Bidirectional Neural Networks for Global Nucleon-Nucleus Optical Model Calculations

This paper presents a differentiable Bidirectional Liquid Neural Network emulator that accurately maps optical potential parameters to nucleon-nucleus scattering wave functions across a broad energy range and diverse nuclei, enabling efficient gradient-based parameter optimization and uncertainty quantification while successfully generalizing to unseen targets.

The Gist

Imagine you are trying to predict how a billiard ball (a proton or neutron) will bounce off a complex, fuzzy target (an atomic nucleus). In the world of nuclear physics, this is called "scattering." To do this accurately, scientists use a set of rules called the "Optical Model," which involves solving a very difficult math problem known as the Schrödinger equation.

Traditionally, solving this equation is like trying to walk through a dark forest one step at a time using a very precise, but slow, map-reading method (called the Numerov algorithm). You have to take every single step carefully to get to the other side. While accurate, this process is rigid. If you want to know how the path changes when you tweak the forest's layout slightly, you have to restart the whole walk from scratch. This makes it very hard to do "what-if" scenarios or to find the perfect forest layout that matches real-world experiments.

The Big Idea: A "Magic" Shortcut
The author of this paper, Jin Lei, has built a "neural network emulator." Think of this not as a faster walker, but as a super-smart GPS that has memorized the entire forest. Instead of walking step-by-step, you give the GPS the layout of the forest (the potential), and it instantly tells you exactly where the ball will be at every point.

But here is the magic trick: This GPS is differentiable. In plain English, this means it doesn't just give you the answer; it can also tell you how the answer would change if you tweaked the forest layout. It's like having a GPS that not only shows you the route but also whispers, "If you move that tree 1 inch to the left, your arrival time changes by 0.2 seconds." This allows scientists to use powerful computer algorithms to automatically fine-tune their models, something the old step-by-step method couldn't do easily.

The Two Big Hurdles (and How They Were Solved)
Building this GPS was tricky because of two major problems:

1. The "Zoom" Problem: At low energy, the billiard ball moves slowly and has a long "wavelength" (it wiggles slowly). At high energy, it moves fast and wiggles very quickly. It's like trying to teach a single camera to take clear photos of both a slow-moving snail and a speeding race car. The patterns look completely different.

   • The Solution: The author invented a new way to measure distance called "phase-space coordinates." Instead of measuring distance in meters (which changes the pattern), they measure it in "wiggles." Imagine stretching a rubber band so that one full wiggle always takes up the same amount of space, no matter how fast the ball is moving. This makes the pattern look the same to the computer, regardless of speed, allowing one single network to handle energies from very slow to very fast.

2. The "Two-Way Street" Problem: The physics problem has rules at both ends: the ball starts at zero at the center of the nucleus, and it behaves in a specific way far away from the nucleus. A standard computer program usually reads from left to right. It knows the start, but it doesn't "know" the finish until it gets there, which makes it hard to get the middle part right.

   • The Solution: The author used a Bidirectional Liquid Neural Network. Imagine two people reading a book to solve a mystery. One reads from the beginning (the center of the nucleus) forward, and the other reads from the end (far away) backward. They meet in the middle and combine their notes. This "two-way" approach ensures the solution respects the rules at both ends simultaneously, leading to much higher accuracy.

What Did They Find?
The author trained this "GPS" on data for 12 different types of atomic nuclei (from light Carbon to heavy Lead) and for both protons and neutrons.

• Accuracy: The GPS is incredibly accurate, with an error rate of only 0.6%. It can predict the path of the ball so well that it reproduces complex "diffraction patterns" (the ripples and shadows created by the scattering) across a massive range of energies.
• Generalization: The real test was whether the GPS could handle a nucleus it had never seen before. The author tested it on three new nuclei (Magnesium, Copper, and Tungsten) that were not in the training data. The GPS got them right with similar accuracy. This proves the computer didn't just "memorize" the training data; it actually learned the underlying physics rules.

Why Does This Matter?
The paper emphasizes that the main goal wasn't just to make calculations faster (though it is fast). The main goal was to create a tool that is mathematically smooth and differentiable.

Think of the old method as a jagged, rocky path where you can't easily slide down to find the lowest point. The new method is a smooth, slippery slide. This allows scientists to use advanced mathematical techniques to automatically adjust their models to match experimental data and to understand the uncertainty in their predictions.

What It Doesn't Do (Yet)
The paper is clear about its limits:

• It currently ignores a specific interaction called "spin-orbit coupling" (a subtle twist in the physics), though the author notes this could be added later.
• It is a "proof of concept." The author built the engine and proved it runs, but hasn't yet used it to solve specific real-world nuclear data problems or medical applications.
• It is an emulator of a specific mathematical model (KD02), not a direct replacement for all experimental data.

In short, the author has built a smart, flexible, and mathematically friendly "surrogate" for a difficult physics problem, allowing scientists to finally use gradient-based optimization to understand nuclear reactions in a way that was previously impossible.

