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Related Concept Videos

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...
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...
Neural Circuits01:25

Neural Circuits

Neural circuits and neuronal pools are two of the main structures found in the nervous system. Neural circuits are networks of neurons that work together to carry out a specific task or process. They consist of interconnected neurons and glial cells, which provide structural and metabolic support.
Neuronal pools are collections of nerve cells with similar functions and interact through chemical and electrical signals. These pools include both interneurons (the central neural circuit nodes that...
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 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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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...

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Related Experiment Videos

A quantifier-reversal approximation paradigm for recurrent neural networks.

Clemens Hutter1, Valentin Abadie2, Helmut Bölcskei2

  • 1Swiss National Bank, Börsenstrasse 15, Zürich, 8001, Switzerland; Chair for Mathematical Information Science, ETH Zürich, Sternwartstrasse 7, Zürich, 8092, Switzerland.

Neural Networks : the Official Journal of the International Neural Network Society
|May 7, 2026
PubMed
Summary

This study introduces a novel recurrent neural network (RNN) approach for function approximation. Unlike traditional methods, this fixed-weight RNN uses temporal computation to achieve any desired accuracy, reducing approximation error with increased runtime.

Keywords:
Approximation theoryQuantifier reversalRecurrent neural networks

Related Experiment Videos

Area of Science:

  • Artificial Intelligence
  • Machine Learning
  • Computational Theory

Background:

  • Classical neural network approximation theory requires architecture and weights to adapt to error tolerance (epsilon).
  • This approach necessitates complex network designs that scale with desired precision.
  • Hardware constraints, particularly limited memory, pose challenges for deep or wide network implementations.

Purpose of the Study:

  • To introduce a new paradigm in neural network approximation by reversing the quantifier order.
  • To develop recurrent neural networks (RNNs) with fixed topology and weights for function approximation.
  • To demonstrate that approximation accuracy can be controlled by runtime rather than network complexity.

Main Methods:

  • Constructing a single recurrent neural network (RNN) with fixed architecture and weights.
  • Utilizing temporal computation and weight sharing as the core mechanisms for approximation.
  • Emulating deep feed-forward ReLU network principles: parallelization, linear combinations, affine transformations, and function composition via a clocked mechanism.

Main Results:

  • Achieved exponentially decaying approximation error as a function of runtime.
  • Demonstrated that RNN size is independent of the error tolerance (epsilon).
  • Showcased hidden-state dimensions that scale linearly with the degree of the approximated univariate polynomial.

Conclusions:

  • A novel RNN-based function approximation method is presented, controlled by runtime.
  • This temporal computation approach offers a memory-efficient alternative for hardware implementations.
  • The fixed-weight RNN paradigm shows promise for approximating functions, starting with univariate polynomials.