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    Exterior complex scaling enables physics-informed neural networks for nuclear reactions

    Jin Lei*

    • School of Physical Science and Engineering, Tongji University, Shanghai 200092, China and Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences, Huizhou 516000, Guangdong Province, China

    • *Contact author: jinl@tongji.edu.cn

    Phys. Rev. C 113, 064618 – Published 22 June, 2026

    DOI: https://doi.org/10.1103/sjz4-pq6p

    Abstract

    Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving differential equations, yet their application to nuclear scattering has been hindered by the oscillatory, nondecaying nature of scattering wave functions. In this work, I demonstrate that exterior complex scaling (ECS) transforms scattering boundary conditions into exponentially decaying waves suitable for neural network solutions, enabling PINNs to solve nuclear reaction problems for the first time. I develop a driven-equation formulation where the source term is confined to the real axis, avoiding the need to analytically continue nuclear potentials into the complex plane. The method is validated on nucleon-nucleus scattering (n+Ca40 at Elab=20MeV) with 21 partial waves, achieving phase shift accuracy of Δδ≲0.1∘ for the strongly absorbed channels (ℓ≤4) and Δδ≤0.60∘ for all channels up to ℓ=10, when compared with conventional solvers. I further demonstrate the approach on heavy-ion scattering (Li6+Pb208 at 40 MeV) with 41 partial waves and strong Coulomb effects, where an auto-adaptive anchor warm-down for weak-source channels yields a mean S-matrix accuracy of |Δ|Sℓ||≈3×10−3 across the full angular momentum range, including the absorption-to-transparency transition region. This work establishes the foundation for extending PINNs to inverse problems where end-to-end differentiability enables direct fitting of optical potential parameters, coupled-channel reactions, and few-body scattering, where traditional grid methods face exponential scaling.

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