Active learning emulators for nuclear two-body scattering in momentum space

This paper extends active learning emulators with error estimation to coupled-channel nuclear two-body scattering in momentum space by employing Lippmann-Schwinger-based reduced-order models trained via greedy algorithms, demonstrating high accuracy and computational speedup for phase shifts and cross sections to facilitate future Bayesian calibrations of nuclear interactions.

The Gist

Imagine you are trying to predict how two tiny, invisible billiard balls (atomic nuclei) bounce off each other. In the world of nuclear physics, these aren't just simple balls; they are complex clouds of energy and force. To predict exactly how they scatter, scientists use incredibly detailed mathematical models called Full-Order Models (FOMs).

Think of the Full-Order Model as a super-accurate, high-definition weather simulation. It accounts for every single wind gust, temperature shift, and humidity level. It gives you the perfect answer, but it takes a massive supercomputer days to run just one simulation. If you wanted to run this simulation a million times to test different theories, you'd be waiting for centuries.

This paper introduces a clever solution: Active Learning Emulators.

The "Smart Shortcut" Analogy

Imagine you are a master chef trying to perfect a new soup recipe. You have a "Gold Standard" recipe (the Full-Order Model) that takes 10 hours to cook and taste-test. You want to know how the soup tastes if you change the amount of salt, pepper, or garlic.

Instead of cooking the soup 10,000 times (which would take forever), you use a Smart Emulator.

1. The Snapshot Phase: You cook the soup a few times with different ingredients. These are your "snapshots."
2. The Greedy Algorithm (The Smart Taster): Instead of guessing where to cook next, you have a smart taster. After every few soups, the taster looks at the "flavor map" and says, "We know how it tastes with 1 tsp of salt and 2 tsp of pepper, but we have no idea what happens if we use 1.5 tsp of salt and 3 tsp of pepper. That's the biggest gap in our knowledge!"
3. The Learning: You only cook the soup for that specific, most uncertain combination. You add that new taste to your "flavor map."
4. The Result: After just a handful of real cooking sessions, you have built a Reduced-Order Model (ROM). This is a simplified, super-fast version of the recipe that can predict the taste of any ingredient combination in a split second, with almost the same accuracy as the 10-hour version.

What This Paper Actually Did

The authors (A. Giri, J. Kim, and colleagues) took this "Smart Taster" approach and applied it to the complex world of nuclear scattering.

• The Old Way: Previous attempts to do this were like trying to predict the weather by looking at a flat map (coordinate space). It worked okay for simple things, but nuclear forces are tricky and often need to be viewed in "momentum space" (like looking at the speed and direction of the particles rather than just their position).
• The New Way: They built a new emulator that works in momentum space. This is crucial because it allows them to use the most modern, high-tech nuclear theories (called "chiral interactions") that were previously too hard to use with these shortcuts.

Key Features of Their New Tool

1. The "Greedy" Taster
They use an algorithm that acts like a detective. It constantly asks, "Where is our knowledge weakest?" and then runs a single, high-precision calculation exactly there. This ensures they don't waste time calculating things they already understand well.

2. Error Estimation (The "Safety Net")
Usually, when you use a shortcut, you don't know how wrong you might be. This paper is special because the emulator comes with a built-in error meter. It doesn't just give you an answer; it tells you, "I am 99.9% sure this answer is correct, and here is the tiny margin of error." This is vital for scientists who need to know exactly how much they can trust their results.

3. Speed and Efficiency
They wrote their code using a special tool called JAX (think of it as a super-charged engine for math).

• The Result: Their emulator is 100 times faster than the original, slow supercomputer model.
• The Analogy: If the original model was a snail crawling across a continent, the new emulator is a jet plane. It can run thousands of simulations in the time it takes the old one to run one.

4. The Bayesian Calibration (The "Fine-Tuning")
The ultimate goal is to use these fast emulators to "calibrate" nuclear theories. Imagine you have a radio that is slightly out of tune. You want to find the exact knob settings (parameters) that make the music perfect.

• Because the emulator is so fast and knows its own error margins, scientists can now use Bayesian statistics (a method of updating beliefs based on new evidence) to find the perfect settings for nuclear forces.
• They tested this by trying to match their model to real-world data on how neutrons and protons scatter. The emulator successfully found the "perfect tune" for the nuclear forces, proving the method works.

Why Does This Matter?

In the past, calibrating nuclear theories was like trying to tune a radio by turning the knob once a day and waiting for the sun to rise to see if the music improved. It was too slow.

Now, with these Active Learning Emulators, scientists can tune the radio in real-time. They can:

• Test thousands of nuclear theories in minutes.
• Quantify exactly how uncertain their predictions are.
• Move closer to understanding the fundamental forces that hold the universe together, from the smallest atoms to the largest stars.

In a nutshell: The authors built a "smart, fast, and self-aware" calculator that learns nuclear physics faster than ever before, allowing scientists to finally tune their theories with the precision they've been dreaming of.

Technical Summary

1. Problem Statement

Modern nuclear theory relies heavily on Bayesian statistical methods to rigorously quantify uncertainties in nuclear interactions (e.g., chiral Effective Field Theory). These methods require repeated evaluations of high-fidelity models (Full-Order Models, or FOMs) to calibrate Low-Energy Constants (LECs) against experimental scattering data.

• The Bottleneck: Solving the scattering equations (typically the Lippmann-Schwinger integral equation) for two-body systems is computationally expensive. Performing thousands of these calculations for Bayesian inference is often prohibitively slow.
• The Gap: While reduced-order models (ROMs) and emulators have been developed for coordinate-space scattering, there is a need for efficient, accurate emulators in momentum space that can handle coupled-channel scattering and provide quantified error estimates. Momentum space is essential for accessing modern chiral interactions and optical models.

