Machine learning the deuteron: new architectures and uncertainty quantification

TL;DR

This paper employs neural networks to accurately solve the deuteron ground state in momentum space, exploring new architectures and uncertainty quantification methods, with findings applicable to broader quantum mechanics problems.

Contribution

It introduces improved neural network architectures and a refined uncertainty estimation approach for solving quantum systems like the deuteron.

Findings

Simple one-layer network performs best.

Two-layer networks tend to overfit.

Uncertainty from model oscillations exceeds initial stochastic uncertainties.

Abstract

We solve the ground state of the deuteron using a variational neural network ansatz for the wave function in momentum space. This ansatz provides a flexible representation of both the $S$ and the $D$ states, with relative errors in the energy which are within fractions of a percent of a full diagonalisation benchmark. We extend the previous work on this area in two directions. First, we study new architectures by adding more layers to the network and by exploring different connections between the states. Second, we provide a better estimate of the numerical uncertainty by taking into account the final oscillations at the end of the minimisation process. Overall, we find that the best performing architecture is the simple one-layer, state-independent network. Two-layer networks show indications of overfitting, in regions that are not probed by the fixed momentum basis where calculations are performed. In all cases, the error associated to the model oscillations around the real minimum is larger than the stochastic initialisation uncertainties. The conclusions that we draw can be generalised to other quantum mechanics settings.