Deep Extreme Learning Machines Based Two-Phase Spatiotemporal Modeling for Distributed Parameter Systems
Kangkang Xu, Haidong Yang, Chengjiu Zhu, Xi Jin, Bi Fan, Luoke Hu
Abstract
Accurate and robust modeling of complex distributed parameter systems (DPSs) is a challenge for three reasons: 1) they have infinite-dimensional characteristics; 2) they are time/space coupled; and 3) there are model uncertainties. In this article, a two-phase spatiotemporal (S/T) modeling framework based on deep extreme learning machine (DELM) is proposed for DPSs. The modeling process consists of two S/T models in two phases: Phase I: a DELM model and Phase II: a Karhunen–Loève (KL) based ELM (KL-ELM) model. In phase I, the DELM model is constructed by combing the multilayer ELM (ML-ELM), ELM, and kernel-based ELM (K-ELM) to approximate the dominant S/T dynamics of DPSs. Since DPSs have an infinite-dimensional characteristic that can hardly be handled directly, ML-ELM is first employed to transform the infinite-dimensional systems into finite-dimensional systems. Then, the ELM model is adopted to further approximate the finite-dimensional systems to ensure the model can predict future dynamic behavior. Finally, the K-ELM is used to reconstruct the infinite-dimensional systems, which can be considered as the inverse process of ML-ELM. Thus, the final DELM model can be used for prediction in both space and time directions. In phase II, a KL-ELM model is constructed to compensate for modeling errors caused by reconstruction error or unknown nonlinear dynamics. By integrating the obtained DELM and KL-ELM models, the proposed two-phase S/T model can be constructed. Experiments on a typical industrial thermal process verified that the proposed method may work better in complex DPSs.