Functional tensor train neural network for solving high-dimensional PDEs
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911211135696896 |
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| author | Feng, Yani Ng, Michael K. Tang, Kejun Zhang, Zhiwen |
| author_facet | Feng, Yani Ng, Michael K. Tang, Kejun Zhang, Zhiwen |
| contents | Discrete tensor train decomposition is widely employed to mitigate the curse of dimensionality in solving high-dimensional PDEs through traditional methods. However, the direct application of the tensor train method typically requires uniform grids of regular domains, which limits its application on non-uniform grids or irregular domains. To address the limitation, we develop a functional tensor train neural network (FTTNN) for solving high-dimensional PDEs, which can represent PDE solutions on non-uniform grids or irregular domains. An essential ingredient of our approach is to represent the PDE solutions by the functional tensor train format whose TT-core functions are approximated by neural networks. To give the functional tensor train representation, we propose and study functional tensor train rank and employ it into a physics-informed loss function for training. Because of tensor train representation, the resulting high-dimensional integral in the loss function can be computed via one-dimensional integrals by Gauss quadrature rules. Numerical examples including high-dimensional PDEs on regular or irregular domains are presented to demonstrate that the performance of the proposed FTTNN is better than that of Physics Informed Neural Networks (PINN). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13386 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Functional tensor train neural network for solving high-dimensional PDEs Feng, Yani Ng, Michael K. Tang, Kejun Zhang, Zhiwen Numerical Analysis Discrete tensor train decomposition is widely employed to mitigate the curse of dimensionality in solving high-dimensional PDEs through traditional methods. However, the direct application of the tensor train method typically requires uniform grids of regular domains, which limits its application on non-uniform grids or irregular domains. To address the limitation, we develop a functional tensor train neural network (FTTNN) for solving high-dimensional PDEs, which can represent PDE solutions on non-uniform grids or irregular domains. An essential ingredient of our approach is to represent the PDE solutions by the functional tensor train format whose TT-core functions are approximated by neural networks. To give the functional tensor train representation, we propose and study functional tensor train rank and employ it into a physics-informed loss function for training. Because of tensor train representation, the resulting high-dimensional integral in the loss function can be computed via one-dimensional integrals by Gauss quadrature rules. Numerical examples including high-dimensional PDEs on regular or irregular domains are presented to demonstrate that the performance of the proposed FTTNN is better than that of Physics Informed Neural Networks (PINN). |
| title | Functional tensor train neural network for solving high-dimensional PDEs |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2510.13386 |