PaperlayerView exact revision on arXiv

Source-linked AI summary

Bidirectional Neural Networks for Global Nucleon-Nucleus Optical Model Calculations

Jin Lei

arXiv:2512.22500v4nucl-th

TL;DR

Scattering calculations need efficient, differentiable methods to support optical-potential optimization and uncertainty quantification. This paper presents a BiLNN emulator using phase-space coordinates and evaluates its wave functions and derived scattering observables against Numerov calculations. It reports accurate observables and generalization to nuclei excluded from training, while noting that comparison with experiment is outside the study’s scope.

  • Problem

    Scattering wave functions combine energy-dependent oscillation scales with amplitudes that can vary by orders of magnitude, challenging a single neural network across the parameter space.

    Source: §I · p. 2

  • Method

    A BiLNN maps optical-potential parameters to wave functions using phase-space coordinates that normalize the oscillation wavelength and radial processing in both directions.

    Source: §I · p. 2, §III · pp. 4–5

  • Results

    Predicted wave functions reproduce Numerov-based scattering angular distributions within a few percent over most angles, capture diffraction patterns spanning more than four orders of magnitude, and generalize to three nuclei outside training.

    Source: §V · p. 13

  • Takeaways & Limitations

    Learning the wave function provides a single predicted quantum state from which phase shifts, S-matrix elements, and elastic and reaction cross sections can be extracted.

    Source: §V · p. 13

    Comparison with experimental data, which would test the optical model’s adequacy, is beyond the scope of this proof-of-concept study.

    Source: §V · p. 14

Abstract

from arXiv · show

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.

I. INTRODUCTION

The introduction motivates a differentiable, broadly generalizing emulator for nucleon–nucleus scattering, where conventional solvers and existing emulators leave practical challenges. It presents phase-space coordinates and a bidirectional network, trained across nuclei, projectiles, energies, and partial waves.

I. INTRODUCTION

  • Numerov is accurate and stable, but its step-by-step procedure is not naturally differentiable, while finite-difference gradients scale with parameter count and can lose accuracy where observables change rapidly.Automatic differentiation can provide exact gradients at a cost comparable to one forward evaluation.Source: §I · p. 1
  • Reduced-order emulators can be highly accurate but require a problem-specific basis, whereas this work trains one network to generalize across nuclei, energies, and partial waves.The introduction frames differentiability as useful for efficient gradients in inverse problems and uncertainty quantification.Source: §I · p. 2
  • Scattering wave functions challenge a single network because their wavelengths vary with energy and their amplitudes can span orders of magnitude through absorption.The de Broglie wavelength is reported to vary from about 9 fm at 10 MeV to about 2 fm at 200 MeV.Source: §I · p. 2
  • The proposed BiLNN processes radial coordinates in both directions, while phase-space coordinates make the wave function oscillate with a universal period across projectile energies.Bidirectional processing incorporates the origin and asymptotic boundary conditions.Source: §I · p. 2
  • 0.6% average relative error is reported for Numerov wave functions across the training domain, spanning twelve nuclei, protons and neutrons, 1–200 MeV, and partial waves 0–30.The training solutions use the KD02 optical potential.Source: §I · p. 2

II. THEORETICAL FRAMEWORK

The framework describes nucleon–nucleus scattering with the radial Schrödinger equation and the central KD02 optical potential. It also specifies a spin-orbit simplification, an extrapolation beyond KD02’s fitted mass range for light nuclei, and numerical convergence checks.

II. THEORETICAL FRAMEWORK

  • The radial Schrödinger equation is solved for partial-wave functions u_l(r), with k and the reduced potential U(r) defined; asymptotic matching extracts S_l using Coulomb functions for protons and Riccati-Hankel functions for neutrons.The potential comprises real and imaginary parts associated with elastic scattering and absorption.Source: §II · p. 3
  • The KD02 implementation uses central volume, surface, and proton Coulomb terms, but deliberately omits spin-orbit coupling, which mainly affects spin observables and backward-angle angular distributions.Including spin-orbit would require treating j = l ± 1/2 channels separately.Source: §II · p. 3
  • KD02 is extrapolated below its fitted lower bound A = 24 for 12C and 16O, so those cases test network handling of a wider mass range rather than physically accurate scattering predictions.The extrapolation extends the smooth mass dependence to lighter nuclei.Source: §II · p. 3
  • Numerov uses a 0.05 fm step and matches near 15 fm; checks at 0.025 fm and 20 fm change training-set S-matrix values by less than 10^-3.Reported comparisons use a multi-radius average.Source: §II · p. 3

III. NEURAL NETWORK ARCHITECTURE

The BiLNN learns a differentiable mapping from optical-potential inputs and physical parameters to radial scattering wave functions. Its spatial, bidirectional liquid architecture combines information from both boundaries, while phase-space coordinates and physics-informed features support generalization across energies and nuclei.

  • The network maps the optical potential and physical parameters to radial wave functions, from which observables such as cross sections and S-matrix elements can be derived.Learning the wave function retains richer physical information than directly fitting cross sections.Source: §III · pp. 3–4
  • Using ρ = kr makes the wavelength 2π in ρ-space regardless of projectile energy, enabling a shared oscillation pattern across energies.The common dimensionless scale uses ρmax = 45.95, computed from the training set's maximum energy and radial extent.Source: §III · p. 4
  • The bidirectional architecture is the dominant factor, while WKB-based phase and absorption features contribute modestly.The recurrent architecture can implicitly learn phase accumulation from potential features, while bidirectional processing incorporates asymptotic boundary information.Source: §III · p. 5
  • Liquid layers treat radius as a spatial propagation coordinate, evolving a hidden state through a learnable ODE aligned with the differential-equation problem.Closed-form continuous-time layers provide analytical solutions and stable gradient propagation.Source: §III · pp. 5–6
  • Forward and backward liquid layers propagate information from the origin and asymptotic boundaries, then combine it to predict the real and imaginary wave-function components.The five components are an encoder, two liquid layers, a combiner, and a decoder.Source: Figure 1 · p. 4, §III · p. 6
  • Training uses Numerov solutions for 12 nuclei, both projectile types, energies from 1 to 200 MeV, and partial waves through l = 30.Approximately 148,800 examples are used; test loss largely flattens by about epoch 350, with training and test curves closely overlapping.Source: §III · pp. 6–7

