Solving an inverse source problem for nonlocal diffusion-wave equations through Laplace-based physics-informed neural networks

This work investigates the nonlocal diffusion-wave equation governed by the time-fractional Caputo derivative. We first establish that the solution is $ t $-analytic through the application of Fourier expansion and the properties of the Mittag-Leffler functions. Building on this, we apply the Laplace transform to demonstrate the subordination principle for solutions of parabolic and hyperbolic equations in the context of the nonlocal diffusion-wave equation. Second, we prove the uniqueness and conditional stability of a solution to an inverse problem involving the determination of the spatially varying source term based on interior information from a subdomain. We alslo introduce a novel framework termed Laplace-based physics-informed neural networks (L-PINNs), which is tailored for determining source terms in nonlocal diffusion-wave systems. We substantiate the proposed approach through a series of numerical experiments, demonstrating its superior accuracy and computational efficiency.