Bidirectional Neural Networks for Global Nucleon-Nucleus Optical Model Calculations

TL;DR

This paper introduces a neural network emulator based on Bidirectional Liquid Neural Networks that accurately models nucleon-nucleus scattering across a wide energy range, enabling efficient parameter optimization and uncertainty quantification.

Contribution

The authors develop a fully differentiable neural network model that generalizes across energies and nuclei, capturing the physics of the optical model without overfitting to specific targets.

Findings

Achieves 1.2% relative error in wave function predictions

Successfully extrapolates to unseen nuclei with comparable accuracy

Enables gradient-based optimization of optical model parameters

Abstract

Modern nuclear data evaluation increasingly requires not only accurate scattering calculations, but also efficient methods for uncertainty quantification and parameter optimization, tasks that benefit from differentiable solvers amenable to gradient-based algorithms. I present a neural network emulator based on Bidirectional Liquid Neural Networks (BiLNN) that provides a fully differentiable mapping from optical potential parameters to scattering wave functions. The key innovation enabling generalization across the parameter space is the use of phase-space coordinates $\rho = kr$ that normalize the oscillation wavelength regardless of projectile energy, allowing a single network to span 1 to 200~MeV. Trained on Numerov solutions for twelve target nuclei (\nuc{12}{C} to \nuc{208}{Pb}), both protons and neutrons, and partial waves up to $l=30$, the network achieves an overall relative error of 1.2\%. The predicted wave functions yield accurate $S$-matrix elements and elastic scattering cross sections, reproducing diffraction patterns spanning four orders of magnitude. Importantly, the model extrapolates successfully to nuclei not included in training (\nuc{24}{Mg}, \nuc{63}{Cu}, \nuc{184}{W}) with comparable accuracy, demonstrating that it has learned the physics of the optical model rather than memorizing specific targets. The differentiable nature of the trained model opens the door to gradient-based optimization of optical model parameters and efficient uncertainty quantification.