Bidirectional Neural Networks for Global Nucleon-Nucleus Optical Model Calculations

A neural network emulator based on Bidirectional Liquid Neural Networks (BiLNN) provides a differentiable mapping from an optical potential to scattering wave functions, enabling gradient-based optimization and uncertainty quantification for nucleon-nucleus scattering. This work is significant because it introduces a single network capable of generalizing across the full parameter space—energies, partial waves, and target nuclei—while maintaining accuracy comparable to traditional Numerov solutions.

How it works

The core innovation is the use of phase-space coordinates, specifically phase-space coordinates ρ = kr, which normalize the oscillation wavelength regardless of projectile energy. This transformation allows a single network to span a vast energy range from 1 to 200 MeV by making the wave function oscillate with a universal period in this new coordinate system. The network is trained on wave functions computed using the Numerov method with the KD02 optical potential, spanning twelve target nuclei (12C to 208Pb), both protons and neutrons, and partial waves up to l = 30.

Bidirectional Liquid Neural Network (BiLNN) Architecture

The network architecture is a Bidirectional Liquid Neural Network (BiLNN) based on Closed-form Continuous-time (CfC) layers. This bidirectional structure processes the radial coordinate sequence in both forward and backward directions, naturally incorporating the two boundary conditions of the scattering problem: the wave function vanishing at the origin and the asymptotic Coulomb behavior at large distances. The architecture consists of five components: an encoder, a forward liquid layer integrating from r = 0 outward, a backward liquid layer integrating from rmax inward, a combiner that merges information from both directions, and a decoder to produce the real (ψR) and imaginary (ψI) parts of the radial wave function.

Input Features and Learning Mechanism

The network is trained to learn a mapping from the optical potential V(r) and physical parameters (energy E, partial wave l, target mass A) to the radial wave function ul(r). At each radial point, the network receives a 9-dimensional feature vector designed to capture specific physical information. Key features include:

**/ρ/ρmax: The normalized phase-space coordinate. 1) **

VR/E and W/E: The real and imaginary parts of the optical potential, normalized by projectile energy, which determines whether the particle is in a classically allowed or forbidden region. 2)

η/ηmax: The Sommerfeld parameter, characterizing Coulomb interaction strength relative to kinetic energy. 3)

sin ϕWKB and cos ϕWKB: WKB phase information derived from the local wave number, providing guidance on expected oscillation phase. 4)

D/Dmax: The cumulative WKB absorption integral, capturing the amplitude envelope due to the imaginary potential. 5)

l/lmax and A1/3/6: Normalized partial wave quantum number and a scale factor related to the nuclear radius.

Training, Optimization, and Generalization

The network is trained by minimizing a cost function defined as the mean squared error between predicted and true wave functions: L = 1/N Σ h(ψ pred R,i − ψ true R,i) squared + (ψ pred I,i − ψ true I,i) squared. Optimization is performed using the AdamW algorithm with a cosine annealing schedule. The training data consists of approximately 148,800 examples generated by solving the Schrödinger equation numerically and interpolating the Numerov solutions onto a 100-point uniform grid. The model achieves an overall relative error of 0.6% across the training domain and successfully generalizes to nuclei not included in training (24Mg, 63Cu, 184W) with comparable accuracy.

Physical Insights and Hierarchy of Design Choices

The study reveals several physical insights:

1. The effectiveness of the phase-space coordinate ρ = kr demonstrates that scattering wave functions possess a universal structure when viewed in the natural units of the de Broglie wavelength, exploiting scale invariance to generalize across energy variations.

2. The success in generalizing to unseen nuclei validates that the network has learned the smooth dependence of the scattering solution on target mass and charge as encoded in the KD02 parameterization rather than memorizing specific targets.

3. The ablation study shows a hierarchy: The bidirectional architecture proves most critical, while WKB input features provide only a modest improvement. Furthermore, the Sommerfeld parameter η and mass encoding A1/3/6 are found to be more impactful than hand-crafted semiclassical features.

Conclusion

The BiLNN serves as a differentiable surrogate Schr¨odinger equation solver, fundamentally different from traditional numerical integrators because it allows for "end-to-end gradient propagation through the surrogate.

