论文标题
基于优化的分数类型非本地模型的参数学习方法
An optimization-based approach to parameter learning for fractional type nonlocal models
论文作者
论文摘要
分数类型的非局部运算符是不遵守经典扩散行为的应用的流行建模选择；但是，非局部模拟中的一个主要挑战是模型参数的选择。在这项工作中，我们提出了一种基于优化的方法来识别具有可选截断半径的分数模型的参数识别方法。我们将推断问题提出为最佳控制问题，其中目标是最大程度地减少观察到的数据与模型的近似解决方案之间的差异，而控制变量是分数顺序和截断长度。对于最小化问题的数值解决方案，我们提出了一种基于梯度的方法，在其中，我们通过近似状态方程的双线性形式及其相对于分数顺序的衍生物来增强数值性能。一个和二维中的几个数值测试说明了理论结果，并显示了我们方法的鲁棒性和适用性。
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.
