We develop a theory of flows in the space of Riemannian metrics induced by
neural network gradient descent. This is motivated in part by recent advances
in approximating Calabi-Yau metrics with neural networks and is enabled by
recent advances in understanding flows in the space of neural networks. We
derive the corresponding metric flow equations, which are governed by a metric
neural tangent kernel, a complicated, non-local object that evolves in time.
However, many architectures admit an infinite-width limit in which the kernel
becomes fixed and the dynamics simplify. Additional assumptions can induce
locality in the flow, which allows for the realization of Perelman's
formulation of Ricci flow that was used to resolve the 3d Poincar\'e
conjecture. We apply these ideas to numerical Calabi-Yau metrics, including a
discussion on the importance of feature learning.

通过神经网络梯度下降在 Riemannian 度量空间中建立流的理论，以近似 Calabi-Yau 度量为动机，并且通过理解神经网络空间中的流进而实现。通过推导相应的度量流方程，我们发现其受到度量神经切向核的控制，这是一个在时间中演化的复杂的非局部对象。然而，许多体系结构可以进行无限宽度的极限，其中核固定且动力学简化。额外的假设可以引入流动的局部性，从而实现 Perelman 的 Ricci 流形式，该流形式曾被用于解决 3D Poincaré 猜想。我们将这些思想应用于数值 Calabi-Yau 度量，包括对特征学习重要性的讨论。

基于神经网络的度量流

Metric Flows with Neural Networks

We extend previous computations of Calabi-Yau metrics on projective
hypersurfaces to free quotients, complete intersections, and free quotients of
complete intersections. In particular, we construct these metrics on generic
quintics, four-generation quotients of the quintic, Schoen Calabi-Yau complete
intersections and the quotient of a Schoen manifold with Z_3 x Z_3 fundamental
group that was previously used to construct a heterotic standard model. Various
numerical investigations into the dependence of Donaldson's algorithm on the
integration scheme, as well as on the Kahler and complex structure moduli, are
also performed.

本文应用 Donaldson 算法对 Calabi-Yau 指标及其在 complete intersections 和 Schoen 流形上的应用进行数值计算，发现由它们的 integration scheme、Kähler 结构和复杂结构模数所决定的许多重要性质。