Interpolating Neural Network-Tensor Decomposition (INN-TD): a scalable and interpretable approach for large-scale physics-based problems

Jiachen Guo, Xiaoyu Xie, Chanwook Park, Hantao Zhang, Matthew J. Politis, Gino Domel, Wing Kam Liu
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:21138-21162, 2025.

Abstract

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 their 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-guo25p, title = {Interpolating Neural Network-Tensor Decomposition ({INN}-{TD}): a scalable and interpretable approach for large-scale physics-based problems}, author = {Guo, Jiachen and Xie, Xiaoyu and Park, Chanwook and Zhang, Hantao and Politis, Matthew J. and Domel, Gino and Liu, Wing Kam}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {21138--21162}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/guo25p/guo25p.pdf}, url = {https://proceedings.mlr.press/v267/guo25p.html}, abstract = {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 their 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.} }
Endnote
%0 Conference Paper %T Interpolating Neural Network-Tensor Decomposition (INN-TD): a scalable and interpretable approach for large-scale physics-based problems %A Jiachen Guo %A Xiaoyu Xie %A Chanwook Park %A Hantao Zhang %A Matthew J. Politis %A Gino Domel %A Wing Kam Liu %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-guo25p %I PMLR %P 21138--21162 %U https://proceedings.mlr.press/v267/guo25p.html %V 267 %X 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 their 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.
APA
Guo, J., Xie, X., Park, C., Zhang, H., Politis, M.J., Domel, G. & Liu, W.K.. (2025). Interpolating Neural Network-Tensor Decomposition (INN-TD): a scalable and interpretable approach for large-scale physics-based problems. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:21138-21162 Available from https://proceedings.mlr.press/v267/guo25p.html.

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