Bridging Autoencoders and Dynamic Mode Decomposition for Reduced-order Modeling and Control of PDEs
This work addresses the need for efficient dimensionality reduction in PDE-driven systems for computational modeling and control, presenting an incremental advancement by bridging autoencoders and dynamic mode decomposition.
The paper tackles the problem of modeling and controlling complex spatiotemporal dynamical systems governed by PDEs by developing a deep autoencoding method for reduced-order modeling and control, showing that a linear autoencoding objective yields results similar to dynamic mode decomposition with control and extending it to a nonlinear framework with controllers designed using stability-constrained deep neural networks, validated through numerical experiments on a reaction-diffusion system.
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.