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

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.