Learning geometries beyond asymptotic AdS

A bstract We present a data-driven method for holographic bulk reconstruction that works even when the spacetime is not asymptotically AdS. Given the data of boundary Green functions within a finite frequency window, we iteratively adjust a bulk metric with a finite radial cutoff until its holographic Green functions reproduce the boundary data. Based on the holographic Wilsonian renormalization group for the Klein-Gordon equation in an undetermined curve space, we construct a radial flow equation and transform it into a Neural ODE, which is an infinite-depth neural network for modeling continuous dynamics. Assuming the double-trace coupling h in the Wilsonian action is real, we demonstrate that the Neural ODE can effectively learn the metrics with AdS, Lifshitz, and hyperscaling-violating asymptotics. In particular, we apply the algorithm to the Sachdev-Ye-Kitaev (SYK) model, which slightly deviates from the conformal limit. In the hyperparameter space spanned by the rescaled temperature $$ \overline{T} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> and the radial cutoff ϵ , we identify a critical curve along which the learned metric is close to an AdS 2 black hole with finite cutoff. We derive an approximate analytical expression for this curve, from which an effective bulk dual of the SYK coupling v is established. Our work provides a promising way for using machine learning to depict the novel bulk geometry dual to the non-conformal boundary systems in the real world.