A deep generative framework for data-driven surrogate modeling and visualization of parameterized nonlinear dynamical systems

Nonlinear dynamical systems in applications such as design and control typically depend on a set of variable parameters that represent system geometry, boundary conditions, material properties, etc. Such a parameterized dynamical system requires a parameterized model (e.g., a parameterized differential equation) to describe. On the one hand, to discover the wide variety of the parameter-dependent dynamical behaviors, repeated simulations with the parameterized model are often required over a large range of parameter values, leading to significant computational burdens especially when the system is complex (strongly nonlinear and/or high-dimensional) and the high-fidelity model is inefficient to simulate. Thus, seeking surrogate models that mimic the behaviors of high-fidelity parameterized models while being efficient to simulate is critically needed. On the other hand, the governing equations of the parameterized nonlinear dynamical system (e.g., an aerodynamic system with a physical model (full-scale or scaled in the laboratory) for optimization or design tasks) may be unknown or partially unknown due to insufficient physics knowledge, leading to an inverse problem where we need to identify the models from measurement data only. Accordingly, this work presents a novel deep generative framework for data-driven surrogate modeling/identification of parameterized nonlinear dynamical systems from data only. Specifically, the presented framework learns the direct mapping from simulation parameters to visualization images of dynamical systems by leveraging deep generative convolutional neural networks, yielding two advantages: (i) the surrogate simulation is efficient because the calculation of transient dynamics over time is circumvented; (ii) the surrogate output retains characterizing ability and flexibility as the visualization image is customizable and supports any visualization scheme for revealing and representing high-level dynamics feature (e.g., Poincaré map). We study and demonstrate the framework on Lorenz system, forced pendulum system, and forced Duffing system. We present and discuss the prediction performance of the obtained surrogate models. It is observed that the obtained model has promising performance on capturing the sensitive parameter dependence of the nonlinear dynamical behaviors even when the bifurcation occurs. We also discuss in detail the limitation of this work and potential future work.