标题： 用深度学习粒子法求解McKean-Vlasov方程 显示英文标题

标题： Solving McKean-Vlasov Equation by deep learning particle method

摘要： 我们介绍了一种基于深度学习的新型无网格模拟方法，用于求解麦凯恩-弗拉索夫随机微分方程（MV-SDE），适用于自相互作用和相互作用场景。传统上，该方程的数值方法依赖于交互粒子法与基于伊藤-泰勒展开的技术相结合。这种方法的收敛率由两个参数决定：每次欧拉迭代的粒子数 $N$ 和时间步长 $h$。然而，对于长时间范围或具有较大利普希茨系数的方程，这种方法通常受到限制，因为它需要显著增加欧拉迭代次数以达到期望的精度 $\epsilon$。为了解决连续交互粒子系统模拟难以并行化的挑战，这些系统涉及求解高维耦合SDE，我们提出了一种基于物理信息神经网络（PINNs）的无网格MV-SDE求解器，无需依赖传播混沌结果。我们的方法利用伊藤微积分构造一个伪MV-SDE，然后量化此方程与原始MV-SDE之间的差异，通过损失函数最小化误差。该损失通过一种独立于时间步长的优化算法控制，并且我们提供了损失函数的误差估计。通过相应的仿真验证了我们方法的优势。 显示英文摘要

摘要： We introduce a novel meshless simulation method for the McKean-Vlasov Stochastic Differential Equation (MV-SDE) utilizing deep learning, applicable to both self-interaction and interaction scenarios. Traditionally, numerical methods for this equation rely on the interacting particle method combined with techniques based on the It\^o-Taylor expansion. The convergence rate of this approach is determined by two parameters: the number of particles $N$ and the time step size $h$ for each Euler iteration. However, for extended time horizons or equations with larger Lipschitz coefficients, this method is often limited, as it requires a significant increase in Euler iterations to achieve the desired precision $\epsilon$. To overcome the challenges posed by the difficulty of parallelizing the simulation of continuous interacting particle systems, which involve solving high-dimensional coupled SDEs, we propose a meshless MV-SDE solver grounded in Physics-Informed Neural Networks (PINNs) that does not rely on the propagation of chaos result. Our method constructs a pseudo MV-SDE using It\^o calculus, then quantifies the discrepancy between this equation and the original MV-SDE, with the error minimized through a loss function. This loss is controlled via an optimization algorithm, independent of the time step size, and we provide an error estimate for the loss function. The advantages of our approach are demonstrated through corresponding simulations.