Nonlinear controllability and function representation by neural stochastic differential equations

TL;DR

This paper explores how neural stochastic differential equations can represent nonlinear functions and relates their controllability to optimal steering problems, providing bounds relevant to control theory and function approximation.

Contribution

It introduces a novel perspective on neural SDEs for function representation and links their nonlinear capabilities to deterministic control problems, deriving bounds on control effort.

Findings

Neural SDEs can realize nonlinear functions of initial conditions.

Controllability of neural SDEs relates to optimal steering of deterministic systems.

Bounds on control effort are derived for steering neural SDEs between states.

Abstract

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 It\^o 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.