Efficient Data-Driven Modeling of Nonlinear Dynamical Systems via Metalearning

Data-driven modeling of nonlinear dynamical systems is essential because of the need for a trade-off among complexity, efficiency, and reliability in analytical or numerical studies as well as the difficulty of deriving fully physics-based models. An important limitation is commonly seen in existing works: modeling of a new dynamical system typically starts from scratch, requiring a large amount of data and intensive computation, though some prior experience or knowledge is available from a previously collected database of similar but different systems. However, on the one hand, the data amount for the new dynamical system is often limited, especially for real-world dynamical systems. On the other hand, the computational resource is also limited and a data-driven modeling task is usually computationally expensive, especially for large-scale systems. To improve data efficiency and computational efficiency in data-driven modeling of nonlinear systems, we present an enhanced data-driven modeling approach by incorporating metalearning into a physics-integrated deep learning framework. The core idea is to learn the metaknowledge about how to model a new system from a previously collected database of similar but different systems. Then this metaknowledge is leveraged to enable efficient modeling of a new system with limited data. For validations we conducted numerical experiments on three sets of fundamental nonlinear systems, including Duffing oscillators, nonlinear pendulums, and van der Pol oscillators. We performed both interpolation and extrapolation modelings to investigate the generalization ability of the presented approach. Furthermore, we conducted a quantitative analysis on data efficiency, addressing two critical issues: how few data are sufficient for the new system modeling and how much prior experience (previously collected database of similar but different systems) is needed for the metalearning. The results show that the presented approach improves both data efficiency and computational efficiency, compared with the conventional data-driven modeling approach (without leverage of the prior database) and the pretraining-based approach (simply using the prior database but without metalearning idea). We also discuss the limitations of this work and potential future study.