In the present work, we develop a novel particle method for a general class of mean field control problems, with source and terminal constraints. Specific examples of the problems we consider include the dynamic formulation of the p-Wasserstein metric, optimal transport around an obstacle, and measure transport subject to acceleration controls. Unlike existing numerical approaches, our particle method is meshfree and does not require global knowledge of an underlying cost function or of the terminal constraint. A key feature of our approach is a novel way of enforcing the terminal constraint via a soft, nonlocal approximation, inspired by recent work on blob methods for diffusion equations. We prove convergence of our particle approximation to solutions of the continuum mean-field control problem in the sense of Gamma-convergence. A byproduct of our result is an extension of existing discrete-to-continuum convergence results for mean field control problems to more general state and measure costs, as arise when modeling transport around obstacles, and more general constraint sets, including controllable linear time invariant systems. Finally, we conclude by implementing our method numerically and using it to compute solutions the example problems discussed above. We conduct a detailed numerical investigation of the convergence properties of our method, as well as its behavior in sampling applications and for approximation of optimal transport maps.

翻译：本文针对一类具有源项和终端约束的通用均值场控制问题，提出了一种新颖的粒子方法。我们考虑的具体问题包括p- Wasserstein度量的动态公式、障碍物周围的最优传输以及受加速度控制约束的测度传输。与现有数值方法不同，我们的粒子方法无需网格，也不需要全局已知的代价函数或终端约束信息。该方法的核心创新在于：受近期扩散方程斑点方法研究的启发，我们通过软非局部近似来施加终端约束。我们证明了该粒子近似在伽马收敛意义下收敛到连续均值场控制问题的解。作为副产品，我们的结果将现有的均值场控制问题离散到连续收敛结果扩展至更一般的状态与测度代价（例如在障碍物传输建模中出现的情况）以及更一般的约束集（包括可控线性时不变系统）。最后，我们通过数值实验验证了该方法，并计算了上述实例问题的解。我们对该方法的收敛性及其在采样应用和最优传输映射逼近中的表现进行了详细的数值研究。