As a fastidious researcher, I have analyzed the provided paper, Bidirectional Neural Networks for Global Nucleon-Nucleus Optical Model, and identified several high-impact areas where this methodology can be leveraged to significantly improve existing AI systems.

Here are the specific improvements and capabilities of an improved AI system:

)

1. Improve gradient-based optimization in classical/physical simulations (e.g., fluid dynamics, molecular modeling, quantum chemistry).

2. Enhance uncertainty quantification in high-dimensional parameter spaces (e.g., Bayesian inference for complex physical models).

3. Enable rapid, differentiable exploration of large model parameter sets without requiring expensive numerical differentiation at every step.

)

Improving AI Systems via BiLNN Integration: Specific Improvements and Capabilities

The core innovation presented—a Bidirectional Liquid Neural Network (BiLNN) that learns a mapping from an optical potential to wave functions while maintaining differentiability—offers transformative capabilities for scientific AI systems, particularly those dealing with complex, high-dimensional physical parameter spaces.

Here are the specific improvements and what the improved AI system can achieve:

1. Enhanced Gradient-Based Optimization for Inverse Problems:

2. Differentiable Surrogate Modeling for High-Dimensional Parameter Sweeps:

3. Robust Uncertainty Quantification via Gradient-Informed Sampling:

)

Improving AI Systems via BiLNN Integration: Specific Improvements and Capabilities (Detailed)

1. Enhanced Gradient-Based Optimization for Inverse Problems:

The BiLNN acts as a differentiable surrogate Schrödinger equation solver, replacing opaque numerical integrators (like Numerov). This allows the AI system to compute exact gradients of physical observables (cross sections, S-matrix elements) with respect to the optical model parameters.

• Specific Capability: End-to-end gradient propagation through the surrogate. The AI can use this for efficient parameter optimization of complex potentials where traditional adjoint methods fail or are computationally prohibitive.

• Benefit: Enables rapid and precise tuning of physical models (e.g., optimizing Woods-Saxon parameters in nuclear physics, or interaction strengths in fluid dynamics) by using standard gradient descent algorithms (like AdamW) directly on the objective function, avoiding the linear scaling cost of numerical differentiation.

2. Differentiable Surrogate Modeling for High-Dimensional Parameter Sweeps:

The system learns the underlying functional dependence of a complex physical solution across a vast parameter space (e.g., energy, target mass, partial wave). The phase-space coordinate transformation is key here, as it normalizes oscillation wavelengths universally.

• Specific Capability: Rapid evaluation of millions of potential configurations. Instead of running a computationally expensive forward model for every point in the parameter space to estimate an observable (like cross sections), the AI system can query the trained BiLNN surrogate instantaneously via its differentiable mapping.

• Benefit: Enables efficient Bayesian inference and frequentist uncertainty quantification across high-dimensional spaces (e.g., nuclear data evaluation, materials science). This allows for systematic exploration of parameter uncertainties with high throughput, crucial for training robust machine learning models in physics.

3. Robust Uncertainty Quantification via Gradient-Informed Sampling:

The differentiability of the network provides exact gradients at a cost comparable to a single forward evaluation.

• Specific Capability: Gradient-informed sampling and sensitivity analysis. The AI system can use these exact gradients to guide sampling strategies (e.g., using techniques like gradient-based importance sampling) to efficiently explore the posterior distribution of model parameters, rather than relying on slow, brute-force methods (like parametric bootstrap).

• Benefit: Provides systematic uncertainty propagation in complex physical models with high fidelity. This is vital for determining which parameters have the most significant impact on a physical outcome, leading to more reliable and trustworthy scientific predictions.

)

Summary of AI System Improvements:

The BiLNN transforms the AI system from a black-box predictor into an active, gradient-aware solver surrogate. The improved system can now perform:

1. Optimization of complex physical models using efficient gradient descent on the learned mapping, bypassing costly numerical differentiation.

2. High-throughput Bayesian uncertainty quantification across vast parameter spaces by leveraging instantaneous forward evaluations and exact gradients for efficient sampling.

3. Physics-guided discovery: The AI can learn universal functional relationships (like scale invariance via the phase-space coordinate) encoded in the solution structure, allowing it to generalize effectively to unseen physical configurations (e.g., new nuclei or energy regimes).

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 (C to 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 (Mg, Cu, 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.