2. Methodology

The authors extend active learning emulators from coordinate space to momentum space using the following framework:

A. Full-Order Model (FOM)

• Governing Equation: The Lippmann-Schwinger (LS) integral equation for the scattering $t$-matrix is used as the high-fidelity solver, rather than the radial Schrödinger equation.
• Formalism: The LS equation is discretized using a quadrature rule for principal value integration, converting the integral equation into a system of coupled linear equations ($A(\theta)t(\theta) = b(\theta)$).
• Affine Decomposition: To enable efficient ROM construction, the nuclear potential $V$ is assumed to have an affine dependence on parameters $\theta$ (the LECs):
  $$V(k, k'; \theta) = \sum_{a=0}^{n_\theta} h_a(\theta) V_a(k, k')$$
  This allows the separation of parameter-dependent functions from momentum-dependent functions.

B. Reduced-Order Models (ROMs)

Two projection-based ROMs are constructed:

1. Galerkin ROM (G-ROM): Projects the residual of the LS equation onto the reduced basis space.
2. Least-Squares Petrov-Galerkin ROM (LSPG-ROM): Minimizes the norm of the residual in a semi-reduced space, offering robustness against singularities.

• Basis Construction: The reduced basis is formed from "snapshots" (high-fidelity solutions) selected via a Greedy Algorithm.
• Active Learning: Instead of random sampling (e.g., Latin Hypercube Sampling), the greedy algorithm iteratively selects new snapshot locations where the estimated emulator error is maximized. This minimizes the number of required high-fidelity calculations.

C. Error Estimation and Propagation

• Residual-Based Estimation: The emulator error is estimated using the norm of the residual vector in the semi-reduced space, avoiding expensive full-space calculations during the online stage.
• Propagation: The error in the discretized $t$-matrix is propagated to physical observables (phase shifts and total cross-sections) using the triangle inequality to establish upper bounds.
• Bayesian Framework: The emulator error is treated as a uniform distribution and combined with EFT truncation errors (modeled via the BUQEYE collaboration's framework) to construct a likelihood function for Bayesian parameter estimation.

D. Implementation

• The entire framework (FOM and ROM) is implemented in Python using Google's JAX library.
• JAX enables Just-In-Time (JIT) compilation and automatic vectorization, allowing for highly efficient offline-online decomposition and parallel evaluation across multiple energies and channels.

3. Key Contributions

1. Momentum Space Extension: Successfully extended active learning emulators to coupled-channel scattering in momentum space, enabling the use of a wider range of modern chiral interactions (including non-local potentials).
2. Error Quantification: Developed a rigorous method to estimate emulator errors for both the $t$-matrix and total cross-sections, allowing these errors to be included in Bayesian calibration.
3. Greedy vs. POD Comparison: Demonstrated that the greedy active learning approach achieves accuracy comparable to the Proper Orthogonal Decomposition (POD) method but requires significantly fewer high-fidelity snapshots (computational savings).
4. Proof-of-Principle Bayesian Calibration: Performed a Bayesian calibration of a chiral NN potential (GT+ potential) against total cross-section data, successfully inferring LECs while accounting for emulator errors.
5. High-Performance Software: Provided a publicly available, JAX-based software framework that achieves speedup factors of up to 100x compared to the optimized FOM solver.

4. Results

• Convergence: The greedy algorithm showed exponential convergence for both uncoupled ($^1S_0$) and coupled ($^3S_1$–$^3D_1$) channels. Accurate results ($\epsilon < 10^{-6}$) were achieved with basis sizes of $N_b \approx 7$ for uncoupled and $N_b \approx 20$ for coupled channels, far fewer than the candidate snapshot pool (150 points).
• Accuracy:
  • Phase Shifts: Emulators reproduced phase shifts and mixing angles with relative errors $< 10^{-5}$ (for tolerance $\alpha=10^{-4}$) and $< 10^{-7}$ (for $\alpha=10^{-7}$).
  • Cross Sections: Total cross-sections were accurately reproduced up to $E_{lab} = 300$ MeV.
  • Unitarity: The emulators preserved the unitarity of the S-matrix to within machine precision (uncoupled) or emulator error tolerance (coupled).
• Speedup:
  • The ROMs achieved speedup factors ($S = T_{FOM}/T_{ROM}$) ranging from 26 to 92 for typical grid sizes ($N=100$), scaling roughly as $N^2$.
  • The speedup increases with stricter grid requirements, making the emulators particularly valuable for high-resolution calculations.
• Bayesian Calibration: The inferred posterior distributions for the LECs using the emulator were consistent with those obtained using the full FOM solver, even when emulator errors were intentionally set large ($\alpha=10^{-2}$) to simulate computationally expensive scenarios (like 3-body scattering).

5. Significance and Outlook

• Enabling Rigorous Uncertainty Quantification: By providing fast, accurate emulators with quantified errors, this work removes the computational barrier to performing full Bayesian calibrations of chiral nuclear forces against scattering data.
• Scalability to Many-Body Systems: This work, combined with recent advances in 3-body scattering emulators (Refs. [24, 25]), lays the foundation for a unified framework to calibrate both NN and 3N forces simultaneously.
• Future Directions: The authors plan to extend these methods to differential cross-sections, spin observables, and non-affine parameters (e.g., momentum cutoffs) using hyperreduction techniques. The ultimate goal is to apply this framework to next-generation chiral interactions to resolve open questions in chiral EFT regarding power counting and nuclear structure predictions.

In summary, this paper establishes a robust, high-performance computational framework that bridges the gap between expensive nuclear scattering calculations and the data-driven requirements of modern Bayesian nuclear theory.