  We address the inverse problem of identifying nonlocal interaction potentials
in nonlinear aggregation-diffusion equations from noisy discrete trajectory
data. Our approach involves formulating and solving a regularized variational
problem, which requires minimizing a quadratic error functional across a set of
hypothesis functions, further augmented by a sparsity-enhancing regularizer. We
employ a partial inversion algorithm, akin to the CoSaMP [57] and subspace
pursuit algorithms [31], to solve the Basis Pursuit problem. A key theoretical
contribution is our novel stability estimate for the PDEs, validating the error
functional ability in controlling the 2-Wasserstein distance between solutions
generated using the true and estimated interaction potentials. Our work also
includes an error analysis of estimators caused by discretization and
observational errors in practical implementations. We demonstrate the
effectiveness of the methods through various 1D and 2D examples showcasing
collective behaviors.
