In this paper, we establish that for a wide class of controlled stochastic
differential equations (SDEs) with stiff coefficients, the value functions of
corresponding zero-sum games can be represented by a deep artificial neural
network (DNN), whose complexity grows at most polynomially in both the
dimension of the state equation and the reciprocal of the required accuracy.
Such nonlinear stiff systems may arise, for example, from Galerkin
approximations of controlled stochastic partial differential equations (SPDEs),
or controlled PDEs with uncertain initial conditions and source terms. This
implies that DNNs can break the curse of dimensionality in numerical
approximations and optimal control of PDEs and SPDEs. The main ingredient of
our proof is to construct a suitable discrete-time system to effectively
approximate the evolution of the underlying stochastic dynamics. Similar ideas
can also be applied to obtain expression rates of DNNs for value functions
induced by stiff systems with regime switching coefficients and driven by
general L\'{e}vy noise.

该研究表明，对于具有强刚性系数的广泛控制随机微分方程，相关零和博弈的价值函数可以由深度人工神经网络来表示，该网络的复杂度在状态方程的维度和所需精度的倒数方面都呈多项式增长。

修正的深度神经网络克服非线性僵硬系统的零和博弈中非平滑价值函数的维度诅咒

Rectified deep neural networks overcome the curse of dimensionality for  nonsmooth value functions in zero-sum games of nonlinear stiff systems

Rough path theory is focused on capturing and making precise the interactions
between highly oscillatory and non-linear systems. It draws on the analysis of
LC Young and the geometric algebra of KT Chen. The concepts and the uniform
estimates, have widespread application and have simplified proofs of basic
questions from the large deviation theory and extended Ito's theory of SDEs;
the recent applications contribute to (Graham) automated recognition of Chinese
handwriting and (Hairer) formulation of appropriate SPDEs to model randomly
evolving interfaces. At the heart of the mathematics is the challenge of
describing a smooth but potentially highly oscillatory and vector valued path
$x_{t}$ parsimoniously so as to effectively predict the response of a nonlinear
system such as $dy_{t}=f(y_{t})dx_{t}$, $y_{0}=a$. The Signature is a
homomorphism from the monoid of paths into the grouplike elements of a closed
tensor algebra. It provides a graduated summary of the path $x$. Hambly and
Lyons have shown that this non-commutative transform is faithful for paths of
bounded variation up to appropriate null modifications. Among paths of bounded
variation with given Signature there is always a unique shortest
representative. These graduated summaries or features of a path are at the
heart of the definition of a rough path; locally they remove the need to look
at the fine structure of the path. Taylor's theorem explains how any smooth
function can, locally, be expressed as a linear combination of certain special
functions (monomials based at that point). Coordinate iterated integrals form a
more subtle algebra of features that can describe a stream or path in an
analogous way; they allow a definition of rough path and a natural linear
"basis" for functions on streams that can be used for machine learning.

粗糙路径理论通过均匀估计来简化对高度振荡和非线性系统之间交互的描述，该理论具有广泛的应用，并通过坐标迭代积分可描述路径的复杂细节，并为机器学习提供了天然的线性基础。