Black Hole Black Boxes: Numerical Black Hole Metrics via AInstein Neural Networks
For numerical relativity researchers, this provides a novel method to discover black hole metrics, but the results are preliminary and only validated on a known solution.
The paper extends the AInstein PINN architecture to Lorentzian signature, recovering the Schwarzschild metric and demonstrating a method to search for novel black hole solutions by enforcing constraints like the Petrov speciality index and trapped-surface conditions.
The AInstein architecture introduced an unsupervised neural method for solving the Riemannian Einstein equations on arbitrary manifolds. This Physics Informed Neural Network approach (PINN) is extended here to Lorentzian signature, validated by recovering the maximally extended Schwarzschild geometry, and tested as novel search method for arbitrary black hole solutions. The topology is built into the architecture by treating $S^{2}$ globally through its standard embedding, such that the network learns an ambient metric on the manifold $\mathbb{R}^{2} \times \mathbb{R}^{3}$, where Penrose coordinates are chosen for $\mathbb{R}^2$ and the metric on $S^{2}$ is obtained by pullback. The architecture is first trained with the objective of recovering the Schwarzschild metric via losses encoding the vacuum Einstein equation, a quadratic Weyl scalar constraint, and the $SO(3)$ symmetry of the resultant metric; directly motivated by the Birkhoff--Jebsen theorem. Following this, the objective is generalised to use the Petrov speciality index, a horizon curvature anchor, and a trapped-surface constraint, to allow search for algebraically general Petrov type I solutions, finding potentially novel general-type Lorentzian Einstein metrics with a genuinely trapped interior.