  Aggregation equations are broadly used to model population dynamics with
nonlocal interactions, characterized by a potential in the equation. This paper
considers the inverse problem of identifying the potential from a single noisy
spatial-temporal process. The identification is challenging in the presence of
noise due to the instability of numerical differentiation. We propose a robust
model-based technique to identify the potential by minimizing a regularized
data fidelity term, and regularization is taken as the total variation and the
squared Laplacian. A split Bregman method is used to solve the regularized
optimization problem. Our method is robust to noise by utilizing a Successively
Denoised Differentiation technique. We consider additional constraints such as
compact support and symmetry constraints to enhance the performance further. We
also apply this method to identify time-varying potentials and identify the
interaction kernel in an agent-based system. Various numerical examples in one
and two dimensions are included to verify the effectiveness and robustness of
the proposed method.
