Geometric partial differential equations (GPDEs), defined on Riemannian manifolds, play a fundamental role in modeling surface evolution processes in science and engineering. Traditional numerical methods, however, often require costly remeshing and recomputation when handling dynamically evolving surfaces or mesh refinements. Physics-informed neural networks (PINNs), which leverage automatic differentiation to compute derivatives and embed physical constraints directly into neural network training, offer a promising alternative. Yet, unlike classical PDEs defined on fixed domains, GPDEs involve time-varying computational surfaces, where differential operators defined on irregular discrete meshes cannot be directly handled through automatic differentiation. This poses unique challenges for applying conventional PINNs to irregular geometric domains. To address these issues, we propose an attention-based discrete physics-informed neural network (ADPINet) architecture for explicitly solving GPDEs defined on triangular mesh surfaces. The proposed method adopts a dual-discrete neural framework in both time and space, enabling direct learning on irregular spatiotemporal domains. An attention-based backbone is designed to extract global correlations among mesh vertices, while a physics-informed loss formulated via discrete differential geometry ensures physical consistency with governing equations without requiring labeled data. Once trained, ADPINet can efficiently predict solutions on refined meshes or newly introduced vertices by directly inputting their coordinates, without retraining or interpolation. Furthermore, ADPINet demonstrates shape-level generalization, being able to predict GPDE solutions for different initial surfaces with similar topological and geometric structures. Extensive numerical experiments, including comparisons with traditional numerical solvers and alternative neural architectures, demonstrate the superior accuracy, mesh-quality preservation, computational efficiency, and generalization capability of the proposed method.
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