A mesh-free neural approach to fractional diffusion in Li$$^{+}$$-ion battery materials

Abstract Li $$^{+}$$ -ion batteries (LIBs) are pivotal for advancing electric vehicles and green energy systems. These applications require high reliability and performance, which requires accurate modeling of Li $$^{+}$$ -ion diffusion within electrode materials. In this study, we developed a fractional-order solid-phase diffusion model for thin-film electrodes in accordance with Fick’s second law in its non-dimensional form. To estimate the solution, we employ Physics-Informed Neural Networks (PINNs), which embed the governing fractional partial differential equation (FPDE), initial, and boundary conditions directly into the training loss function, where loss gradients are computed via backpropagation and the network parameters are updated using the Adam optimizer. The PINN framework offers a flexible, mesh-free approach to solving complex FPDEs, eliminating the need for transform-based or discretization techniques. We validate the effectiveness of the PINN method by comparing the neural network-based predictions with the exact analytical solution, Laplace transform-based differential transform method (LT-DTM) and Laplace transform-based $$\alpha $$ -parametrized differential transform method (LT- $$\alpha $$ PDTM). The results demonstrate high accuracy and smooth convergence across different fractional orders, highlighting the potential of PINNs for solving fractional diffusion problems in electrochemical systems. This approach offers a powerful tool for the analysis, control, and optimization of Li $$^{+}$$ -ion battery performance under realistic operational conditions.