论文标题
基于迭代观点的神经网络的Fokker-Planck方程的严格理由
Rigorous justification of the Fokker-Planck equations of neural networks based on an iteration perspective
论文作者
论文摘要
在这项工作中，主要目标是建立神经网络的Fokker-Planck方程与微观模型之间的严格联系：扩散 - 跳跃随机过程，该过程捕获了集成与传火模型中神经元集合的平均田间行为。证明基于一种新颖的迭代方案：具有辅助随机变量来计算点火事件，随机过程的密度函数和PDE问题允许级数表示序列表示的解决方案，因此在每个子问题中验证每个子问题中的密度函数和PDE解决方案之间的链接的难度得到了极大的减轻。迭代方法为将概率方法与PDE技术整合在一起提供了通用框架，我们证明，扩散 - 跳跃随机过程的密度函数确实是Fokker-Planck方程的经典解，具有独特的磁通结构。
In this work, the primary goal is to establish rigorous connection between the Fokker-Planck equation of neural networks with its microscopic model: the diffusion-jump stochastic process that captures the mean field behavior of collections of neurons in the integrate-and-fire model. The proof is based on a novel iteration scheme: with an auxiliary random variable counting the firing events, both the density function of the stochastic process and the solution of the PDE problem admit series representations, and thus the difficulty in verifying the link between the density function and the PDE solution in each sub problem has been greatly mitigated. The iteration approach provides a generic frame in integrating the probability approach with PDE techniques, with which we prove that the density function of the diffusion-jump stochastic process is indeed the classical solution of the Fokker-Planck equation with a unique flux-shift structure.
