Adaptive Coordinate Transforms for Neural Operators
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , , , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909021851615232 |
|---|---|
| author | Liu, Chaoyu Li, Zhonghao Chen, Gaohang Shumaylov, Zakhar Deng, Zhongying Zhang, Qian Qiao, Zhonghua Schönlieb, Carola-Bibiane |
| author_facet | Liu, Chaoyu Li, Zhonghao Chen, Gaohang Shumaylov, Zakhar Deng, Zhongying Zhang, Qian Qiao, Zhonghua Schönlieb, Carola-Bibiane |
| contents | Neural operators have achieved promising performance on partial differential equations (PDEs), but most existing models are built on fixed Eulerian coordinates. This mismatch between evolving physical structures and static coordinates creates spatial misalignment, leading to unnecessarily non-local operator mappings and reinforcing a smoothness preference near sharp transitions. Inspired by adaptive coordinate transformations in classical PDE analysis, we propose the Adaptive Coordinate Transform (ACT) block, a plug-and-play module for data-driven geometric adaptation in neural operators. ACT blocks resolve this structural limitation by learning adaptive coordinate systems within the operator learning pipeline. Specifically, given an input feature, the ACT block learns a coordinate transformation and represents the same feature under the transformed coordinates via differentiable sampling. This operation preserves the underlying signal while changing its spatial representation, equivalent to expressing the same physical quantity in different coordinate systems. By adapting the coordinate system to the data, ACT allows the network to better track evolving structures, reduce operator complexity, and dynamically focus on critical features to improve learning. We evaluate the proposed approach across diverse PDE benchmarks and multiple neural operator architectures. Experimental results demonstrate consistent and significant improvements in predictive accuracy, indicating that learning coordinate systems provides a powerful mechanism for enhancing operator learning. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06203 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Adaptive Coordinate Transforms for Neural Operators Liu, Chaoyu Li, Zhonghao Chen, Gaohang Shumaylov, Zakhar Deng, Zhongying Zhang, Qian Qiao, Zhonghua Schönlieb, Carola-Bibiane Computational Engineering, Finance, and Science Neural operators have achieved promising performance on partial differential equations (PDEs), but most existing models are built on fixed Eulerian coordinates. This mismatch between evolving physical structures and static coordinates creates spatial misalignment, leading to unnecessarily non-local operator mappings and reinforcing a smoothness preference near sharp transitions. Inspired by adaptive coordinate transformations in classical PDE analysis, we propose the Adaptive Coordinate Transform (ACT) block, a plug-and-play module for data-driven geometric adaptation in neural operators. ACT blocks resolve this structural limitation by learning adaptive coordinate systems within the operator learning pipeline. Specifically, given an input feature, the ACT block learns a coordinate transformation and represents the same feature under the transformed coordinates via differentiable sampling. This operation preserves the underlying signal while changing its spatial representation, equivalent to expressing the same physical quantity in different coordinate systems. By adapting the coordinate system to the data, ACT allows the network to better track evolving structures, reduce operator complexity, and dynamically focus on critical features to improve learning. We evaluate the proposed approach across diverse PDE benchmarks and multiple neural operator architectures. Experimental results demonstrate consistent and significant improvements in predictive accuracy, indicating that learning coordinate systems provides a powerful mechanism for enhancing operator learning. |
| title | Adaptive Coordinate Transforms for Neural Operators |
| topic | Computational Engineering, Finance, and Science |
| url | https://arxiv.org/abs/2605.06203 |