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Physics-Informed Deep Ai Simulation For Fractal Integro-Differential Equation

Author

Listed:
  • XUEJUAN LI

    (School of Science, Xi’an University of Architecture and Technology, Xi’an, P. R. China)

  • RUI ZHAO

    (School of Science, Xi’an University of Architecture and Technology, Xi’an, P. R. China)

Abstract

Fractal integro-differential equations (IDEs) can describe the effect of local microstructure on a complex physical problem, however, the traditional numerical methods are not suitable for solving the new-born models with the fractal integral and fractal derivative. Here we show that deep learning can be used to solve the bottleneck. By the two-scale transformation, the fractal IDE is first approximately converted to its traditional integro-differential partner, which is further converted to a differential equation system by introducing an auxiliary variable to remove the integral operation. Moreover, a flexible adaptive technology is adopted to deal with the loss weights of a deep learning neural network. A fractal Volterra IDE is used to show the effectiveness and simplicity of this new physics-informed deep AI simulation model. All results indicate the AI simulation model has good robustness and convergence, and the fractal Volterra IDE might explore the different properties of viscoelasticity for a porous medium.

Suggested Citation

  • Xuejuan Li & Rui Zhao, 2024. "Physics-Informed Deep Ai Simulation For Fractal Integro-Differential Equation," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 32(01), pages 1-8.
    • Handle: RePEc:wsi:fracta:v:32:y:2024:i:01:n:s0218348x24500221
    DOI: 10.1142/S0218348X24500221
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