Exterior complex scaling enables physics-informed neural networks for quantum scattering

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, non-decaying 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+${}^{40}$Ca at $E_{\text{lab}}=20$~MeV) with 21 partial waves, achieving phase shift accuracy of $\Delta \delta \lesssim 0.1^{\circ }$ for the strongly absorbed channels ($\ell \le 4$) and $\Delta \delta \le 0.60^{\circ }$ for all channels up to $\ell =10$, when compared to conventional solvers. I further demonstrate the approach on heavy-ion scattering (${}^6$Li+${}^{208}$Pb 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 $|\Delta |S_{\ell }||\approx 3\times 10^{-3}$ 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.