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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.00872 |
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| _version_ | 1866914303171362816 |
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| author | Wang, Shihao Qian, Qipeng Wang, Jingquan |
| author_facet | Wang, Shihao Qian, Qipeng Wang, Jingquan |
| contents | We study solution learning for heat-based equations in self-similar variables (SSV). We develop an SSV training framework compatible with standard neural-operator training. We instantiate this framework on the two-dimensional incompressible Navier-Stokes equations and the one-dimensional viscous Burgers equation, and perform controlled comparisons between models trained in physical coordinates and in the corresponding self-similar coordinates using two simple fully connected architectures (standard multilayer perceptrons and a factorized fully connected network). Across both systems and both architectures, SSV-trained networks consistently deliver substantially more accurate and stable extrapolation beyond the training window and better capture qualitative long-time trends. These results suggest that self-similar coordinates provide a mathematically motivated inductive bias for learning the long-time dynamics of heat-based equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_00872 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Learning Heat-based Equations in Self-similar variables Wang, Shihao Qian, Qipeng Wang, Jingquan Machine Learning Mathematical Physics We study solution learning for heat-based equations in self-similar variables (SSV). We develop an SSV training framework compatible with standard neural-operator training. We instantiate this framework on the two-dimensional incompressible Navier-Stokes equations and the one-dimensional viscous Burgers equation, and perform controlled comparisons between models trained in physical coordinates and in the corresponding self-similar coordinates using two simple fully connected architectures (standard multilayer perceptrons and a factorized fully connected network). Across both systems and both architectures, SSV-trained networks consistently deliver substantially more accurate and stable extrapolation beyond the training window and better capture qualitative long-time trends. These results suggest that self-similar coordinates provide a mathematically motivated inductive bias for learning the long-time dynamics of heat-based equations. |
| title | Learning Heat-based Equations in Self-similar variables |
| topic | Machine Learning Mathematical Physics |
| url | https://arxiv.org/abs/2602.00872 |