  Nonlocal operators of fractional type are a popular modeling choice for
applications that do not adhere to classical diffusive behavior; however, one
major challenge in nonlocal simulations is the selection of model parameters.
In this work we propose an optimization-based approach to parameter
identification for fractional models with an optional truncation radius. We
formulate the inference problem as an optimal control problem where the
objective is to minimize the discrepancy between observed data and an
approximate solution of the model, and the control variables are the fractional
order and the truncation length. For the numerical solution of the minimization
problem we propose a gradient-based approach, where we enhance the numerical
performance by an approximation of the bilinear form of the state equation and
its derivative with respect to the fractional order. Several numerical tests in
one and two dimensions illustrate the theoretical results and show the
robustness and applicability of our method.
