Seed-Robust PINN Determination of $s$-Wave Bound States and Jost-Function-Based vertex constants in $_Λ^{208}$Pb

Abstract: We investigate physics-informed neural networks (PINNs) for computing the $s$-wave bound states of the hypernucleus $_Λ^{208}$Pb, modeled as a two-body system composed of a $Λ$ hyperon and a $^{207}$Pb core. The interaction is described by a Woods--Saxon potential without spin--orbit coupling. In the PINN formulation, the radial bound-state wave function is represented by an artificial neural network, and the eigenenergies are obtained from the Rayleigh--Ritz variational quotient. Because PINN eigenvalue calculations can depend on the residual-loss formulation and random-seed initialization, two residual losses are compared across four independent random seeds. Their performance is assessed using eigenvalue accuracy, coefficient of variation, signal-to-noise ratio, bias--variance decomposition, and Hermitian spectral-ordering consistency. The normalized residual loss gives the most stable and physically consistent bound-state spectrum for the four seeds considered. With this loss, the computed bound-state energies and root-mean-square radii are in very good agreement with the corresponding theoretical values. The resulting wave functions are used to construct the Jost functions through the Wronskian with incoming and outgoing Riccati--Hankel functions. From these Jost functions, the residue of the partial-wave $S$-matrix at the bound-state pole, the Asymptotic Normalization and the Nuclear Vertex Constants are extracted. These quantities show reasonable agreement with theoretical values, although with larger standard deviations than those obtained for the eigenenergies and radii. The results indicate that the Rayleigh--Ritz formulation combined with a seed-robust normalized residual loss provides a stable PINN framework for bound-state and Jost-function-based calculations in hypernuclear two-body systems.