AInstein: Machine Learning "Special" (Pseudo)-Riemannian Metrics
by
Sala Paoluzi
A numerical scheme based on semi-supervised machine learning, "AInstein", was recently introduced (see https://iopscience.iop.org/article/10.1088/3050-287X/ae1117) to approximate generic Riemannian Einstein metrics on a given manifold. Its versatility stems from encoding the differentiable structure directly in the loss function, making the method applicable to manifolds constructed in a "bottom-up" fashion that admit no natural embedding in R^n. A limitation, however, is that the resulting numerical metric is not inherently global.
To address this, we introduce a new approach for all real (n-1)-dimensional manifolds that can be embedded in R^n, in which the neural-network ansatz is automatically globally defined. After a brief review of the original AInstein model, the talk presents novel results obtained with the new architecture, including applications to two open problems: the Kazdan–Warner (prescribed curvature) problem on S^2 and the search for negative-curvature metrics on S^4 and S^5. Finally, we focus on a further extension of the method to Lorentzian metrics, presenting some preliminary results concerning black holes.