The use of Physics-informed neural networks (PINNs) has shown promise in solving forward and inverse problems of fractional diffusion equations. However, due to the fact that automatic differentiation is not applicable for fractional derivatives, solving fractional diffusion equations using PINNs requires addressing additional challenges. To address this issue, this paper proposes an extension to PINNs called Laplace-based fractional physics-informed neural networks (Laplace-fPINNs), which can effectively solve the forward and inverse problems of fractional diffusion equations. This approach avoids introducing a mass of auxiliary points and simplifies the loss function. We validate the effectiveness of the Laplace-fPINNs approach using several examples. Our numerical results demonstrate that the Laplace-fPINNs method can effectively solve both the forward and inverse problems of high-dimensional fractional diffusion equations.

翻译：物理信息神经网络（PINNs）在求解分数阶扩散方程的正反问题方面展现出潜力。然而，由于自动微分不适用于分数阶导数，使用PINNs求解分数阶扩散方程需要应对额外的挑战。为解决这一问题，本文提出了一种PINNs的扩展方法——基于拉普拉斯变换的分数阶物理信息神经网络（Laplace-fPINNs），该方法能够有效求解分数阶扩散方程的正反问题。该方案避免了引入大量辅助点，并简化了损失函数。我们通过多个算例验证了Laplace-fPINNs方法的有效性。数值结果表明，Laplace-fPINNs方法能够有效求解高维分数阶扩散方程的正反问题。