension-independent approximation rates for polynomially decaying non-sigmoidal activation functions.
  • Weaker, but still dimension-independent, rates for bounded, integrable activation functions.
  • Demonstration that stratified sampling can improve approximation rates under certain conditions.

Conclusions:

  • The approximation capabilities of neural networks are more general than previously shown.
  • Activation function properties significantly influence approximation rates.
  • Stratified sampling offers a potential method for enhancing neural network performance.