Data-Driven Modeling of Parameterized Nonlinear Dynamical Systems with a Dynamics-Embedded Conditional Generative Adversarial Network

Abstract

Nonlinear dynamical systems in applications such as design and control generally depend on a set of variable parameters that represent system geometry, boundary conditions, material properties, etc. Modeling of such parameterized nonlinear systems from first principles is often challenging due to insufficient knowledge of the underlying physics (e.g., damping), especially when the physics-associated parameters are considered to be variable. In this study, we present a dynamics-embedded conditional generative adversarial network (Dyn-cGAN) for data-driven modeling and identification of parameterized nonlinear dynamical systems, capturing transient dynamics conditioned on the system parameters. Specifically, a dynamics block is embedded in a modified conditional generative adversarial network, thereby identifying temporal dynamics and its dependence on the system parameters, simultaneously. The data-driven Dyn-cGAN model is learned to perform long-term prediction of the dynamical response of a parameterized nonlinear dynamical system (equivalently a family of nonlinear systems with different parameter values), given any initial conditions and system parameter values. The capability of the presented Dyn-cGAN is evaluated by numerical studies on a variety of parameterized nonlinear dynamical systems including pendulums, Duffing, and Lorenz systems, considering various combinations of initial conditions and system (physical) parameters as inputs and different ranges of nonlinear dynamical behaviors including chaotic. It is observed that the presented data-driven framework is reasonably effective for predictive modeling and identification of parameterized nonlinear dynamical systems. Further analysis also indicates that its prediction accuracy degrades gracefully as the complexity of the nonlinear system increases, such as strongly nonlinear systems and systems with multiple parameters changing. The limitations of this work and potential future work are also discussed.