On the Recovery of Source Term for Fractional Evolution PDEs by MC-fPINNs

In this paper, we solve the inverse source problem of fractional evolution PDEs by MC-fPINNs. We construct the loss function in terms of the governing equation residual, boundary residual, initial residual and measurement data with noise. Meanwhile, we present a rigorous error analysis of this method. In the experimental section, we present the reconstruction outcomes of the source term for three evolutionary fractional partial differential equations (fPDEs): the evolutionary fractional Laplacian equation, the time-space fractional diffusion equation, and the fractional advection-diffusion equation. These experiments illustrate robust performance of MC-fPINNs in both low-dimensional and high-dimensional scenarios. Our results confirm the effectiveness of MC-fPINNs in solving such inverse source problem, and also provide a theoretical foundation to choose neural networks parameters in this algorithm.