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

arXiv: https://arxiv.org/abs/2409.06101 · HTML: https://mlsanity.com/paper/2409.06101/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 23 references that are in the map.
- First posted on arXiv 9 Sep 2024.

## Builds on (most cited first, 10 of 10)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Multiscale Simulations of Complex Systems by Learning their Effective Dynamics](https://mlsanity.com/paper/2006.13431/) (2020)
- [Deep Dynamical Modeling and Control of Unsteady Fluid Flows](https://mlsanity.com/paper/1805.07472/) (2018)
- [Deep Neural Networks for Nonlinear Model Order Reduction of Unsteady Flows](https://mlsanity.com/paper/2007.00936/) (2020)
- [Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs](https://mlsanity.com/paper/1702.04400/) (2017)
- [Neural Identification for Control](https://mlsanity.com/paper/2009.11782/) (2020)
- [Deep Learning-Aided Model Predictive Control of Wind Farms for AGC Considering the Dynamic Wake Effect](https://mlsanity.com/paper/2006.09656/) (2020)

## Built on by (most cited first, 1 of 1)

- [RRAEDy: Adaptive Latent Linearization of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2512.07542/) (2025)

## Abstract (arXiv)

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.