nLab gradient flow

Idea

For $(X,g)$ a Riemannian manifold and $f : X \to \mathbb{R}$ a smooth function, let

$$\nabla f := g^{-1}(d_{dR} f) \in \Gamma(T X)$$

be the gradient vector field of $X$. The flow induced by this on $X$ is the gradient flow of $f$.

Examples

• Yang-Mills instantons are the gradient flow trajectories of the Chern-Simons action functional.

• Ricci flow is the gradient flow of the action functional of dilaton gravity.

  (This is a key part of Perelman’s proof of the Poincare conjecture.)

• the learning algorithm of a neural network is a gradient descent for the loss function (Thierry & Mieg 2018)

References

• Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré: Gradient Flows – In Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Springer (2008) [doi10.1007/978-3-7643-8722-8]

• Philippe Clément, Introduction to Gradient Flows in Metric Spaces (II) (pdf)

See also:

• Wikipedia: Gradient descent

• Wikipedia: Method of steepest descent