A Machine Learning Approach to the Nirenberg Problem
This provides a computational tool for exploring existence questions in geometric analysis, offering a quantitative perspective on a longstanding mathematical problem.
The paper tackles the Nirenberg problem of prescribing Gaussian curvature on a sphere by developing a mesh-free physics-informed neural network to parametrize the conformal factor and enforce the curvature equation. It achieves very low losses (10^-7 to 10^-10) for realizable curvatures and distinguishes realizable from non-realisable cases, enabling assessment of unknown functions.
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.