AI for an Inverse Problem: Physical Model Solving Quantum Gravity

Abstract

Mathematical inverse problems of determining a governing differential equation for given solution data remain a fundamental challenge. To find a working example of AI for math, we provide a concrete example using a physical setup of a quantum gravity problem. We present a novel sparse Neural Network (NN) model which is interpretable, to solve the inverse problem: the AdS/CFT correspondence. According to the conjectured correspondence, a special condensed matter system on a ring is equivalent to a gravity system on a bulk disk. The inverse problem is to reconstruct the higher-dimensional gravity metric from the data of the condensed matter system. We use the response functions of a condensed matter system as our data, and by supervised machine learning, we successfully train the neural network which is equivalent to a scalar field equation on an emergent geometry of the bulk spacetime. The developed method may work as a ground for generic bulk reconstruction, i.e. a solution to the inverse problem of the AdS/CFT correspondence. From a technical perspective, to achieve better numerical control, our neural network model incorporates a novel layer that implements the Runge-Kutta method.