Calabi-Yau manifolds play a crucial role in string compactifications. Yau's theorem guarantees the existence of a metric that satisfies the string's equation of motion. However, Yau's proof is non-constructive, and no analytic expressions for metrics on Calabi-Yau threefolds are known. We use machine learning, more precisely neural networks, to learn Calabi-Yau metrics and their Kahler and complex structure moduli dependence.
I will start with an introduction to Calabi-Yau manifolds and their moduli. I will then illustrate in an example how we train neural networks to find Calabi-Yau metrics by solving a Monge-Ampere type partial differential equation. The approach generalizes to manifolds with reduced structure, such as SU(3) structure or G2 manifolds, which feature in string compactifications with flux and in the M-theory formulation of string theory, respectively. I will illustrate this generalization for a particular SU(3) structure metric and compare the machine learning result to the known, analytic expression.