DeepSPoC：基于深度学习的混沌序列传播
DeepSPoC: A Deep Learning Based Sequential Propagation of Chaos
AI总结：
针对高维平均场问题中粒子方法计算成本高的问题，提出DeepSPoC神经粒子方法，利用神经网络表示粒子分布，以粒子-网络交互替代粒子-粒子交互，降低内存并提升可扩展性。
摘要：
基于混沌传播（PoC）的经典粒子方法已被开发用于求解平均场随机微分方程及其相关的非线性福克-普朗克方程。然而，直接的PoC实现难以应用于高维问题，因为它们需要模拟和存储大量相互作用的粒子，且粒子间相互作用成本往往很高。受这些局限性的推动，我们基于最近提出的混沌序列传播（SPoC）框架，该框架用序列相互作用机制取代了PoC中完全相互作用的粒子系统。基于这一结构，我们提出了DeepSPoC，一种将神经密度表示嵌入序列粒子动力学的神经粒子方法。DeepSPoC逐批模拟粒子，同时神经网络表示演化的经验定律，并将其代入平均场SDE的系数中，从而用粒子-网络相互作用取代直接的粒子-粒子相互作用。在DeepSPoC中，使用最近开发的称为KRnet的正则化流模型来近似粒子的经验测度。与直接粒子实现相比，DeepSPoC大幅降低了内存消耗，并更高效地评估相互作用项，从而提高了高维问题的可扩展性。我们将DeepSPoC应用于广泛的平均场方程，并验证了其有效性和计算优势。
英文摘要：
Classical particle methods based on propagation of chaos (PoC) have been developed for solving mean-field stochastic differential equations and their associated nonlinear Fokker--Planck equations. However, direct PoC implementations are difficult to apply to high-dimensional problems because they require simulating and storing large numbers of interacting particles, often with high particle-particle interaction costs. Motivated by these limitations, we build on the recently proposed sequential propagation of chaos (SPoC) framework, which replaces the fully interacting particle system in PoC with a sequential interaction mechanism. Based on this structure, we present DeepSPoC, a neural particle method that embeds a neural density representation into the sequential particle dynamics. DeepSPoC simulates particles batch by batch, while the neural network represents the evolving empirical law and is substituted into the coefficients of the mean-field SDE, thereby replacing direct particle-particle interactions with particle-network interactions. In DeepSPoC, a recently developed normalizing flow model called KRnet is used to approximate the empirical measure of particles. Compared with direct particle implementations, DeepSPoC substantially reduces memory consumption and evaluates interaction terms more efficiently, thereby improving scalability for high-dimensional problems. We apply DeepSPoC to a wide range of mean-field equations and verify its effectiveness and computational advantages.
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