Koopman Theory-Based Missing Value Imputation Model for Nonlinear Dynamical Systems

In this paper, we propose a novel deep learning model based on Koopman theory to learn the partial differential equations (PDEs) inherent in data observed from nonlinear dynamical systems for missing data imputation. Since nonlinear PDEs such as the Navier-Stokes equations in fields like fluid dynamics and quantum mechanics still lack solutions, this paper addresses the long-term prediction problem of nonlinear dynamical systems by leveraging the Koopman Autoencoder (KAE) model. To improve the long-term prediction performance of KAE on nonlinear systems, we propose a multi-input-based KAE model that utilizes high temporal resolution multi-input data instead of lowering the temporal resolution of the model prediction. We validated the effectiveness of the proposed method through MSE, MAPE, and SMAPE metrics on three nonlinear dynamical system datasets- Navier-Stokes (smoke), Navier-Stokes (viscous flow), and Shallow-Water, showing significant improvement over baseline models.