As physics-informed neural networks (PINNs) continue to evolve, the limitations of traditional PINNs are becoming increasingly apparent. One major drawback is their reliance on automatic differentiation, which imposes constraints on solving partial differential equations, especially for those with fractional derivatives. For fractional-order equations, automatic differentiation cannot be directly applied, making it a challenge to use PINNs effectively. To address this, we propose a novel Laplace based physics-informed neural networks (L-PINNs) that combines PINNs with the Laplace transform to solve the coefficient inverse problem for acoustic equation with fractional derivative. This approach transforms fractional derivatives, enabling efficient handling of time-fractional inverse problems. Moreover, the nonlinear coefficient functions can be determined efficiently by the L-PINNs for the inverse problem of acoustic equations or time-fractional differential equations. We validate the effectiveness of the L-PINNs method by solving the one-dimensional and two-dimensional coefficient inverse problem for the acoustic equation with a fractional derivative, demonstrating its ability to solve the time-fractional coefficient inverse problem efficiently.
Chen et al. (Tue,) studied this question.