Online Spatiotemporal Extreme Learning Machine for Complex Time-Varying Distributed Parameter Systems

Many industrial processes are complex nonlinear distributed parameter systems (DPSs) with time-varying spatiotemporal dynamics. However, time-varying spatiotemporal dynamics and the nonlinear relationships between spatial points are currently not given much consideration in the existing data-driven modeling methods. Thus, accurately modeling a nonlinear DPS with time-varying spatiotemporal dynamics using these current methods is challenging. Here, we propose a spatiotemporal extreme learning machine (ELM) to accurately model time-varying and nonlinear DPSs. First, we develop the nonlinear spatial activation function to describe the nonlinear relationships between spatial points. As a result, in contrast to the traditional ELM method which is only used to model the temporal dynamics, the spatiotemporal ELM inherently takes the spatial information into consideration. Next, an online time coefficient model is developed, which accounts for the time-varying temporal dynamics of the DPS. After the integration of the spatial activation function with the time coefficient model, this modeling method is able to adapt to real-time spatiotemporal variation. Unlike the existing data-driven DPS modeling approaches, the proposed method has the capability to accurately represent the nonlinear relationships between spatial points and has the adaptive ability for modeling time-varying dynamics. Finally, through application on practical curing experiments, the proposed method can improve the modeling precision for an unknown, time-varying, and nonlinear DPS due to smaller modeling error as compared to the several commonly used DPS modeling methods.