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| Hauptverfasser: | , , , , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2407.06664 |
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| _version_ | 1866912204821889024 |
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| author | Ye, Zhanhong Huang, Xiang Chen, Leheng Liu, Zining Wu, Bingyang Liu, Hongsheng Wang, Zidong Dong, Bin |
| author_facet | Ye, Zhanhong Huang, Xiang Chen, Leheng Liu, Zining Wu, Bingyang Liu, Hongsheng Wang, Zidong Dong, Bin |
| contents | This paper introduces PDEformer-1, a versatile neural solver capable of simultaneously addressing various partial differential equations (PDEs). With the PDE represented as a computational graph, we facilitate the seamless integration of symbolic and numeric information inherent in a PDE. A graph Transformer and an implicit neural representation (INR) are employed subsequently to generate mesh-free predicted solutions. We generated a dataset with up to three million samples involving diverse one-dimensional PDEs to pretrain our model. Compared with baseline models trained specifically on benchmark datasets, our pretrained model achieves comparable accuracy via zero-shot inference, and the advantage expands after finetuning. For PDEs new or unseen in the pretraining stage, our model can adapt quickly by finetuning on a relatively small set of examples from the target equation. Additionally, PDEformer-1 demonstrates promising results in the inverse problem of PDE scalar coefficient recovery and coefficient field recovery. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06664 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | PDEformer-1: A Foundation Model for One-Dimensional Partial Differential Equations Ye, Zhanhong Huang, Xiang Chen, Leheng Liu, Zining Wu, Bingyang Liu, Hongsheng Wang, Zidong Dong, Bin Numerical Analysis This paper introduces PDEformer-1, a versatile neural solver capable of simultaneously addressing various partial differential equations (PDEs). With the PDE represented as a computational graph, we facilitate the seamless integration of symbolic and numeric information inherent in a PDE. A graph Transformer and an implicit neural representation (INR) are employed subsequently to generate mesh-free predicted solutions. We generated a dataset with up to three million samples involving diverse one-dimensional PDEs to pretrain our model. Compared with baseline models trained specifically on benchmark datasets, our pretrained model achieves comparable accuracy via zero-shot inference, and the advantage expands after finetuning. For PDEs new or unseen in the pretraining stage, our model can adapt quickly by finetuning on a relatively small set of examples from the target equation. Additionally, PDEformer-1 demonstrates promising results in the inverse problem of PDE scalar coefficient recovery and coefficient field recovery. |
| title | PDEformer-1: A Foundation Model for One-Dimensional Partial Differential Equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2407.06664 |