Addressing the ground state of the deuteron by physics-informed neural networks

Machine learning techniques have proven to be effective in addressing the structure of atomic nuclei. Physics$-$Informed Neural Networks (PINNs) are a promising machine learning technique suitable for solving integro-differential problems such as the many-body Schr\"odinger problem. So far, there has been no demonstration of extracting nuclear eigenstates using such method. Here, we tackle realistic nucleon-nucleon interaction in momentum space, including models with strong high-momentum correlations, and demonstrate highly accurate results for the deuteron. We further provide additional benchmarks in coordinate space. We introduce an expression for the variational energy that enters the loss function, which can be evaluated efficiently within the PINNs framework. Results are in excellent agreement with proven numerical methods, with a relative error between the value of the predicted binding energy by the PINN and the numerical benchmark of the order of $10^{-6}$. Our approach paves the way for the exploitation of PINNs to solve more complex atomic nuclei.