A Machine Learning Approach to the Nirenberg Problem

TL;DR

This work uses a mesh-free physics-informed neural network to tackle the Nirenberg problem of prescribing Gaussian curvature on $S^2$ within the conformal class of the round metric. By directly learning the conformal factor $u$ and enforcing the curvature equation through a geometry-aware loss, the approach distinguishes realisable from obstructed prescribed curvatures, validated by Gauss–Bonnet checks. The method yields accurate reconstructions for solvable cases and provides quantitative evidence for unknown cases, while yielding interpretable closed-form harmonic expansions in some instances. The findings illustrate that neural solvers can serve as exploratory tools in geometric analysis, offering computational insights that complement classical existence theory and potentially informing computer-assisted proofs.

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.

Figures (14)

• Figure 1: Stereographic map and inverse between a sphere and disks
• Figure 2: Training dynamics of the network without RFFs as a function of spherical harmonic order $l$.
• Figure 3: Topology of the Nirenberg Neural Network architecture.
• Figure 4: Minimum losses for spherical harmonic prescribers. Known realisable and unrealisable solutions are shown as ticks and crosses respectively. Unknown solutions are represented by '?'. Band represents the space between highest loss of known realisable and lowest loss of known unrealisable solutions from ensemble of all runs in this work (presented in Figure \ref{['fig:general_loss_scatter']}).
• Figure 5: Gauss-Bonnet violations for the spherical harmonic prescribers. The integral of the predicted scalar curvature over the sphere is performed numerically, using the conformal volume element. The plot shows the relative deviation of this quantity, $\int_{S^2} K_g e^{2u} \, dA_{g_0}$, from the expected value of $2 \pi$. The notation and stylistic choices are the same as in Figure \ref{['fig:sh_loss_scatter']}; accordingly, the band represents the space between highest loss of known realisable and lowest loss of known unrealisable solutions from ensemble of all runs in this work (presented in Figure \ref{['fig:general_gauss_bonnet']}).

Theorems & Definitions (12)

• Theorem 2.1: Kazdan-Warner Identity
• Example 2.1: Spherical Harmonics of even degree
• Theorem 2.2: xu1993remarks
• Proposition 2.1: Classification of Zonal Spherical Harmonics
• proof
• Theorem 2.3
• Theorem 2.4: xu1993remarks
• Corollary 2.4.1: ji2009scalar
• Theorem 2.5: anderson2021nirenberg, Theorem 1.4
• Proposition 2.2: Spectral Pairs