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Adaptive Learning Neural Network Method for Solving Time-Fractional Diffusion Equations.

Babak Shiri1, Hua Kong2, Guo-Cheng Wu3

  • 1Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, PRC shiri@njtc.edu.cn.

Neural Computation
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Summary
This summary is machine-generated.

This study introduces a novel neural network approach for solving fractional diffusion equations using an adaptive gradient descent method. The technique simplifies complex gradient calculations, demonstrating effectiveness in numerical examples.

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Artificial Intelligence

Background:

  • Fractional diffusion equations model complex phenomena with memory effects.
  • Solving these equations analytically is often challenging.
  • Neural networks offer a promising alternative for complex differential equations.

Purpose of the Study:

  • To present a novel neural network method for solving fractional diffusion equations.
  • To propose an adaptive gradient descent method for minimizing energy functions.
  • To address the complexity of gradient calculations in fractional calculus.

Main Methods:

  • A neural network architecture is employed.
  • An adaptive gradient descent algorithm is utilized for optimization.
  • A simplified approach is developed to handle the intricate gradients arising from fractional calculus memory effects.

Main Results:

  • The proposed method effectively solves fractional diffusion equations.
  • Numerical examples validate the method's performance.
  • Both one-layer and two-layer neural networks demonstrate the technique's efficacy.

Conclusions:

  • The developed neural network method provides an effective solution for fractional diffusion equations.
  • The simplified gradient calculation is crucial for practical implementation.
  • This approach holds potential for advancing computational methods in fractional calculus.