AI in physics: selected studies in classical and quantum mechanics

Abstract

We explore three questions at the intersection of physics and artificial intelligence (AI): Can we use techniques in AI to solve physics-based initial value problems? Using this framework, can we solve Sturm–Liouville problems such as the general Legendre equation and the time-independent Schrödinger equation? And can we use AI to predict thermodynamic parameters of a Bose-condensed cloud of atoms to solve a current bottleneck in quantum fluids research?

Solving initial value problems using AI has yet to be implemented in the strictest sense. Many approaches in the literature require sparse data from the solution of the initial value problem in order to interpolate between these data points. We present the neural initial value problem—a framework which can approximate solutions to a range of dynamical systems without a priori knowledge of the solution. We first consider the most minimally conceivable neural network to solve an initial value problem describing free particle dynamics. We then extend this framework to solve a range of non-linear, coupled, and chaotic problems including the classical pendulum and the Hénon–Heiles system. We introduce probabilistic activation functions—it is our observation that these are required to find solutions of initial value problems. We also introduce coupled neural networks to solve systems of differential equations. These frameworks further allow us to solve eigenvalue problems such as the time-independent Schrödinger equation in a harmonic trap or the Legendre equation.

Numerical schemes for finite-temperature Bose gases require knowledge of the chemical potential and temperature, which are imprecisely and destructively determined in experiments. We demonstrate a proof-of-principle AI that can predict these parameters given only a two-dimensional atomic density profile. The model produces accurate predictions for harmonically trapped condensates and can also produce accurate predictions on toroidally trapped condensates despite never being trained on such a trap. The model can also predict the values of these parameters during the thermalisation procedure. Since predictions happen in fractions of a second, this model could be used for real-time analysis of these parameters in experiments.