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.

对于不遵循经典扩散行为的应用，分数类型的非局部算符是一种流行的建模选择。然而，非局部模拟的一个主要挑战是模型参数的选择。在这项工作中，我们为具有可选截断半径的分数模型提出了一种基于优化的参数识别方法。我们将推理问题公式化为最佳控制问题，其目的是最大程度地减少观测数据与模型的近似解之间的差异，并且控制变量为分数阶和截断长度。对于最小化问题的数值解，我们提出了一种基于梯度的方法，其中，我们通过近似状态方程的双线性形式及其相对于分数阶的导数来增强数值性能。一维和二维的几个数值试验说明了理论结果，并表明了我们方法的鲁棒性和适用性。