We study a new variant of mathematical prediction-correction model for crowd motion. The prediction phase is handled by a transport equation where the vector field is computed via an eikonal equation $\Vert \nabla\varphi\Vert=f$, with a positive continuous function $f$ connected to the speed of the spontaneous travel. The correction phase is handled by a new version of the minimum flow problem. This model is flexible and can take into account different types of interactions between the agents, from gradient flow in Wassersetin space to granular type dynamics like in sandpile. Furthermore, different boundary conditions can be used, such as non-homogeneous Dirichlet (e.g., outings with different exit-cost penalty) and Neumann boundary conditions (e.g., entrances with different rates). Combining finite volume method for the transport equation and Chambolle-Pock's primal dual algorithm for the eikonal equation and minimum flow problem, we present numerical simulations to demonstrate the behavior in different scenarios.

翻译：我们研究了一种新的群体运动数学预测-校正模型变体。预测阶段通过传输方程实现，其中向量场通过程函方程$\Vert \nabla\varphi\Vert=f$计算，$f$为与自发运动速度相关的正连续函数。校正阶段通过最小流问题的新版本处理。该模型具有灵活性，能够考虑代理之间不同形式的相互作用，从Wasserstein空间中的梯度流到类似沙堆的颗粒型动力学。此外，可应用不同边界条件，如非齐次Dirichlet边界（例如具有不同出口成本惩罚的疏散）和Neumann边界条件（例如具有不同进入速率的入口）。通过结合传输方程的有限体积法、程函方程和最小流问题的Chambolle-Pock原始对偶算法，我们展示了不同场景下的数值模拟行为。