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

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.