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<p>Forward-inverse problems of high-dimensional time-fractional reaction-diffusion (TFRD) equations are investigated in this paper, focusing on data-driven solutions and parameter estimation. A fast L1-fractional physics-informed neural network (FL1-fPINN) method on graded meshes is proposed. First, the neural network is constructed and used to approximate the solutions of TFRD equations. The Caputo fractional derivative is numerically discretized by using fast L1 approximation on graded meshes. The adaptive activation function is employed to improve the training efficiency of FL1-fPINN. Then initial-boundary conditions (IBCs) as hard and soft constraints are imposed in the loss function. An optimization problem is provided based on the discretization scheme. Finally, by solving the problem and training the neural network, the FL1-fPINN solutions gradually converge to the analytical solutions of the TFRD equations. The numerical experiments with comprehensive sensitivity analysis are given. By solving forward, inverse and noisy problems of the TFRD equations, numerical results demonstrate the remarkable time-saving capabilities and robustness of the proposed method.</p>

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Data-driven solutions and parameter estimation of the high-dimensional time-fractional reaction-diffusion equations using an improved fPINN method

  • 期刊论文

  • 发表日期：

    2025年1月17日

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  • Jieyu Shi,
  • Xinlong Liu,
  • Xiaozhong Yang

摘要

Forward-inverse problems of high-dimensional time-fractional reaction-diffusion (TFRD) equations are investigated in this paper, focusing on data-driven solutions and parameter estimation. A fast L1-fractional physics-informed neural network (FL1-fPINN) method on graded meshes is proposed. First, the neural network is constructed and used to approximate the solutions of TFRD equations. The Caputo fractional derivative is numerically discretized by using fast L1 approximation on graded meshes. The adaptive activation function is employed to improve the training efficiency of FL1-fPINN. Then initial-boundary conditions (IBCs) as hard and soft constraints are imposed in the loss function. An optimization problem is provided based on the discretization scheme. Finally, by solving the problem and training the neural network, the FL1-fPINN solutions gradually converge to the analytical solutions of the TFRD equations. The numerical experiments with comprehensive sensitivity analysis are given. By solving forward, inverse and noisy problems of the TFRD equations, numerical results demonstrate the remarkable time-saving capabilities and robustness of the proposed method.