用神经网络求解球面曲率分布的可实现性问题
A Machine Learning Approach to the Nirenberg Problem
- 构建无网格物理信息神经网络，直接学习曲率对应的共形因子
- 可实现曲率损失低至10⁻⁷~10⁻¹⁰，不可实现则显著升高
- 适合几何分析中存在性问题的快速验证与探索
本文提出Nirenberg神经网络：一种针对球面$S^2$上共形度量下指定高斯曲率的数值方法。该无网格物理信息神经网络（PINN）全局参数化共形因子，并通过几何感知损失强制满足曲率方程。利用高斯-邦内特定理进行一致性检验，对学习到的模型进行球谐展开以增强可解释性。对于已知可实现的曲率函数，神经网络损失极低（10⁻⁷~10⁻¹⁰）；不可实现曲率则导致显著更高的损失。该差异可用于评估未知情况，区分可能可实现与不可实现的函数。当前结果表明，神经求解器可在几何分析中作为探索工具，为长期存在的存在性问题提供量化计算视角。
原文摘要 · Abstract (English)
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.
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