Universal approximation of flows of control systems by recurrent neural networks

TL;DR

This paper proves that discrete-time recurrent neural networks can universally approximate the flow functions of continuous-time dynamical systems with inputs, enabling practical learning from sampled data.

Contribution

It establishes the universal approximation capability of discrete-time RNNs for continuous-time system flows, extending prior continuous-time results.

Findings

Discrete-time RNNs can approximate continuous-time flows.

Assumptions hold for well-behaved ODE systems.

Facilitates learning from sampled trajectory data.

Abstract

We consider the problem of approximating flow functions of continuous-time dynamical systems with inputs. It is well-known that continuous-time recurrent neural networks are universal approximators of this type of system. In this paper, we prove that an architecture based on discrete-time recurrent neural networks universally approximates flows of continuous-time dynamical systems with inputs. The required assumptions are shown to hold for systems whose dynamics are well-behaved ordinary differential equations and with practically relevant classes of input signals. This enables the use of off-the-shelf solutions for learning such flow functions in continuous-time from sampled trajectory data.