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

Priyabrata Saha, Saibal Mukhopadhyay

TL;DR

This paper analytically shows 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, enabling the development of a nonlinear reduced-order model.

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.

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

TL;DR

This paper analytically shows 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, enabling the development of a nonlinear reduced-order model.

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.
Paper Structure (20 sections, 3 theorems, 33 equations, 5 figures)

This paper contains 20 sections, 3 theorems, 33 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem and Preliminaries
  3. Problem statement
  4. Dynamic mode decomposition with control
  5. Method
  6. Learning a reduced order model
  7. Analysis of the linear reduced-order model for a fixed encoder
  8. The connection between the solutions of the linear autoencoding model and DMDc
  9. Extending the linear model to a deep model
  10. Learning control
  11. Numerical Experiments
  12. Conclusion
  13. Proofs
  14. Proof of theorem \ref{['theorem:lin_opt']}
  15. An alternative representation of (\ref{['eqn:lin_loss_pred_sol_2']})
  16. ...and 5 more sections

Key Result

Theorem 1

Consider the following objective function where ${\bm{G}} = [{\bm{A}}_{\normalfont\text{R}} \ \ {\bm{B}}_{\normalfont\text{R}}] \in \mathbb{R}^{r_{\bm{x}} \times (r_{\bm{x}}+d_{\bm{u}})}, {\bm{E}}_{{\bm{x}} {\bm{u}}} = \in \mathbb{R}^{(r_{\bm{x}} + d_{\bm{u}}) \times (d_{\bm{x}} + d_{\bm{u}})}$, ${\bm{I}}_{d_{\bm{u}}}$ being the identity where ${\bm{Y}}$ and $\bm{\mathit{\Omega}}$ are the data

Figures (5)

  • Figure 1: (a): Autoencoding architecture for reduced-order modeling. The state encoder $\mathcal{E}_{\bm{x}}$ reduces the dimension of the state. The ROM $\mathcal{F}$ takes the current reduced state and actuation to predict the next reduced state, which is then uplifted to the full state by the state decoder $\mathcal{D}_{\bm{x}}$. All modules are trained together using a combined loss involving $\mathcal{L}_{\text{pred}}$ and $\mathcal{L}_{\text{recon}}$. (b): The control learning process. Given a reduced state, $\mathcal{F}_s$ predicts a target dynamics for the closed-loop system, and the controller $\Pi$ predicts an actuation to achieve that target. Both the modules are trained jointly using the loss function $\mathcal{L}_{\text{ctrl}}$. Parameters of the dark-shaded modules are kept fixed during this process.
  • Figure 2: The first three dynamic modes of the reaction–diffusion system, obtained using DMDc and LAROM.
  • Figure 3: (a): Prediction performance of different methods in the reaction–diffusion example. The shaded interval shows the $95\%$ confidence interval around the mean from $100$ test sequences and $3$ different training instances. (b,c): Control performance of different methods in the reaction–diffusion example. The shaded interval shows the $1$-standard deviation range around the mean from $3$ different training instances.
  • Figure 4: Qualitative comparison of prediction performance of DMDc, Deep Koopman, and DeepROM for the reaction–diffusion system using one example sequence.
  • Figure 5: Visual comparison of the uncontrolled solution and the controlled solutions of the reaction–diffusion system using DeepROC, Deep Koopman + LQR, and DMDc + LQR.

Theorems & Definitions (5)

  • Theorem 1
  • Remark 1
  • Corollary 1.1
  • Remark 2
  • Lemma 2