卷积神经网络算法与归一化倒向随机偏微分方程
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论文发表信息
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英文摘要
We develop a generic convolutional neural network (CNN) based numerical scheme to simulate the
2-tuple adapted strong solution to a unified system of backward stochastic partial differential
equations (B-SPDEs) driven by Brownian motions, which can be used to model many real-world system
dynamics such as optimal control and differential game problems. The dynamics of the scheme is
modeled by a CNN through conditional expectation projection. It consists of two convolution parts:
W layers of backward networks and $L$ layers of reinforcement iterations. Furthermore, it is a
completely discrete and iterative algorithm in terms of both time and space with mean-square error
estimation and almost sure (a.s.) convergence supported by both theoretical proof and numerical
examples. In doing so, we need to prove the unique existence of the 2-tuple adapted strong solution
to the system under both conventional and Malliavin derivatives with general local Lipschitz and
linear growth conditions.
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中文简介
人工智能卷积神经网络在现代科技中有重要应用并产生重大影响，但其背后的数学机制与理论是什么？这个问题
一直是国际上关注的热点，也是各个国家科学基金资助的焦点。该文在国际上作有关特邀大会嘉宾主旨报告及学
术交流时，也是各国科学家重点提问与关注的问题。本质上，人工智能卷积神经网络是如何确定数据集及如何基
于该数据集构造各种线性或非线性模型而产生的，这在数学上的本质表现就是我们文中介绍的条件数学期望在给
定条件空间上的投影及相应的线性或非线性基的构造问题。当然，具体算法会与倒向随机偏微分方程交互。
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关键词
AI based Monte Carlo simulation, convolution neural network modeling, mean-square and almost sure
convergence, backward stochastic partial differential equation, Cauchy terminal value problem,
random field Malliavin calculus
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国际会议特邀大会嘉宾主旨报告(Keynote Speeches)
- 该文中的主要学术成就曾作为特邀大会嘉宾主旨报告在国际会议上报告。
- The main results in the paper are presented as invited keynote talks in
international conferences.
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