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Learning Networked Dynamical System Models with Weak Form and Graph Neural Networks

Yin Yu, Daning Huang, Seho Park, Herschel C. Pangborn

TL;DR

Facing the challenge of data-driven, control-oriented modeling for complex networked systems, the paper introduces two linked approaches: the weak Latent Dynamics Model (wLDM) and its graph-based extension, the weak Graph Koopman Bilinear Form (wGKBF). By leveraging a weak form of the dynamics, wLDM achieves increased numerical stability and faster training compared to derivative-based or trajectory-based methods, while wGKBF embeds graph topology and Koopman bilinear structure to handle heterogeneous, multi-timescale networks. The authors demonstrate performance on a controlled double pendulum, stiff Brusselator dynamics, and an electrified aircraft energy system, showing superior predictive accuracy and training efficiency over baselines like LSTM, NODE, and biDMDc. The work provides practical guidance on hyperparameters and demonstrates a scalable, control-oriented framework for learning networked dynamical models.

Abstract

This paper presents a sequence of two approaches for the data-driven control-oriented modeling of networked systems, i.e., the systems that involve many interacting dynamical components. First, a novel deep learning approach named the weak Latent Dynamics Model (wLDM) is developed for learning generic nonlinear dynamics with control. Leveraging the weak form, the wLDM enables more numerically stable and computationally efficient training as well as more accurate prediction, when compared to conventional methods such as neural ordinary differential equations. Building upon the wLDM framework, we propose the weak Graph Koopman Bilinear Form (wGKBF) model, which integrates geometric deep learning and Koopman theory to learn latent space dynamics for networked systems, especially for the challenging cases having multiple timescales. The effectiveness of the wLDM framework and wGKBF model are demonstrated on three example systems of increasing complexity - a controlled double pendulum, the stiff Brusselator dynamics, and an electrified aircraft energy system. These numerical examples show that the wLDM and wGKBF achieve superior predictive accuracy and training efficiency as compared to baseline models. Parametric studies provide insights into the effects of hyperparameters in the weak form. The proposed framework shows the capability to efficiently capture control-dependent dynamics in these systems, including stiff dynamics and multi-physics interactions, offering a promising direction for learning control-oriented models of complex networked systems.

Learning Networked Dynamical System Models with Weak Form and Graph Neural Networks

TL;DR

Facing the challenge of data-driven, control-oriented modeling for complex networked systems, the paper introduces two linked approaches: the weak Latent Dynamics Model (wLDM) and its graph-based extension, the weak Graph Koopman Bilinear Form (wGKBF). By leveraging a weak form of the dynamics, wLDM achieves increased numerical stability and faster training compared to derivative-based or trajectory-based methods, while wGKBF embeds graph topology and Koopman bilinear structure to handle heterogeneous, multi-timescale networks. The authors demonstrate performance on a controlled double pendulum, stiff Brusselator dynamics, and an electrified aircraft energy system, showing superior predictive accuracy and training efficiency over baselines like LSTM, NODE, and biDMDc. The work provides practical guidance on hyperparameters and demonstrates a scalable, control-oriented framework for learning networked dynamical models.

Abstract

This paper presents a sequence of two approaches for the data-driven control-oriented modeling of networked systems, i.e., the systems that involve many interacting dynamical components. First, a novel deep learning approach named the weak Latent Dynamics Model (wLDM) is developed for learning generic nonlinear dynamics with control. Leveraging the weak form, the wLDM enables more numerically stable and computationally efficient training as well as more accurate prediction, when compared to conventional methods such as neural ordinary differential equations. Building upon the wLDM framework, we propose the weak Graph Koopman Bilinear Form (wGKBF) model, which integrates geometric deep learning and Koopman theory to learn latent space dynamics for networked systems, especially for the challenging cases having multiple timescales. The effectiveness of the wLDM framework and wGKBF model are demonstrated on three example systems of increasing complexity - a controlled double pendulum, the stiff Brusselator dynamics, and an electrified aircraft energy system. These numerical examples show that the wLDM and wGKBF achieve superior predictive accuracy and training efficiency as compared to baseline models. Parametric studies provide insights into the effects of hyperparameters in the weak form. The proposed framework shows the capability to efficiently capture control-dependent dynamics in these systems, including stiff dynamics and multi-physics interactions, offering a promising direction for learning control-oriented models of complex networked systems.
Paper Structure (33 sections, 1 theorem, 45 equations, 11 figures, 5 tables, 1 algorithm)

This paper contains 33 sections, 1 theorem, 45 equations, 11 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Choice of Model Form
  3. Learning Dynamics from Data
  4. Contributions
  5. Problem Statement and Preliminaries
  6. Problem Statement
  7. Available Algorithms
  8. Direct Methods
  9. Indirect Methods
  10. Applications in Machine Learning
  11. Formulation of Weak Latent Dynamics Model
  12. Weak Form for Learning Dynamics
  13. Representation of Dynamics by Weak Form
  14. Discretization and Integration Matrices
  15. Choice of Test Functions
  16. ...and 18 more sections

Key Result

Theorem 1

Goswami2022 Suppose a set of the Koopman eigenfunctions $\{\varphi_1,\varphi_2,\dots,\varphi_n\}$ of the autonomous system forms a invariant subspace of ${\mathcal{L}}_{{\mathbf{f}}_i},i=1,\dots,m$. Then Eq. (eqn:controlAffine) is bilinearizable with an $n$ dimensional state space.

Figures (11)

  • Figure 1: Structural diagram of the wLDM architecture.
  • Figure 2: Illustration of the double pendulum.
  • Figure 3: Comparison of prediction performance of the wLDM and the two baseline methods for the damped double pendulum dynamics with controls.
  • Figure 4: Comparison of predictive performance for models trained with increasing time window horizon $N$.
  • Figure 5: Trained wLDM performance with varying PO and IO at a fixed $N=61$ with $95\%$ confidence intervals computed using a bootstrap method.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Theorem 1