A Neural Operator Unifying Graph Neural Networks and Point Transformers

Neural operators have emerged as a powerful tool for learning mappings between function spaces, particularly for solving partial differential equations (PDEs). This study introduces a novel framework that unifies Graph Neural Networks (GNNs) and Transformers, combining their complementary strengths to enhance the approximation of operators derived from solutions of PDEs. By integrating the feature extraction capabilities of GNNs with the attention mechanisms of Transformers, this approach efficiently captures information across multiple scales, from local patterns to global structures, thereby enhancing both the accuracy and efficiency of operator learning.The proposed model consistently outperforms existing baseline models in experiments across various datasets, reducing relative errors. Notably, the model demonstrates robustness in handling multi-input/output problems on irregular meshes, whereas other models struggle to converge. The results demonstrate the effectiveness and generalization capacity of the framework across different types and scales of PDEs. Overall, the integration of GNNs with attention mechanisms represents a significant advancement in operator learning for PDEs, offering a reliable and efficient solution for practical applications.

Abstract licence: CC BY-NC-ND

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