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[2602.12368] A Machine Learning Approach to the Nirenberg Problem
Summary
This paper presents a novel machine learning framework, the Nirenberg Neural Network, to address the Nirenberg problem of prescribing Gaussian curvature on spheres, enhancing geometric analysis through neural solvers.
Why It Matters
The Nirenberg problem is a significant challenge in differential geometry, and this research introduces a machine learning approach that could revolutionize how such problems are tackled. By leveraging neural networks, the study provides a new computational perspective that may lead to breakthroughs in understanding geometric properties and their applications.
Key Takeaways
- Introduces the Nirenberg Neural Network for solving curvature problems.
- Utilizes a mesh-free physics-informed neural network approach.
- Achieves low losses for realizable curvatures, aiding in geometric analysis.
- Employs the Gauss-Bonnet theorem for consistency checks.
- Demonstrates potential for neural solvers in exploring geometric existence questions.
Computer Science > Machine Learning arXiv:2602.12368 (cs) [Submitted on 12 Feb 2026] Title:A Machine Learning Approach to the Nirenberg Problem Authors:Gianfranco Cortés, Maria Esteban-Casadevall, Yueqing Feng, Jonas Henkel, Edward Hirst, Tancredi Schettini Gherardini, Alexander G. Stapleton View a PDF of the paper titled A Machine Learning Approach to the Nirenberg Problem, by Gianfranco Cort\'es and 6 other authors View PDF HTML (experimental) Abstract: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 com...