A Hybrid Legendre Polynomial–Physics-Informed Neural Network for Predicting Solutions of Laplace's Equation in Spherical Coordinates

Abstract

 Boundary value problems for Laplace's equation on spherical and shell-shaped domains arise in potential theory, heat conduction, electrostatics, and gravitation, with classical solutions expressed as series in spherical harmonics. Standard physics-informed neural networks (PINNs) represent such solutions with a generic, domain-agnostic multilayer perceptron that ignores the exact angular separability of the Laplacian on spherical domains. This paper examines whether embedding the known Legendre and spherical-harmonic structure of the operator into a physics-informed model changes accuracy, computational cost, and interpretability relative to a standard PINN. We propose a hybrid framework in which a compact network learns only the radial expansion coefficients (r) of a truncated spherical-harmonic series, while the angular dependence is supplied analytically through associated Legendre polynomials; this reduces the PDE residual to an exact, one-dimensional radial ODE residual per mode, removing the need to differentiate with respect to the angular coordinates. A composite loss combining PDE-residual, boundary-condition, and data terms is minimized with gradient descent. On a reproducible spherical-shell Laplace benchmark with a known closed-form multipole solution, the proposed model is trained and evaluated against a matched-budget standard PINN using RMSE, MAE, relative  error, and , with ablations over degree, collocation-point count, and network capacity. The two models achieve comparable accuracy, with relative  error below 4%; the proposed model trains in roughly half the time and is markedly more accurate at low capacity, while the standard PINN is marginally more accurate at larger capacity. The learned coefficients give a direct, per-degree interpretation unavailable from a standard PINN.