论文标题
神经随机微分方程的非线性可控性和功能表示
Nonlinear controllability and function representation by neural stochastic differential equations
论文作者
论文摘要
在学习和近似功能的近似方面，人们对特定非线性的期望就其随机内部参数表示了极大的兴趣。这种表示的例子包括“无限宽”的神经网，其中潜在的非线性由单个神经元的激活函数给出。在本文中，我们将此视角带入了通过神经随机微分方程（SDE）来表示功能表示。神经SDE是一个ITô扩散过程，其漂移和扩散矩阵是某些参数家族的要素。我们表明，神经SDE实现其初始条件的非线性函数的能力可能与在有限的时间中在两个给定点之间在两个给定点之间进行最佳指导的问题有关。通过确定性控制输入正式替换SDE中的布朗运动，可以获得此辅助系统。我们在完成此转向所需的最低控制工作方面得出了上限和下限。在运动计划和确定性最佳控制的背景下，这些界限可能具有独立的兴趣。
There has been a great deal of recent interest in learning and approximation of functions that can be expressed as expectations of a given nonlinearity with respect to its random internal parameters. Examples of such representations include "infinitely wide" neural nets, where the underlying nonlinearity is given by the activation function of an individual neuron. In this paper, we bring this perspective to function representation by neural stochastic differential equations (SDEs). A neural SDE is an Itô diffusion process whose drift and diffusion matrix are elements of some parametric families. We show that the ability of a neural SDE to realize nonlinear functions of its initial condition can be related to the problem of optimally steering a certain deterministic dynamical system between two given points in finite time. This auxiliary system is obtained by formally replacing the Brownian motion in the SDE by a deterministic control input. We derive upper and lower bounds on the minimum control effort needed to accomplish this steering; these bounds may be of independent interest in the context of motion planning and deterministic optimal control.
