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Generalized Quadratic Embeddings for Nonlinear Dynamics using Deep Learning

Pawan Goyal, Peter Benner

TL;DR

The paper addresses learning compact, quadratic representations of nonlinear dynamics by discovering lifted coordinates that render the system's evolution quadratic in latent space. It proposes a deep autoencoder-based approach to simultaneously learn lifted coordinates and quadratic latent dynamics, with a loss that enforces consistency with observed derivatives and reconstructions. The authors establish global asymptotic stability guarantees for the latent quadratic model through a structured parameterization and demonstrate results on a nonlinear pendulum, dissipative Lotka-Volterra, and a high-dimensional Burgers' equation, showing superior accuracy over Koopman-inspired linear embeddings and operator inference. The work provides a scalable, interpretable framework for data-driven modeling of nonlinear dynamics with potential applications in engineering design and control.

Abstract

The engineering design process often relies on mathematical modeling that can describe the underlying dynamic behavior. In this work, we present a data-driven methodology for modeling the dynamics of nonlinear systems. To simplify this task, we aim to identify a coordinate transformation that allows us to represent the dynamics of nonlinear systems using a common, simple model structure. The advantage of a common simple model is that customized design tools developed for it can be applied to study a large variety of nonlinear systems. The simplest common model -- one can think of -- is linear, but linear systems often fall short in accurately capturing the complex dynamics of nonlinear systems. In this work, we propose using quadratic systems as the common structure, inspired by the lifting principle. According to this principle, smooth nonlinear systems can be expressed as quadratic systems in suitable coordinates without approximation errors. However, finding these coordinates solely from data is challenging. Here, we leverage deep learning to identify such lifted coordinates using only data, enabling a quadratic dynamical system to describe the system's dynamics. Additionally, we discuss the asymptotic stability of these quadratic dynamical systems. We illustrate the approach using data collected from various numerical examples, demonstrating its superior performance with the existing well-known techniques.

Generalized Quadratic Embeddings for Nonlinear Dynamics using Deep Learning

TL;DR

The paper addresses learning compact, quadratic representations of nonlinear dynamics by discovering lifted coordinates that render the system's evolution quadratic in latent space. It proposes a deep autoencoder-based approach to simultaneously learn lifted coordinates and quadratic latent dynamics, with a loss that enforces consistency with observed derivatives and reconstructions. The authors establish global asymptotic stability guarantees for the latent quadratic model through a structured parameterization and demonstrate results on a nonlinear pendulum, dissipative Lotka-Volterra, and a high-dimensional Burgers' equation, showing superior accuracy over Koopman-inspired linear embeddings and operator inference. The work provides a scalable, interpretable framework for data-driven modeling of nonlinear dynamics with potential applications in engineering design and control.

Abstract

The engineering design process often relies on mathematical modeling that can describe the underlying dynamic behavior. In this work, we present a data-driven methodology for modeling the dynamics of nonlinear systems. To simplify this task, we aim to identify a coordinate transformation that allows us to represent the dynamics of nonlinear systems using a common, simple model structure. The advantage of a common simple model is that customized design tools developed for it can be applied to study a large variety of nonlinear systems. The simplest common model -- one can think of -- is linear, but linear systems often fall short in accurately capturing the complex dynamics of nonlinear systems. In this work, we propose using quadratic systems as the common structure, inspired by the lifting principle. According to this principle, smooth nonlinear systems can be expressed as quadratic systems in suitable coordinates without approximation errors. However, finding these coordinates solely from data is challenging. Here, we leverage deep learning to identify such lifted coordinates using only data, enabling a quadratic dynamical system to describe the system's dynamics. Additionally, we discuss the asymptotic stability of these quadratic dynamical systems. We illustrate the approach using data collected from various numerical examples, demonstrating its superior performance with the existing well-known techniques.
Paper Structure (10 sections, 1 theorem, 16 equations, 8 figures, 1 table)

This paper contains 10 sections, 1 theorem, 16 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quadratic-embeddings for nonlinear systems
  3. Data-driven discovery of quadratic embeddings using deep learning
  4. Asymptotic stability-guaranteeing quadratic embeddings
  5. Numerical Results
  6. A nonlinear pendulum
  7. Dissipative Lotka-Volterra example
  8. A High-dimensional Example: Nonlinear Burgers' Equations
  9. Discussions
  10. Implementation details

Key Result

Lemma 4.1

Consider a quadratic system as follows: where $\mathbf{A} \in \mathbb{R}^{n\times n}$ and $\mathbf{H} \in \mathbb{R}^{n\times n^2}$. Assume that $\mathbf{A}$ can be written as $\mathbf{J} -\mathbf{R}$ with $\mathbf{J} = -\mathbf{J}^\top$ and $\mathbf{R} = \mathbf{R}^\top \succ 0$, and $\mathbf{H} = $ with $\mathbf{H}_i = -\mathbf{H}_i ^\t

Figures (8)

  • Figure 1.1: The figure illustrates that nonlinear dynamical systems can be written as quadratic dynamical systems in an appropriate finite-dimensional lifted coordinate system. In the right plot, we depict a neural network architecture to learn a lifted coordinate system that has the desired quadratic embeddings.
  • Figure 5.1: Nonlinear pendulum example: A comparison of the trajectories obtained using linear-embeds, quad-embeds, and quad-OpInf methods with the ground truth ones on the testing data is presented.
  • Figure 5.2: Nonlinear pendulum example: The figure shows a qualitative comparison of the performance of linear-embeds, quad-embeds, and quad-OpInf on the testing data based on the measure \ref{['eq:error_measure']}. Note that the error measure \ref{['eq:error_measure']} contains the log; hence, more negative is the values, better the method perform better.
  • Figure 5.3: Lotka-Volterra example: A comparison of the trajectories obtained using linear-embeds, quad-embeds, and quad-OpInf methods with the ground truth ones on the testing data is presented. Note that quad-OpInf yields unstable trajectories; therefore, they are not shown in the figure.
  • Figure 5.4: Lotka-Volterra example: The figure shows a qualitative comparison of linear-embeds, quad-embeds, and quad-OpInf on the testing data. Note that while plotting the error for quad-OpInf, unstable trajectories are removed, which are as many as $50$ out of $100$ testing initial conditions.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Lemma 4.1: goyal2023guaranteed
  • Remark 1