Time–frequency heterogeneous feature based physical-informed neural network for internal multimodal water estimation of fuel cells

Urgent demand for greening transportation significantly drives the development of proton exchange membrane fuel cells (PEMFC). However, the unmeasurable and unobservable characteristics of internal water significantly constrain the optimal control and health status maintenance. To address this problem, an internal multimodal water estimation framework integrates heterogeneous feature extraction and optimized physical-informed neural network (PINN) modeling is proposed. Time–frequency heterogeneous features of multimodal water are extracted through industrially measurable variables and distribution of relaxation times, thereby synchronized via equivalent mapping and self-attention mechanisms, which expands observable dimensions and facilitates reliable target estimation. These heterogeneous features serve as inputs for the data-mechanism dual-driven PINN approach, which leverages multi-physical field equations for generalization and neural networks for accuracy. To further accelerate convergence of PINN, a local adaptive algorithm initializes network weights, which are subsequently fine-tuned using Bayesian optimization. The comparative analysis demonstrated that as the dataset size decreased, the proposed method improved accuracy by 44.83% and convergence speed by 35.29%. Compared to other methods, proposed method achieved the best performance, with a root mean square error of 0.0045 and 0.1142 for liquid saturation and membrane hydration, respectively. This study paves the way for the precise and fast regulation for internal status of proton exchange membrane fuel cell.