Technical Summary

Technical Summary: Bidirectional Neural Networks for Global Nucleon-Nucleus Optical Model Calculations

Problem Statement
Modern nuclear data evaluation and theoretical physics increasingly require efficient methods for uncertainty quantification and parameter optimization in nucleon-nucleus scattering. These tasks rely on differentiable solvers to compute gradients of observables with respect to potential parameters. Traditional numerical approaches, such as the Numerov algorithm for solving the radial Schrödinger equation, are highly accurate but fundamentally discrete and non-differentiable. While numerical differentiation (finite differences) can be applied, it scales linearly with the number of parameters and suffers from accuracy degradation in regions where observables change rapidly. Furthermore, existing reduced-order emulators (e.g., eigenvector continuation) often require problem-specific bases that must be rebuilt for new systems, limiting their global applicability. The challenge is to develop a solver that is not only fast but inherently differentiable, capable of generalizing across a wide parameter space of energies, target nuclei, and partial waves without retraining.

Methodology
The author presents a neural network emulator based on a Bidirectional Liquid Neural Network (BiLNN) architecture designed to map optical potential parameters directly to scattering wave functions. The methodology relies on two primary innovations to enable generalization across the full parameter space (1–200 MeV, $^{12}$C to $^{208}$Pb, $l=0$ to $30$):

1. Phase-Space Coordinate Transformation: To address the challenge of varying de Broglie wavelengths across the energy range (a factor of 4.5 variation), the network utilizes a dimensionless phase-space coordinate $\rho = kr$. This transformation normalizes the oscillation wavelength to a universal period of $2\pi$ regardless of projectile energy, allowing a single network to learn a universal oscillation pattern rather than energy-specific ones.
2. Bidirectional Liquid Architecture: The network employs Closed-form Continuous-time (CfC) layers, originally developed for temporal dynamics, reinterpreted here as spatial propagators along the radial coordinate. A bidirectional structure processes the radial sequence in both forward (from $r=0$) and backward (from $r_{max}$) directions. This design naturally incorporates the two boundary conditions of the scattering problem: the wave function vanishing at the origin and the asymptotic Coulomb behavior at large distances.

The network is trained on a dataset of approximately 148,800 examples generated by solving the radial Schrödinger equation using the Numerov method with the global Koning-Delaroche (KD02) optical potential. The input features include normalized phase-space coordinates, potential components ($V/E, W/E$), the Sommerfeld parameter, WKB phase and absorption integrals, partial wave quantum numbers, and target mass scaling. The model predicts the real and imaginary parts of the radial wave function $u_l(r)$, from which S-matrix elements and cross sections are derived via standard asymptotic matching.

Key Contributions

• Differentiable Surrogate: The work establishes a differentiable mapping from optical potentials to wave functions, enabling gradient-based optimization and uncertainty quantification without the cost of numerical differentiation.
• Global Generalization: Unlike previous emulators that require system-specific bases, this single network generalizes across twelve training nuclei, both proton and neutron projectiles, and a broad energy range. Crucially, it successfully generalizes to unseen nuclei ($^{24}$Mg, $^{63}$Cu, $^{184}$W) with comparable accuracy, demonstrating that it has learned the smooth functional dependence encoded in the KD02 parameterization rather than memorizing specific targets.
• Architectural Design: The paper demonstrates that the bidirectional architecture is the critical component for accuracy, providing a 33% relative error reduction compared to a unidirectional variant. It also shows that explicit WKB input features provide only marginal improvements, suggesting the recurrent layers implicitly learn phase accumulation from potential features.

Results

• Wave Function Accuracy: The trained BiLNN achieves an overall relative error of 0.6% in reproducing Numerov wave functions across the training domain. The error is energy-dependent, ranging from $\sim 2.5\%$ at low energies (10–20 MeV) to $<0.5\%$ at high energies ($>50$ MeV).
• Observables: The predicted wave functions yield accurate S-matrix elements and elastic scattering cross sections. The model reproduces diffraction patterns spanning four orders of magnitude.
  • S-Matrix: The error in S-matrix modulus and phase is approximately 0.6%.
  • Cross Sections: For forward angles, errors are modest (1–10%). At backward angles, where destructive interference between Coulomb and nuclear amplitudes leads to small total amplitudes, errors can reach $\sim 50\%$. The author attributes this to the fundamental numerical sensitivity of backward-angle scattering to small phase errors, not a specific failure of the network. Integral quantities like reaction cross sections remain accurate to well below 1%.
• Generalization: For unseen nuclei ($^{24}$Mg, $^{63}$Cu, $^{184}$W), the wave function errors are 0.51%, 0.55%, and 0.86% respectively, comparable to the in-sample test error.

Significance and Claims
The paper positions this work as a proof-of-concept for a differentiable surrogate solver in nuclear physics. The author explicitly states that the primary advantage is not computational speed (though the network is fast) but differentiability, which bridges the gap between forward modeling and inverse inference.

The significance lies in enabling:

1. Gradient-based optimization of optical model parameters.
2. Systematic uncertainty quantification through gradient-informed sampling.

The author is modest regarding the current scope:

• The accuracy (0.6%) is sufficient for parameter sweeps and Bayesian uncertainty quantification but is lower than conventional Numerov solvers and unsuitable for applications requiring sub-0.1% precision.
• The current implementation is limited to central potentials; spin-orbit coupling is omitted but described as a straightforward extension.
• The model is validated as an emulator of the Numerov solver with KD02 inputs; it is not claimed to be a direct fit to experimental data or a test of the optical model's physical adequacy.
• The work does not demonstrate the downstream applications (optimization, uncertainty quantification) but establishes the necessary differentiable architecture for them.

Future work is identified as utilizing this differentiable surrogate for the gradient-based optimization of optical model parameters and uncertainty quantification.