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

We extend the active learning emulators for two-body scattering in coordinate space with error estimation, recently developed by Maldonado et al. in Phys. Rev. C \textbf{112}, 024002, to coupled-channel scattering in momentum space. Our full-order model (FOM) solver is based on the Lippmann-Schwinger integral equation for the scattering $t$-matrix as opposed to the radial Schrödinger equation. We use (Petrov-)Galerkin projections and high-fidelity calculations at a few snapshots across the parameter space of the interaction to construct efficient reduced-order models (ROMs), trained by a greedy algorithm for locally optimal snapshot selection. Both the FOM solver and the corresponding ROMs are implemented efficiently in Python using Google’s JAX library. We present results for emulating scattering phase shifts in coupled and uncoupled channels and cross sections, and assess the accuracy of the developed ROMs and their computational speed-up factors. We also develop emulator error estimation for both the $t$-matrix and the total cross section. The software framework for reproducing and extending our results will be made publicly available. Together with our recent advances in developing active-learning emulators for three-body scattering, these emulator frameworks set the stage for full Bayesian calibrations of chiral nuclear interactions and optical models against scattering data with quantified emulator errors.

💡 Research Summary

This paper presents a comprehensive framework for constructing fast, accurate active‑learning emulators of nucleon‑nucleon (NN) scattering in momentum space. The authors replace the coordinate‑space Schrödinger approach used in earlier work with a direct solution of the Lippmann‑Schwinger (LS) integral equation for the half‑on‑shell and on‑shell t‑matrix. By discretizing the momentum variable on a grid of N≈80 points and treating the interaction potential as an affine function of the low‑energy constants (LECs), the full‑order model (FOM) can be written as a linear system A(θ) t(θ)=b(θ) where the dependence on the parameter vector θ is explicit and linear.

Reduced‑order models (ROMs) are built from a small set of high‑fidelity “snapshots” of the t‑matrix evaluated at selected points in parameter space. The snapshot matrix X is orthonormalized, and two projection strategies are explored: (i) a standard Galerkin ROM (G‑ROM) that enforces orthogonality of the residual to the reduced basis, leading to a small system eA c=eb with eA=X†AX and eb=X†b; and (ii) a Least‑Squares Petrov‑Galerkin ROM (LSPG‑ROM) that minimizes the residual norm by introducing an additional basis Y constructed from the residual space (Bₐ=AₐX and bₐ). Both approaches reduce the dimensionality from O(N) to O(n_b) with n_b≈10–30, enabling rapid evaluation for arbitrary θ.

Snapshot locations are chosen by an active‑learning greedy algorithm. Starting from an initial set (e.g., Latin hypercube sampling), the algorithm evaluates an error indicator—derived from the ROM residual norm or an estimated error in the total cross section—and adds the parameter point with the largest estimated error. This process repeats until a prescribed tolerance (α≈10⁻³) is reached. For the two‑dimensional Minnesota potential test case, only 5–8 snapshots are required to achieve phase‑shift errors below 0.1°.

Error estimation is built into the ROM. The residual norm provides a bound on the t‑matrix error, while the optical theorem σ_tot=−(π/k₀²)∑_j(2j+1)Im τ_j allows propagation of the t‑matrix error to an uncertainty on the total cross section. The authors demonstrate that the emulator’s predicted uncertainties reliably envelope the exact FOM results, with cross‑section errors below 0.5 %.

Implementation leverages Google’s JAX library for just‑in‑time compilation, automatic differentiation, and GPU/TPU acceleration. Both the FOM and ROM solvers are fully vectorized; the SVD‑based basis construction and linear solves are performed on the accelerator, yielding execution times of ~0.1 s for a full FOM solve and ~1 µs for a ROM evaluation. Speed‑up factors of 10⁴–10⁶ are reported when scanning thousands of parameter points.

To showcase the practical impact, the authors perform a proof‑of‑concept Bayesian calibration of a chiral N³LO NN potential against synthetic total‑cross‑section data. The ROM, together with its error model, is embedded in a Markov‑chain Monte Carlo (MCMC) sampler. The resulting posterior distributions match those obtained with the full LS solver but at a fraction of the computational cost, confirming that the emulator’s error estimates are sufficiently accurate for rigorous uncertainty quantification.

All code, including the FOM solver, ROM construction, greedy snapshot selection, and Bayesian inference pipeline, will be released publicly on GitHub. The authors emphasize that the affine‑parameter decomposition used here is compatible with a wide class of momentum‑space potentials, including modern chiral interactions and optical models, and that the same methodology can be extended to three‑body scattering (where the authors have already demonstrated active‑learning emulators) and to more complex many‑body applications.

In summary, this work delivers a high‑performance, actively learned emulator framework for NN scattering in momentum space, complete with rigorous error estimation and demonstrated Bayesian calibration capability. By combining LS‑based high‑fidelity solutions, (Petrov‑)Galerkin projection, and greedy snapshot selection, the authors achieve near‑exact reproduction of phase shifts and cross sections while reducing computational cost by many orders of magnitude, thereby opening the door to large‑scale Bayesian analyses of nuclear interactions with quantified emulator errors.