A Machine Learning Approach to the Nirenberg Problem
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915795967148032 |
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| author | Cortés, Gianfranco Esteban-Casadevall, Maria Feng, Yueqing Henkel, Jonas Hirst, Edward Gherardini, Tancredi Schettini Stapleton, Alexander G. |
| author_facet | Cortés, Gianfranco Esteban-Casadevall, Maria Feng, Yueqing Henkel, Jonas Hirst, Edward Gherardini, Tancredi Schettini Stapleton, Alexander G. |
| contents | This work introduces the Nirenberg Neural Network: a numerical approach to the Nirenberg problem of prescribing Gaussian curvature on $S^2$ for metrics that are pointwise conformal to the round metric. Our mesh-free physics-informed neural network (PINN) approach directly parametrises the conformal factor globally and is trained with a geometry-aware loss enforcing the curvature equation. Additional consistency checks were performed via the Gauss-Bonnet theorem, and spherical-harmonic expansions were fit to the learnt models to provide interpretability.
For prescribed curvatures with known realisability, the neural network achieves very low losses ($10^{-7} - 10^{-10}$), while unrealisable curvatures yield significantly higher losses. This distinction enables the assessment of unknown cases, separating likely realisable functions from non-realisable ones. The current capabilities of the Nirenberg Neural Network demonstrate that neural solvers can serve as exploratory tools in geometric analysis, offering a quantitative computational perspective on longstanding existence questions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_12368 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Machine Learning Approach to the Nirenberg Problem Cortés, Gianfranco Esteban-Casadevall, Maria Feng, Yueqing Henkel, Jonas Hirst, Edward Gherardini, Tancredi Schettini Stapleton, Alexander G. Machine Learning High Energy Physics - Theory Analysis of PDEs Differential Geometry This work introduces the Nirenberg Neural Network: a numerical approach to the Nirenberg problem of prescribing Gaussian curvature on $S^2$ for metrics that are pointwise conformal to the round metric. Our mesh-free physics-informed neural network (PINN) approach directly parametrises the conformal factor globally and is trained with a geometry-aware loss enforcing the curvature equation. Additional consistency checks were performed via the Gauss-Bonnet theorem, and spherical-harmonic expansions were fit to the learnt models to provide interpretability. For prescribed curvatures with known realisability, the neural network achieves very low losses ($10^{-7} - 10^{-10}$), while unrealisable curvatures yield significantly higher losses. This distinction enables the assessment of unknown cases, separating likely realisable functions from non-realisable ones. The current capabilities of the Nirenberg Neural Network demonstrate that neural solvers can serve as exploratory tools in geometric analysis, offering a quantitative computational perspective on longstanding existence questions. |
| title | A Machine Learning Approach to the Nirenberg Problem |
| topic | Machine Learning High Energy Physics - Theory Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2602.12368 |