A parametric framework for kernel-based dynamic mode decomposition using deep learning

Surrogate modelling is widely applied in computational science and engineering to mitigate computational efficiency issues for the real-time simulations of complex and large-scale computational models or for many-query scenarios, such as uncertainty quantification and design optimisation. In this work, we propose a parametric framework for the kernel-based dynamic mode decomposition method based on the linear and nonlinear disambiguation optimisation (LANDO) algorithm. The proposed parametric framework consists of two stages, offline and online. The offline stage prepares a series of LANDO models that emulate the dynamics of the system with particular parameters. The online stage leverages those LANDO models to generate new data at a desired time instant, and approximate the mapping between parameters and the state with the data using deep learning techniques. For high-dimensional dynamical systems, the framework is further combined with POD to reduce the dimensionality of the state space and improve the efficiency of the learning procedure. The effectiveness of the proposed method is demonstrated through three numerical examples including Lotka–Volterra model, heat equation and reaction–diffusion equation. The work also provides discussion on the main sources of approximation error and on the computational trade-offs associated with the offline-online structure of the framework.