PHYSICS-INFORMED DEEP AI SIMULATION FOR FRACTAL INTEGRO-DIFFERENTIAL EQUATION

Xuejuan Li and Rui Zhao
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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

FRACTALS (fractals), 2024, vol. 32, issue 01, 1-8

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.

Keywords: Machine Learning; Physics-Informed Deep Simulation; Two-Scale Fractal Theory; Fractal Differential; Fractal Integral (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1142/S0218348X24500221

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