Solving the inverse Source Problems for wave equation with final time measurements by a data driven approach

This paper develops a discrete data-driven approach for solving the inverse source problem of the wave equation with final time measurements. Focusing on the $L^2$-Tikhonov regularization method, we analyze its convergence under two different noise models, using noisy discrete spatial observations. By exploiting the spectral decomposition of the forward operator and introducing a noise separation technique into the variational framework, we establish error bounds for the reconstructed solution $u$ and the source term $f$ without requiring classical source conditions. Moreover, an expected convergence rate for the source error is derived in a weaker topology. We also extend the analysis to the fully discrete case with finite element discretization, showing that the overall error depends only on the noise level, regularization parameter, time step size, and spatial mesh size. These estimates provide a basis for selecting the optimal regularization parameter in a data-driven manner, without a priori information. Numerical experiments validate the theoretical results and demonstrate the efficiency of the proposed algorithm.