A Constructive Approach to Function Realization by Neural Stochastic Differential Equations
This work addresses the issue of high-complexity controls in neural dynamical systems for applications, offering a constructive approach to function realization.
The paper tackles the problem of function approximation by neural dynamical systems by imposing structural restrictions on system dynamics to characterize realizable functions, using a cascade of a neural stochastic differential equation, a deterministic dynamical system, and a readout map, with methods including probabilistic and geometric approaches.
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.