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There has been a great deal of recent interest in learning and approximationof functions that can be expressed as expectations of a given nonlinearity withrespect to its random internal parameters. Examples of such representationsinclude "infinitely wide" neural nets, where the underlying nonlinearity isgiven by the activation function of an individual neuron. In this paper, webring this perspective to function representation by neural stochasticdifferential equations (SDEs). A neural SDE is an It\^o diffusion process whosedrift and diffusion matrix are elements of some parametric families. We showthat the ability of a neural SDE to realize nonlinear functions of its initialcondition can be related to the problem of optimally steering a certaindeterministic dynamical system between two given points in finite time. Thisauxiliary system is obtained by formally replacing the Brownian motion in theSDE by a deterministic control input. We derive upper and lower bounds on theminimum control effort needed to accomplish this steering; these bounds may beof independent interest in the context of motion planning and deterministicoptimal control.

Nonlinear controllability and function representation by neural stochastic differential equations