Constructing Inverse Potentials from Scattering Phase Shifts using Physics-Informed Neural Networks: Application to Neutron-Alpha Scattering

Abstract

We develop a physics-informed neural networks (PINNs) framework for the inverse scattering problem in nuclear physics and apply it to the $P_{3/2}$ partial wave of neutron-alpha elastic scattering. The radial potential is represented by a feed-forward network whose output is multiplied by a Gaussian envelope, embedding the finite-range condition directly into the architecture rather than through a soft penalty term. This distinction proves essential: without the envelope, the optimizer produces potentials with non-vanishing tails and the resulting phase shifts remain inconsistent with the data regardless of training duration, demonstrating that hard structural constraints are indispensable for physically meaningful solutions to nuclear inverse problems. Phase shifts are generated at each scattering energy by numerically integrating the variable-phase equation with a fourth-order Runge-Kutta scheme, making the entire pipeline end-to-end differentiable.Training converges stably to a loss near $3\times10^{-4}$ and recovers a smooth, purely attractive central potential with a well depth of $-60.47$~MeV. Adding the centrifugal barrier to the learned potential reveals a well-defined barrier-well structure that naturally accounts for the $P_{3/2}$ resonance. The extracted resonance parameters, $E_{r} = 0.95$~MeV and $Γ_{r} = 0.78$~MeV, together with the P-wave effective-range parameters, are in good agreement with expected values. A leave-one-out analysis confirms that the reconstruction is stable against the removal of any single data point. These results establish physics-guided machine learning as a reliable route to potential reconstruction from nuclear scattering data.