论文标题
物理信息搜索RICCI流动，嵌入和可视化的度量解决方案
A physics-informed search for metric solutions to Ricci flow, their embeddings, and visualisation
论文作者
论文摘要
具有嵌入其损失功能（物理信息的神经网络）的PDE的神经网络被用作函数近似器，以找到RICCI流（基于曲率的进化）的解决方案。开发了一种通用方法并应用于真实的圆环。通过将标量曲率的时间演变与使用标准PDE求解器进行比较，该溶液的有效性得到了验证，该标量曲率在整个歧管上降低到0的恒定值为0。我们还考虑了两个实际维度的RICCI流程方程的某些孤子解。我们通过利用嵌入到$ \ mathbb {r}^3 $中来创建流量的可视化。报告了随着时间的流逝，环形度量的高度精确数值演变的快照。我们为在弦理论的背景下确定Ricci Flat Calabi-YAU指标的问题提供了有关该方法的应用的指南，这是复杂几何形状的长期存在问题。
Neural networks with PDEs embedded in their loss functions (physics-informed neural networks) are employed as a function approximators to find solutions to the Ricci flow (a curvature based evolution) of Riemannian metrics. A general method is developed and applied to the real torus. The validity of the solution is verified by comparing the time evolution of scalar curvature with that found using a standard PDE solver, which decreases to a constant value of 0 on the whole manifold. We also consider certain solitonic solutions to the Ricci flow equation in two real dimensions. We create visualisations of the flow by utilising an embedding into $\mathbb{R}^3$. Snapshots of highly accurate numerical evolution of the toroidal metric over time are reported. We provide guidelines on applications of this methodology to the problem of determining Ricci flat Calabi--Yau metrics in the context of String theory, a long standing problem in complex geometry.
