A Physics-Informed Recurrent Neural Network with Fractional-Order Kinetics for Robust Lithium-Ion Battery State of Charge Estimation

Accurate State of Charge (SOC) estimation is critical for the safety and efficiency of Battery Management Systems (BMS). While data-driven methods have shown promise, they often exhibit limited generalization capability due to the lack of physical constraints. Incorporating symmetry in the battery, such as through the use of Physics-Informed Neural Networks (PINNs), can mitigate this issue. However, PINNs typically rely on integer-order equivalent circuit model differential equations, which fail to accurately describe the complex electrochemical relaxation processes. To bridge this gap, we propose a novel Fractional Differential Physics-Informed Neural Network (FDE-PINN) framework. Unlike traditional approaches, this method embeds a Fractional-Order Equivalent Circuit Model (FO-ECM) into the Gated Recurrent Unit (GRU) architecture to explicitly capture the anomalous diffusion and long-memory effects inherent in battery polarization. Specifically, the network is trained by minimizing a composite loss function that integrates the data fitting error with residuals from fractional-order governing equations, including Coulomb counting and fractional voltage dynamics. Extensive experiments on the Panasonic 18650PF dataset and CALCE A123 dataset verify the method’s superiority. Results demonstrate that the proposed FDE-GRU model achieves an average MSE of 14.29×10−4 (with an MAE of 2.43% and RMSE of 3.23%) on the NCA chemistry and 26.24×10−4 (with an MAE of 3.75% and RMSE of 5.09%) on the LiFePO4 chemistry, significantly outperforming traditional methods by reducing the estimation error by 35.6% and 26.2% compared to the standard GRU, respectively.