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

TL;DR

This work demonstrates that physics-informed neural networks (PINNs) can be successfully deployed to solve the deuteron ground-state problem with realistic nucleon-nucleon interactions in momentum space. By embedding the Schrödinger equation, boundary conditions, normalization, and a variational energy objective into a unified loss function, the authors achieve high-precision benchmarks against exact diagonalization, including a relative energy error on the order of $10^{-6}$ for at least one interaction. The study covers both coordinate-space and momentum-space formulations, using Minnesota, $N^4LO\,$(chiEFT), and CD-Bonn potentials, and highlights training dynamics, mesh choices, and normalization strategies. The results establish PINNs as a promising route for ab initio nuclear structure calculations and lay the groundwork for extending to three-dimensional problems and three-nucleon forces in more complex nuclei.

Abstract

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 Schrödinger 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.

Figures (6)

• Figure 1: Ground state of the deuteron. The red dots show the predictions of the eigenfunction, as obtained after approximately 40000 training epochs. This is compared to the Minnesota potential in the $^3S_1$ channel as a blue line.
• Figure 2: Training history for the Minnesota potential. (a) Behavior of the predicted nuclear eigenvalue. The purple line shows the energy of the deuteron predicted by the PINN, while the blue horizontal line is the experimental energy. (b) Partial contributions to the loss function. The green line is for the integral loss $\mathcal{L}_{integ}$, yellow is for the normalization loss $\mathcal{L}_{norm}$, brown shows the boundary conditions loss $\mathcal{L}_{BCs}$ and the black line is the differential equation loss $\mathcal{L}_{schrod}$. We considered a loss to be converged once it reaches below the dashed line, which corresponds to a hyperparameter that has been set to $100$. (a) Total loss, Eq. \ref{['eq: loss_overall']}, through training.
• Figure 3: Ground state of the deuteron for the $N{}^4LO$$\chi$EFT interaction. The red and green dots are the PINN solutions for the $^3S_1$ and $^3D_1$ wavefunction, obtained after $4 \times 10^5$ epochs of training. The blue and orange lines are the exact solutions for the $^3S_1$ and $^3D_1$ components, respectively.
• Figure 4: Behavior of the PINN network for the $N^4LO$ interaction during training. (a) Ground state energy. The purple line shows the energy of the deuteron predicted by the PINN, while the blue horizontal line is the experimental energy. (b) Fidelity computed with respect to the exact diagonalization. (c) Total loss.
• Figure 5: Behavior of the partial losses through training for the $N^4LO$ interactions. The yellow line is the normalization loss $\mathcal{L}_{norm}$, in brown is the boundary conditions loss $\mathcal{L}_{BCs}$ and the black line is the differential equation loss $\mathcal{L}_{Schrod}$. A loss is considered converged once it passes the blue dashed line, which is a hyperparameter, here set at $1$.