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Related Concept Videos

Classification of Systems-II01:31

Classification of Systems-II

Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,
Sampling Continuous Time Signal01:11

Sampling Continuous Time Signal

In signal processing, a continuous-time signal can be sampled using an impulse-train sampling technique, followed by the zero-order hold method. Impulse-train sampling involves the use of a periodic impulse train, which consists of a series of delta functions spaced at regular intervals determined by the sampling period. When a continuous-time signal is multiplied by this impulse train, it generates impulses with amplitudes corresponding to the signal's values at the sampling points.
In the...
Basic Continuous Time Signals01:22

Basic Continuous Time Signals

Basic continuous-time signals include the unit step function, unit impulse function, and unit ramp function, collectively referred to as singularity functions. Singularity functions are characterized by discontinuities or discontinuous derivatives.
The unit step function, denoted u(t), is zero for negative time values and one for positive time values, exhibiting a discontinuity at t=0. This function often represents abrupt changes, such as the step voltage introduced when turning a car's...
Observational Learning01:12

Observational Learning

Albert Bandura's observational learning, also known as imitation or modeling, occurs when a person observes and imitates another's behavior. It is a quicker process than operant conditioning. A well-known example is the Bobo doll study, where children who saw an adult acting aggressively towards the doll were more likely to act aggressively when left alone, compared to those who observed a nonaggressive adult. Many psychologists view observational learning as a form of latent learning because...
Orthogonal Trajectories01:26

Orthogonal Trajectories

Orthogonal trajectories describe the geometric relationship between two families of curves that intersect each other at right angles. One illustrative case involves a family of parabolas that open sideways along the x-axis. These curves share a common shape but differ by a scaling parameter, resulting in a set of curves that all pass through the origin and widen at different rates.Determining Orthogonal TrajectoriesTo identify the orthogonal trajectories for these parabolas, the first step...
Continuous -time Fourier Transform01:11

Continuous -time Fourier Transform

The Fourier series is instrumental in representing periodic functions, offering a powerful method to decompose such functions into a sum of sinusoids. This technique, however, necessitates modification when applied to nonperiodic functions. Consider a pulse-train waveform consisting of a series of rectangular pulses. When these pulses have a finite period, they can be accurately represented by a Fourier series. Yet, as the period approaches infinity, resulting in a single, isolated pulse, the...

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Related Experiment Videos

Learning continuous trajectories in recurrent neural networks with time-dependent weights.

M Galicki1, L Leistritz, H Witte

  • 1Institute of Medical Statistics, Computer Science and Documentation, Friedrich Schiller University, D-07740 Jena, Germany.

IEEE Transactions on Neural Networks
|February 7, 2008
PubMed
Summary

This study introduces a novel learning algorithm for recurrent neural networks using optimal control. The method efficiently trains networks with time-varying weights for trajectory learning and system tracking.

Related Experiment Videos

Area of Science:

  • Machine Learning
  • Control Theory
  • Artificial Intelligence

Background:

  • Recurrent neural networks (RNNs) are powerful tools for sequential data.
  • Training RNNs with time-varying weights and constraints presents significant challenges.
  • Existing methods often struggle with convergence and finding admissible initial solutions.

Purpose of the Study:

  • To develop a general learning algorithm for continuous trajectory learning in RNNs with arbitrary criteria and state-dependent constraints.
  • To transform the RNN learning process into an optimal control problem.
  • To propose a novel algorithm based on Pontryagin's maximum principle for efficient and optimal weight determination.

Main Methods:

  • Formulating the RNN learning problem as an optimal control framework.
  • Treating time-varying network weights as control variables.
  • Applying a variational formulation of Pontryagin's maximum principle to derive a new learning algorithm.

Main Results:

  • The proposed algorithm is demonstrated to converge to an optimal solution under reasonable conditions.
  • The method enables efficient identification of admissible initial solutions that satisfy state constraints.
  • This admissible solution serves as an effective initial guess for criterion minimization.

Conclusions:

  • The developed optimal control-based learning algorithm offers an efficient approach for training RNNs with time-varying weights.
  • The methodology is applicable to temporal sequence classification and optimal tracking of nonlinear dynamic systems.
  • Numerical examples validate the efficiency and effectiveness of the proposed approach.