Physics-informed neural networks for multi-stage Koopman modeling of microbial fermentation

This paper investigates the modeling problem of microbial fermentation suitable for model-based control design techniques. Given the evident nonlinear and stage characteristics of microbial fermentation processes, a single data-driven model cannot fully capture microbial growth characteristics. Therefore, we propose a multi-stage Koopman modeling method based on physics-informed neural networks. Initially, the fuzzy C-means clustering algorithm is employed to partition the microbial growth stages. Subsequently, the Koopman operator is approximated through physics-informed neural networks. Utilizing the Koopman operator to map the dynamic behavior of the microbial fermentation system into a high-dimensional linear space, and modeling each growth stage separately in the linear space. Compared to conventional neural network methods, physics-informed neural networks integrate the advantages of physical models and neural networks, thereby better preserving the dynamic information of microbial growth and enhancing the model's generalization performance. A penicillin fermentation case study verifies the effectiveness of our proposed method.