# A Machine Learning Approach to the Nirenberg Problem

arXiv: https://arxiv.org/abs/2602.12368 · HTML: https://mlsanity.com/paper/2602.12368/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 56 references that are in the map.
- First posted on arXiv 12 Feb 2026.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Approximating Continuous Functions by ReLU Nets of Minimal Width](https://mlsanity.com/paper/1710.11278/) (2017)
- [Neural Network Approximations for Calabi-Yau Metrics](https://mlsanity.com/paper/2012.15821/) (2020)
- [Learning Size and Shape of Calabi-Yau Spaces](https://mlsanity.com/paper/2111.01436/) (2021)
- [Machine learning Calabi-Yau metrics](https://mlsanity.com/paper/1910.08605/) (2019)
- [Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks](https://mlsanity.com/paper/2201.06780/) (2022)
- [Machine Learned Calabi-Yau Metrics and Curvature](https://mlsanity.com/paper/2211.09801/) (2022)
- [A Physics-Informed Neural Network Framework For Partial Differential Equations on 3D Surfaces: Time-Dependent Problems](https://mlsanity.com/paper/2103.13878/) (2021)
- [Approximating High-Dimensional Minimal Surfaces with Physics-Informed Neural Networks](https://mlsanity.com/paper/2309.02589/) (2023)
- [Aspects of holographic entanglement using physics-informed-neural-networks](https://mlsanity.com/paper/2509.25311/) (2025)
- [Machine Learning Gravity Compactifications on Negatively Curved Manifolds](https://mlsanity.com/paper/2501.00093/) (2024)
- [The Mathematician's Assistant: Integrating AI into Research Practice](https://mlsanity.com/paper/2508.20236/) (2025)
- [Search for Z/2 eigenfunctions on the sphere using machine learning](https://mlsanity.com/paper/2507.13122/) (2025)
- [Learn and Verify: A Framework for Rigorous Verification of Physics-Informed Neural Networks](https://mlsanity.com/paper/2601.19818/) (2026)

## Built on by (most cited first, 5 of 5)

- [Minimising Willmore Energy via Neural Flow](https://mlsanity.com/paper/2604.04321/) (2026)
- [Minimal surfaces, Knots, and Neural Networks](https://mlsanity.com/paper/2605.26234/) (2026)
- [PINNs in More General Geometry](https://mlsanity.com/paper/2604.25020/) (2026)
- [Lean-Quantum: Toward AI-Assisted Formalization of Quantum Information](https://mlsanity.com/paper/2607.05492/) (2026)
- [A user's guide to PINNs in geometric analysis: lessons from the asymptotic Plateau problem](https://mlsanity.com/paper/2607.28733/) (2026)

## Abstract (arXiv)

This work introduces the Nirenberg Neural Network: a numerical approach to the Nirenberg problem of prescribing Gaussian curvature on S2 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.