| Abstract: |
| Physics-informed Neural Networks (PINNs) have emerged as a popular method for solving both forward and inverse differential equations. However, the automatic differentiation techniques employed by PINNs face challenges when solving fractional-order equations. To address this issue, we propose the spectral method based fractional PINNs, termed spectral-fPINNs. This method adopts a more efficient global discretization approach based on Jacobi polynomials, which reduces the need for auxiliary points. Meanwhile, the transformation between physical value and expansion coefficients is computed efficiently by a standard matrix-vector multiplication, thereby increasing the efficiency of the algorithm. The performance of spectral-fPINNs is validated via several examples. We first consider the accuracy, stability and efficiency of our method for solving steady-state fractional partial differential equations. We also analyze the errors under different parameters. Subsequently, experiments are conducted on more complex time-dependent equations. Additionally, an application of the tempered equations on finance and the inverse problems are presented. These results demonstrate the advantages of spectral-fPINNs in terms of efficiency in solving tempered fractional partial differential equations. |
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