Universal Time-Uniform Trajectory Approximation for Random Dynamical Systems with Recurrent Neural Networks

TL;DR

This paper proves that certain recurrent neural networks can uniformly approximate trajectories of random dynamical systems over an infinite time horizon, even on non-compact domains, with simple feedback structures.

Contribution

It establishes universal approximation capabilities of deep recurrent neural networks for infinite-horizon stochastic trajectories on non-compact spaces, extending prior finite-time results.

Findings

Recurrent neural networks can approximate random trajectories uniformly over infinite time.

The approximation holds on non-compact domains with mild, natural conditions.

The proof is simple and contrasts with existing literature limited to finite intervals.

Abstract

The capability of recurrent neural networks to approximate trajectories of a random dynamical system, with random inputs, on non-compact domains, and over an indefinite or infinite time horizon is considered. The main result states that certain random trajectories over an infinite time horizon may be approximated to any desired accuracy, uniformly in time, by a certain class of deep recurrent neural networks, with simple feedback structures. The formulation here contrasts with related literature on this topic, much of which is restricted to compact state spaces and finite time intervals. The model conditions required here are natural, mild, and easy to test, and the proof is very simple.