FPINNs Algorithm For Solving Caputo-Hadamard Time-fractional Diffusion Equation On Multiple Spatial Dimensions

Fractional Physics-Informed Neural Networks(fPINNs)is a novel deep learning solver for fractional partial differential equations that discretizes time fractional derivative operator using L1 formula and spatia.l fractional Laplacian using the Grünwald-Letnikov(GL)formula.It overcomes the problem of automatic differentiation methods not being suitable for fractional operators and extends Physics-Informed Neural Networks(PINNs)to spatial fractional problems.Compared with Riemann-Liouville and Caputo fractional derivatives,the CaputoHadamard fractional derivative has the advantage of characterizing the logarithmic creep law,which can be effectively used in the study of discrete time fractional derivatives.For time fractional diffusion equations,due to their historical dependence,it is necessary to calculate the difference quotient information of diffe rent time steps multiple times.Traditional numerical methods may face the problem of dimensionality disaster.Therefore,this paper proposes an improved fPINNs model for fast computation of high-dimensional time fractional problems.By numerically discretizing the time fractional derivative term using the L1 discretization formula of Caputo-Hadamard fractional derivative and using the original equation’s right-hand side term and boundary conditions as physical information,a computationally efficient loss function is constructed and inputed into the neural network for training.For this model,we compare the results of multiple loss functions and optimizers and conclude that the sigmoid activation function and Adam optimizer have stronger versatility and higher accuracy.In one-dimensional space,we verify that the fPINNs algorithm still has good training performance for sparse grids or randomly distributed data.To verify the model’s generalization ability,we add variable coefficients,convection terms,and fractional derivatives with oddness conditions to the equation and obtained good results.By adding acertain amount of noise pollution to the right-hand side term,the computational results still achieve a relatively low relative error.Finally,we extend these results to high-dimensional space to verify the algorithm’s effectiveness in handling highdimensional problems.