IV. RESULTS

On held-out parameter combinations, BiLNN reproduces Numerov wave functions and S-matrix elements with approximately 0.6% relative error, while accuracy varies most with projectile energy. It also generalizes to unseen nuclei and supports differentiable optical-model calculations, though downstream optimization and uncertainty applications are not demonstrated here.

Held-out accuracy

  • 0.6% overall relative error on held-out wave functions and approximately 0.6% corresponding S-matrix error were reported for unseen parameter combinations.The test set contains 20% of the data, approximately 29,760 samples; S-matrix extraction averages over 15 asymptotic matching radii.Source: §IV · p. 7

Wave functions

  • Below 1% point-wise agreement held across the radial range for representative proton and neutron cases, including s-waves and centrifugally suppressed partial waves.Figure 3 contrasts BiLNN predictions with Numerov solutions across four projectile, target, energy, and partial-wave cases.Source: §IV · p. 8, Figure 3 · p. 8

Scattering observables

  • For p+208Pb, S-matrix predictions agreed within 0.5% for strongly absorbed waves and 0.3% for peripheral waves across energies of 18, 75, and 197 MeV.The grazing angular momentum shifts from about 8 at 18 MeV to about 22 at 197 MeV.Source: Figure 4 · p. 8, §IV · p. 9
  • Proton diffraction minima were reproduced across up to four orders of magnitude, while neutron predictions captured the cross-section magnitude and oscillatory structure across six targets.Proton discrepancies reach about 50% at the deepest backward-angle minima; neutron scattering lacks Coulomb-nuclear cancellation but remains sensitive at deep minima.Source: §IV · p. 10, Figure 5 · p. 11, Figure 6 · p. 12

Generalization

  • For unseen 24Mg, 63Cu, and 184W, wave-function errors were 0.51%, 0.55%, and 0.86%, respectively, comparable to the 0.6% in-sample test error.The nuclei were selected between training targets in mass, testing interpolation across different proton-to-neutron ratios.Source: §IV · pp. 10–11

Ablations

  • The unidirectional model had 0.81% overall error versus 0.61% for BiLNN; reducing training nuclei from 12 to 6 raised error from 0.6% to 0.73%.The ablation also found 0.53% error for a 256-unit model, with 56% more parameters; a 200-point grid performed worse than the baseline at 0.72%.Source: §IV · pp. 11–12, Table I · p. 12

Differentiability

  • The trained BiLNN provides gradients at a cost comparable to a forward pass, but optimization and uncertainty-quantification applications are not demonstrated in this work.Its gradients are exact for the surrogate but approximate the true gradients to the surrogate's training accuracy.Source: §IV · pp. 12–13

Error distribution

  • S-matrix error averaged 0.6%, with the strongest variation by energy: about 2.5% at 10–20 MeV versus 0.2–0.5% above 50 MeV.Errors were about 1.0–1.3% at low partial waves and generally 0.3–0.8% across targets, rising to about 1% for the heaviest.Source: §IV · p. 10, Figure 7 · p. 13

V. CONCLUSIONS

The BiLNN emulates Numerov wave functions across a broad nucleon–nucleus parameter space and generalizes to unseen nuclei. Its predicted wave functions reproduce scattering observables, while the study identifies limits in precision and scope.

V. CONCLUSIONS

  • Across twelve nuclei, protons and neutrons, 1–200 MeV, and l = 0–30, average relative error is 0.6%; unseen-nucleus wave-function errors are 0.51%, 0.55%, and 0.86%.The unseen-nucleus errors correspond respectively to 24Mg, 63Cu, and 184W.Source: §V · p. 13, Figure 8 · p. 14
  • The phase-space coordinate ρ = kr normalizes oscillation wavelength across energies, while the bidirectional architecture incorporates boundary conditions at both ends.The paper relates this coordinate to generalization across a factor of 4.5 in de Broglie wavelength using one parameter set.Source: §V · pp. 13–14
  • Predicted wave functions yield S-matrix elements and elastic scattering distributions matching Numerov within a few percent over most angles, with larger deviations at backward angles and deep minima.Predictions capture diffraction patterns spanning more than four orders of magnitude; deviations reach 10–20% at backward angles and about 50% at the deepest minima.Source: §V · p. 13
  • The proof of concept emulates the Numerov solver with KD02 inputs, but does not test the optical model against experimental data and omits spin-orbit coupling.The reported 0.6% accuracy is stated to be unsuitable where sub-0.1% precision is required.Source: §V · pp. 13–15
  • Removing bidirectionality increases error by 33%, whereas removing WKB features changes it by about 0.01 percentage point; removing WKB, Sommerfeld, and mass encodings raises error from 0.61% to 0.75%.The authors attribute the smaller WKB contribution to recurrent layers implicitly learning phase accumulation, while global Coulomb-strength and nuclear-size encodings are more impactful.Source: §V · p. 14
  • The BiLNN is positioned for rapid parameter sweeps and Bayesian uncertainty quantification, while PINN-ECS is described as suited to high-accuracy single calculations and requires retraining for each (E, l).The paper distinguishes the approaches by scope rather than a common accuracy metric.Source: §V · pp. 14–15
Loading 2512.22500v4…