Dynamic Mode Decomposition-inspired Autoencoders for Reduced-order Modeling and Control of PDEs : Theory and Design
Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
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Keywords: PDEs, Autoencoders, Reduced-order modeling, Control, Dynamic mode decomposition
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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 studies a deep autoencoding learning method for controlling 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, yielding a solution that closely resembles the result obtained through the $\textit{dynamic mode decomposition with control}$ algorithm. Subsequently, we 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. Our framework operates without prior knowledge of the governing equations of the underlying system, relying solely on time series data of observations and actuations. Empirical analyses are presented to validate the efficacy of our approach in both modeling and controlling spatiotemporal dynamical systems, exemplified through applications to reaction-diffusion systems and fluid flow systems.
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Submission Number: 8629
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