Recurrent Neural Networks for Partially Observed Dynamical Systems

TL;DR

This paper introduces an algebraic delay embedding method that can be implemented with RNNs, improving interpretability and enabling structured modeling of complex nonlinear dynamical systems with unobserved variables.

Contribution

It provides a novel algebraic approach to delay embedding with explicit error approximation, linking delay embedding theory with RNN implementation.

Findings

Explicit error approximation for delay embedding

Asymptotic error dependence on system size

RNN implementation enhances interpretability

Abstract

Complex nonlinear dynamics are ubiquitous in many fields. Moreover, we rarely have access to all of the relevant state variables governing the dynamics. Delay embedding allows us, in principle, to account for unobserved state variables. Here we provide an algebraic approach to delay embedding that permits explicit approximation of error. We also provide the asymptotic dependence of the first order approximation error on the system size. More importantly, this formulation of delay embedding can be directly implemented using a Recurrent Neural Network (RNN). This observation expands the interpretability of both delay embedding and RNN and facilitates principled incorporation of structure and other constraints into these approaches.