Bridging Autoencoders and Dynamic Mode Decomposition for Reduced-order Modeling and Control of PDEs
<|reference_start|>Bridging Autoencoders and Dynamic Mode Decomposition for Reduced-order Modeling and Control of PDEs: 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.<|reference_end|>
@article{saha2024bridging,
    title={Bridging Autoencoders and Dynamic Mode Decomposition for Reduced-order
  Modeling and Control of PDEs},
    author={Priyabrata Saha and Saibal Mukhopadhyay},
    journal={arXiv preprint arXiv:2409.06101},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.06101},
    primaryClass={eess.SY cs.LG cs.SY math.OC}
}