  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.
