GCDO-Net: Differential operator graph network for gradient computation on unstructured meshes

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Accurate gradient computation is indispensable for numerical simulations, playing a pivotal role in the discretization of partial differential equations (PDEs) and the post-processing of field data. However, achieving such accuracy remains a challenge on triangular unstructured meshes, where geometric distortions and irregularities often severely degrade the performance of conventional numerical operators. Therefore, this study proposes a Gradient‑Corrected Differential Operator Graph Network (GCDO‑Net). The proposed GCDO-Net architecture integrates conventional numerical priors as a global physical foundation, which is subsequently augmented by learned local residuals for data-driven error correction. The methodology is implemented through three core components: First, graph features are constructed by encoding traditional differential operators into node and edge attributes, capturing essential local geometry and physical coupling. Second, a Graph Neural Network (GNN) learns residual corrections to the baseline operators. This process incorporates a novel direction‑decoupled dynamic gating (DDG) mechanism, which adaptively fuses physical priors with data-driven insights by assigning direction-specific weights, thereby resolving the conflict between isotropic message passing and directional gradient sensitivity. Third, the model incorporates boundary-enhanced regularization to mitigate errors in critical regions. Numerical experiments on both analytical functions and partial differential equations (PDEs) demonstrate the superiority of GCDO-Net. For analytical test functions, the framework significantly improves gradient estimation accuracy, especially for second-order derivatives on low-quality and coarse unstructured meshes. When applied to the solution of Poisson, transient heat conduction, and 2D short beam bending elasticity problems, the corrected operator achieves an average relative error reduction exceeding 11% for second-order derivative terms and achieves significant precision gains in displacement gradients and strain fields (e.g., a 42.15% reduction in relative error for engineering shear strain γ xy under unseen loading conditions) compared to baseline operators, while exhibiting superior frequency adaptability, spatio-temporal stability, and load generalization. These results demonstrate the effectiveness of GCDO-Net for gradient estimation, with its generalization performance validated across a range of nearby source frequencies on the training mesh topology.