Artificial Intelligence (AI) can be used for finding structures in data from scratch, i.e. without knowing about them beforehand. One might wonder whether AI can be used to decide whether a dynamical system is integrable or not? In this talk, I report on how we can set up machines to successfully search for the Lax pair and Lax connection in some classical examples. We shall also see how symbolic regression can be used to render the output interpretable for a mathematical physicist.

I briefly give an overview on the key Machine Learning frameworks involved in this analysis (neural networks, auto-differentiation, representation learning). This talk is mainly based on 2103.07475, and further related work can be found in 2104.14444, 2003.13679.

We consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat-Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a $p$-adic version of entropy which obeys a Ryu-Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one $p$-adic backgrounds, along with a Bekenstein-Hawking-type formula for black hole entropy. We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated).

Using the tensor Radon transform and related numerical methods, we study how bulk geometries can be explicitly reconstructed from boundary entanglement entropies in the specific case of AdS$_3$/CFT$_2$. We find that, given the boundary entanglement entropies of a 2d CFT, this framework provides a quantitative measure that detects whether the bulk dual is geometric in the perturbative (near AdS) limit. In the case where a well-defined bulk geometry exists, we explicitly reconstruct the unique bulk metric tensor once a gauge choice is made. We then examine the emergent bulk geometries for static and dynamical scenarios in holography and in many-body systems. Apart from the physics results, our work demonstrates that numerical methods are feasible and effective in the study of bulk reconstruction in AdS/CFT.