A Machine Learning Approach to the Nirenberg Problem

TL;DR

This paper presents a neural network-based numerical method for solving the Nirenberg problem of prescribing Gaussian curvature on the sphere, demonstrating high accuracy and potential for exploring geometric existence questions.

Contribution

Introduces a mesh-free physics-informed neural network approach for the Nirenberg problem, enabling assessment of curvature realisability and offering a new computational tool in geometric analysis.

Findings

Achieves very low loss for realisable curvatures

Distinguishes realisable from non-realisable curvatures based on loss

Provides a quantitative method for geometric existence questions

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 computational perspective on longstanding existence questions.