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

TL;DR

This work tackles the barrier that oscillatory scattering boundary conditions pose for physics-informed neural networks by marrying exterior complex scaling (ECS) with a driven-equation PINN approach. ECS converts outgoing waves into exponentially decaying functions, allowing a neural network to represent the scattered wave with decaying boundaries while the nuclear interaction remains on the real axis. The method achieves high-accuracy phase shifts and S-matrix elements for nucleon-nucleus scattering ($n+$ $^{40}$Ca at $E_{ ext{lab}}=20$ MeV) and heavy-ion scattering ($^6$Li+$^{208}$Pb at 40 MeV), demonstrating conditions under which PINNs can compete with conventional solvers and enabling differentiable inverse problems. The approach offers a pathway to CMS for multi-channel systems and few-body scattering, where grid-based methods suffer from scaling, by leveraging end-to-end differentiability and a mesh-free representation of the wave function.

Abstract

Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving differential equations, yet their application to quantum scattering problems 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 scattering 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 $Δδ< 0.1^\circ$ for most channels 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. 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.

Figures (6)

• Figure 1: Neural network architecture for PINN-ECS. The network takes the radial coordinate $r$ as input and passes it through four hidden layers of 128 neurons each with sinusoidal activation functions. The raw network output $\tilde{u}(r)$ is multiplied by the boundary condition factor $g(r) = r^{\ell+1}(1 - r/R_{\text{max}})\sigma_c(r)$, where the sigmoid cap $\sigma_c(r)$ prevents numerical overflow for high angular momentum channels. This hard-coded construction ensures $u(0) = u(R_{\text{max}}) = 0$ exactly, regardless of the learned weights.
• Figure 2: Normalized boundary condition factor $g(r)$ showing the sigmoid capping that prevents growth beyond $R_{\text{nuc}}^{\ell+1}$: $\ell = 0$ (solid), $\ell = 1$ (dashed), $\ell = 2$ (dash-dotted), and $\ell = 3$ (dotted). The vertical dashed line marks $R_{\text{nuc}}$. For $\ell > 0$, the factor transitions smoothly from $r^{\ell+1}$ behavior near the origin to a bounded value $R_{\text{nuc}}^{\ell+1}$ in the asymptotic region.
• Figure 3: PINN-ECS results for n+$^{40}$Ca scattering at $E_{\text{lab}} = 20$ MeV with the KD02 optical potential. (a) Phase shifts $\delta$ comparing PINN-ECS (circles) to COLOSS reference (squares). (b) S-matrix magnitudes $|S|$. (c) Phase shift errors $\Delta\delta = \delta_{\text{PINN}} - \delta_{\text{COLOSS}}$.
• Figure 4: Auto-adaptive anchor scale $s_a$ versus angular momentum $\ell$ for n+$^{40}$Ca. The scale is computed from $s_a = \max(\langle|V_{\text{short}} F_\ell|^2\rangle, 0.1)$ and decreases with $\ell$ due to centrifugal suppression of the source term. Circles with solid line: $j = \ell + 1/2$; squares with dashed line: $j = \ell - 1/2$. The floor value of 0.1 is applied for $\ell \geq 7$.
• Figure 5: $^6$Li+$^{208}$Pb elastic scattering at $E_{\text{lab}} = 40$ MeV. (a) Rutherford ratio versus center-of-mass angle: COLOSS (solid line) and PINN-ECS (dashed line). (b) S-matrix magnitude $|S_\ell|$ versus partial wave $\ell$: COLOSS (solid line) and PINN-ECS (circles). The horizontal dashed line marks the unitarity limit $|S| = 1$.