对Ricci流的度量解决方案，其嵌入和可视化的物理信息搜索
A physics-informed search for metric solutions to Ricci flow, their embeddings, and visualisation
摘要：
神经网络与偏微分方程嵌入在其损失函数（物理信息神经网络）被用作函数逼近器，以找到解决方案的里奇流（曲率为基础的演变）的黎曼度量。一个通用的方法，并应用于真正的环面。通过将标量曲率的时间演化与使用标准PDE求解器发现的时间演化进行比较，验证了该解决方案的有效性，该解决方案在整个流形上减小到常数值0。我们还考虑了某些孤子解的Ricci流方程在两个真正的维度。我们通过嵌入到$\mathbb{R}^3$中来创建流的可视化。快照的高度精确的数值演变的环形度规随时间的推移报告。我们提供的应用程序的指导方针，这种方法的问题，确定里奇平坦卡拉比-丘度量的背景下，弦论，一个长期存在的问题，在复杂的几何。
英文摘要：
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.
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