Bidirectional Neural Networks for Global Nucleon-Nucleus Optical Model Calculations

TL;DR

The paper develops a differentiable surrogate for nucleon-nucleus scattering within the optical-model framework by introducing a Bidirectional Liquid Neural Network (BiLNN) that maps phase-space coordinates $\rho = k r$ and optical-potential parameters to the radial wave function. This approach generalizes across energies $E$ from $1$ to $200$ MeV, projectile types (protons and neutrons), and target nuclei, achieving a $1.2\%$ relative error on wave functions and accurately reproducing $S$-matrix elements and elastic cross sections. Importantly, it extrapolates to unseen nuclei, indicating learning of the underlying physics rather than memorization, and provides a fully differentiable surrogate suitable for gradient-based inverse problems and uncertainty quantification in nuclear data evaluation.

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 $ρ= kr$ 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.

Figures (7)

• Figure 1: Schematic of the Bidirectional Liquid Neural Network (BiLNN) architecture for predicting scattering wave functions. Left panel: The problem definition, showing the Schrödinger equation with effective potential $\tilde{U} = U + l(l+1)/r^2$, the boundary conditions at the origin and at infinity, and the phase-space coordinate transformation $\rho = kr$. Main panel: The network architecture. The input potential $V(r)$ and associated features are first processed by an encoder network. The encoded representations feed into two parallel liquid layers: the forward layer integrates from $r=0$ toward $r_{\max}$, incorporating the boundary condition $\psi(0)=0$, while the backward layer integrates from $r_{\max}$ toward $r=0$, incorporating asymptotic information. The outputs are concatenated in the combiner, then decoded to produce the real ($\psi_R$) and imaginary ($\psi_I$) parts of the radial wave function $u_l(r)$.
• Figure 2: Comparison of wave functions predicted by BiLNN (dashed red lines) with numerically exact Numerov solutions (solid blue lines) for representative test cases. (a) p+^208Pb${}^{208}$Pb at $E=83$ MeV, $l=0$: $s$-wave showing full penetration to the nuclear interior with $\sim$5 oscillations; (b) p+^208Pb${}^{208}$Pb at $E=72$ MeV, $l=15$: high-$l$ partial wave with centrifugal barrier suppression at small $r$; (c) n+^40Ca${}^{40}$Ca at $E=87$ MeV, $l=0$: neutron $s$-wave without Coulomb effects; (d) n+^40Ca${}^{40}$Ca at $E=36$ MeV, $l=10$: lower-energy neutron with centrifugal suppression. The network accurately reproduces both amplitude and phase across all cases.
• Figure 3: $S$-matrix comparison for p+^208Pb${}^{208}$Pb scattering at three energies: $E = 27$ MeV (left), $E = 83$ MeV (center), and $E = 157$ MeV (right). Top row [(a)--(c)]: $|S_l|$ showing oscillatory absorption at low $l$ with sharp transition to transparency ($|S_l| \to 1$) at the grazing angular momentum $l_g$, which increases with energy from $l_g \approx 12$ at 27 MeV to $l_g \approx 20$ at 157 MeV. Bottom row [(d)--(f)]: $\arg(S_l)$ showing the nuclear and Coulomb phase shifts, with rapid $l$-dependence at low energies that smooths out at higher energies. Blue circles: Numerov reference; red squares: BiLNN prediction.
• Figure 4: Proton elastic scattering angular distributions $d\sigma/d\sigma_{\text{Ruth}}$ for three target nuclei at two energies each. (a,d) ^40Ca${}^{40}$Ca at $E = 27$ MeV and $E = 97$ MeV; (b,e) ^90Zr${}^{90}$Zr at $E = 31$ MeV and $E = 56$ MeV; (c,f) ^208Pb${}^{208}$Pb at $E = 27$ MeV and $E = 83$ MeV. The top row shows lower-energy cases with broader diffraction features; the bottom row shows higher-energy cases with more compressed, sharper minima. Blue solid lines: Numerov reference; red dashed lines: BiLNN prediction. The diffraction patterns, including positions, depths, and widths of multiple minima, are reproduced across up to four orders of magnitude in cross section ratio.
• Figure 5: Neutron elastic scattering angular distributions $d\sigma/d\Omega$ for six target nuclei spanning the training set. (a) ^12C${}^{12}$C at $E = 49$ MeV; (b) ^40Ca${}^{40}$Ca at $E = 34$ MeV; (c) ^90Zr${}^{90}$Zr at $E = 22$ MeV; (d) ^56Fe${}^{56}$Fe at $E = 83$ MeV; (e) ^120Sn${}^{120}$Sn at $E = 74$ MeV; (f) ^208Pb${}^{208}$Pb at $E = 63$ MeV. Blue solid lines: Numerov reference; red dashed lines: BiLNN prediction.