Bridging Autoencoders and Dynamic Mode Decomposition for Reduced-order Modeling and Control of PDEs

TL;DR

This paper introduces a deep autoencoding approach that bridges linear autoencoders and dynamic mode decomposition, enabling efficient nonlinear reduced-order modeling and control of PDE-driven systems, validated on a reaction-diffusion example.

Contribution

It analytically connects linear autoencoders with DMDc and extends to deep autoencoders for nonlinear PDE systems, incorporating neural network-based control design.

Findings

Deep autoencoding models closely match DMDc solutions.

The approach effectively models complex PDE systems.

Controllers designed with stability constraints improve system regulation.

Abstract

Modeling and controlling complex spatiotemporal dynamical systems driven by partial differential equations (PDEs) often necessitate dimensionality reduction techniques to construct lower-order models for computational efficiency. This paper explores a deep autoencoding learning method for reduced-order modeling and control of dynamical systems governed by spatiotemporal PDEs. We first analytically show that an optimization objective for learning a linear autoencoding reduced-order model can be formulated to yield a solution closely resembling the result obtained through the dynamic mode decomposition with control algorithm. We then extend this linear autoencoding architecture to a deep autoencoding framework, enabling the development of a nonlinear reduced-order model. Furthermore, we leverage the learned reduced-order model to design controllers using stability-constrained deep neural networks. Numerical experiments are presented to validate the efficacy of our approach in both modeling and control using the example of a reaction-diffusion system.