Interpolating Neural Network-Tensor Decomposition (INN-TD): a scalable and interpretable approach for large-scale physics-based problems
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arXiv
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| Format: | Preprint |
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2025
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| author | Guo, Jiachen Xie, Xiaoyu Park, Chanwook Zhang, Hantao Politis, Matthew Domel, Gino Hughes, Thomas J. R. Liu, Wing Kam |
| author_facet | Guo, Jiachen Xie, Xiaoyu Park, Chanwook Zhang, Hantao Politis, Matthew Domel, Gino Hughes, Thomas J. R. Liu, Wing Kam |
| contents | Deep learning has been extensively employed as a powerful function approximator for modeling physics-based problems described by partial differential equations (PDEs). Despite their popularity, standard deep learning models often demand prohibitively large computational resources and yield limited accuracy when scaling to large-scale, high-dimensional physical problems. Their black-box nature further hinders the application in industrial problems where interpretability and high precision are critical. To overcome these challenges, this paper introduces Interpolating Neural Network-Tensor Decomposition (INN-TD), a scalable and interpretable framework that has the merits of both machine learning and finite element methods for modeling large-scale physical systems. By integrating locally supported interpolation functions from finite element into the network architecture, INN-TD achieves a sparse learning structure with enhanced accuracy, faster training/solving speed, and reduced memory footprint. This makes it particularly effective for tackling large-scale high-dimensional parametric PDEs in training, solving, and inverse optimization tasks in physical problems where high precision is required. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02041 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Interpolating Neural Network-Tensor Decomposition (INN-TD): a scalable and interpretable approach for large-scale physics-based problems Guo, Jiachen Xie, Xiaoyu Park, Chanwook Zhang, Hantao Politis, Matthew Domel, Gino Hughes, Thomas J. R. Liu, Wing Kam Computational Engineering, Finance, and Science Deep learning has been extensively employed as a powerful function approximator for modeling physics-based problems described by partial differential equations (PDEs). Despite their popularity, standard deep learning models often demand prohibitively large computational resources and yield limited accuracy when scaling to large-scale, high-dimensional physical problems. Their black-box nature further hinders the application in industrial problems where interpretability and high precision are critical. To overcome these challenges, this paper introduces Interpolating Neural Network-Tensor Decomposition (INN-TD), a scalable and interpretable framework that has the merits of both machine learning and finite element methods for modeling large-scale physical systems. By integrating locally supported interpolation functions from finite element into the network architecture, INN-TD achieves a sparse learning structure with enhanced accuracy, faster training/solving speed, and reduced memory footprint. This makes it particularly effective for tackling large-scale high-dimensional parametric PDEs in training, solving, and inverse optimization tasks in physical problems where high precision is required. |
| title | Interpolating Neural Network-Tensor Decomposition (INN-TD): a scalable and interpretable approach for large-scale physics-based problems |
| topic | Computational Engineering, Finance, and Science |
| url | https://arxiv.org/abs/2503.02041 |