Geometry‐Aware Transferable Neural Networks for Solving Partial Differential Equations on Surfaces

In this paper, we propose a shallow neural network method specifically designed for solving partial differential equations (PDEs) on surfaces, called the Geometry‐Aware Transferable Neural Network (GATransNet). The method effectively addresses the challenge of determining hidden‐layer neurons in fixed‐feature networks. By introducing a geometry‐aware selection strategy, our approach closely integrates the geometric characteristics of neurons with the tangent‐plane structure of the surface, significantly enhancing the model's adaptability and stability. Furthermore, we combine the backward differentiation formula in time with the proposed GATransNet in space to construct a fully discrete numerical scheme, termed BDF‐GATransNet, for solving time‐dependent problems. Extensive numerical experiments involving the Laplace–Beltrami operator and related surface PDEs on both static and evolving surfaces are conducted to demonstrate the accuracy, efficiency, and flexibility of the proposed method. These results highlight its potential as a robust and versatile approach for solving surface PDEs.