A Constructive Approach to Function Realization by Neural Stochastic Differential Equations

TL;DR

This paper introduces a constructive method for realizing functions using neural stochastic differential equations with structural restrictions, characterizing the functions achievable under these constraints.

Contribution

It presents a novel constructive approach that characterizes function classes realizable by neural SDE-based systems with imposed structural restrictions.

Findings

Characterizes functions realizable by neural SDE systems.

Uses probabilistic and geometric methods for analysis.

Provides a framework for practical, restricted neural dynamical systems.

Abstract

The problem of function approximation by neural dynamical systems has typically been approached in a top-down manner: Any continuous function can be approximated to an arbitrary accuracy by a sufficiently complex model with a given architecture. This can lead to high-complexity controls which are impractical in applications. In this paper, we take the opposite, constructive approach: We impose various structural restrictions on system dynamics and consequently characterize the class of functions that can be realized by such a system. The systems are implemented as a cascade interconnection of a neural stochastic differential equation (Neural SDE), a deterministic dynamical system, and a readout map. Both probabilistic and geometric (Lie-theoretic) methods are used to characterize the classes of functions realized by such systems.