论文标题
生物模式形成的平均最佳控制
Mean-field optimal control for biological pattern formation
论文作者
论文摘要
我们为给定模式的参数识别提出了一个平均场最佳控制问题。成本功能基于建模模式的概率度量与所需模式之间的余地距离。对应于最佳控制问题的一阶最佳条件是使用平均场级别的拉格朗日方法得出的。基于这些条件，我们提出了一种梯度下降方法来识别相关参数，例如旋转角度和力缩放，可能在空间上是不均匀的。我们将一阶最佳条件离散，以便在粒子水平上采用算法。此外，由于用于离散化的粒子数量倾向于无穷大，因此我们证明了对照的收敛速率。空间均匀案例的数值结果证明了该方法的可行性。
We propose a mean-field optimal control problem for the parameter identification of a given pattern. The cost functional is based on the Wasserstein distance between the probability measures of the modeled and the desired patterns. The first-order optimality conditions corresponding to the optimal control problem are derived using a Lagrangian approach on the mean-field level. Based on these conditions we propose a gradient descent method to identify relevant parameters such as angle of rotation and force scaling which may be spatially inhomogeneous. We discretize the first-order optimality conditions in order to employ the algorithm on the particle level. Moreover, we prove a rate for the convergence of the controls as the number of particles used for the discretization tends to infinity. Numerical results for the spatially homogeneous case demonstrate the feasibility of the approach.
