用深度学习粒子方法求解 McKean-Vlasov 方程
Solving McKean-Vlasov Equation by deep learning particle method
AI总结：
本文提出一种基于物理信息神经网络的无网格深度学习方法求解 McKean-Vlasov 随机微分方程，通过构造伪 MV-SDE 并优化损失函数，避免传统粒子方法对时间步长和粒子数的依赖，实现高维耦合系统的可并行高效模拟。
摘要：
我们引入一种利用深度学习的新型无网格模拟方法，用于求解 McKean-Vlasov 随机微分方程（MV-SDE），适用于自相互作用和相互作用两种情形。传统上，该方程的数值方法依赖于相互作用粒子方法，并结合基于 Itô-Taylor 展开的技术。该方法的收敛速度由两个参数决定：粒子数 $N$ 和每次 Euler 迭代的时间步长 $h$。然而，对于较长的时间范围或具有较大 Lipschitz 系数的方程，这种方法往往受到限制，因为要达到期望精度 $ε$，需要显著增加 Euler 迭代次数。连续相互作用粒子系统的模拟涉及求解高维耦合 SDE，难以并行化，为克服由此带来的挑战，我们提出一种基于物理信息神经网络（PINNs）的无网格 MV-SDE 求解器，该方法不依赖于混沌传播结果。我们的方法利用 Itô 微积分构造一个伪 MV-SDE，然后量化该方程与原始 MV-SDE 之间的差异，并通过损失函数将误差最小化。该损失通过优化算法控制，与时间步长无关，并且我们给出了损失函数的误差估计。相应模拟展示了我们方法的优势。
英文摘要：
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 Itô-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 $ε$. 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 Itô 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.
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