Data-Driven Reduced Modeling of Delayed Dynamical Systems via Spectral Submanifolds

TL;DR

This paper extends spectral submanifold theory to delay differential equations, enabling data-driven reduction of nonlinear delay systems into low-dimensional models that are predictive even in chaotic regimes.

Contribution

It introduces a data-driven SSM reduction method for delay systems that requires no prior knowledge of the system's delay structure or form.

Findings

Data-driven SSM models accurately reduce delay systems without prior delay information.

Reduced models remain predictive in chaotic regimes.

Parametric SSM reduction captures bifurcations in complex delay systems.

Abstract

We show how the recent extension of spectral submanifold (SSM) theory to delay differential equations (DDEs) enables data-driven model reduction of nonlinear delay systems. First, using a scalar DDE with a single discrete delay, we compare equation-based and data-driven SSM reductions, to illustrate the need for the latter. We then use the same algorithm to obtain purely data-driven, SSM-reduced, delay-free ODE models for several nonlinear delayed systems. Our approach requires no information about the form of the underlying DDE, or about the number and magnitude of the delays it contains. Our SSM-reduced, low-dimensional models remain predictive even for chaotic dynamics. We also illustrate the use of parametric SSM-reduction to capture bifurcations in systems with both distributed and discrete delays. Finally we extend the theoretical underpinning of delayed SSM-reductions to non-autonomous systems with periodic delays, and apply these results to experimental data from a control system with feedback delay and quantization.