Learning-guided Kansa collocation for forward and inverse PDEs beyond linearity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917311339823104 |
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| author | Hu, Zheyuan Chen, Weitao Öztireli, Cengiz Zhou, Chenliang Zhong, Fangcheng |
| author_facet | Hu, Zheyuan Chen, Weitao Öztireli, Cengiz Zhou, Chenliang Zhong, Fangcheng |
| contents | Partial Differential Equations are precise in modelling the physical, biological and graphical phenomena. However, the numerical methods suffer from the curse of dimensionality, high computation costs and domain-specific discretization. We aim to explore pros and cons of different PDE solvers, and apply them to specific scientific simulation problems, including forwarding solution, inverse problems and equations discovery. In particular, we extend the recent CNF (NeurIPS 2023) framework solver to coupled and non-linear settings, together with down-stream applications. The outcomes include implementation of selected methods, self-tuning techniques, evaluation on benchmark problems and a comprehensive survey of neural PDE solvers and scientific simulation applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07970 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Learning-guided Kansa collocation for forward and inverse PDEs beyond linearity Hu, Zheyuan Chen, Weitao Öztireli, Cengiz Zhou, Chenliang Zhong, Fangcheng Computational Engineering, Finance, and Science Artificial Intelligence Machine Learning Numerical Analysis 65N35, 65D15, 65Z05 G.1.8; I.2.6; I.6.1; J.2 Partial Differential Equations are precise in modelling the physical, biological and graphical phenomena. However, the numerical methods suffer from the curse of dimensionality, high computation costs and domain-specific discretization. We aim to explore pros and cons of different PDE solvers, and apply them to specific scientific simulation problems, including forwarding solution, inverse problems and equations discovery. In particular, we extend the recent CNF (NeurIPS 2023) framework solver to coupled and non-linear settings, together with down-stream applications. The outcomes include implementation of selected methods, self-tuning techniques, evaluation on benchmark problems and a comprehensive survey of neural PDE solvers and scientific simulation applications. |
| title | Learning-guided Kansa collocation for forward and inverse PDEs beyond linearity |
| topic | Computational Engineering, Finance, and Science Artificial Intelligence Machine Learning Numerical Analysis 65N35, 65D15, 65Z05 G.1.8; I.2.6; I.6.1; J.2 |
| url | https://arxiv.org/abs/2602.07970 |