Seed-robust PINN determination of s-wave bound states and jost-function-based vertex constants in $$_{\Lambda }^{208}$$Pb

Abstract We investigate Physics-Informed Neural Networks (PINNs) for computing the s -wave bound states of the hypernucleus $$_{\Lambda }^{208}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow> <mml:mi>Λ</mml:mi> </mml:mrow> <mml:mn>208</mml:mn> </mml:mmultiscripts> </mml:math> Pb, modeled as a two-body system of a $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> hyperon and a $$^{207}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mn>207</mml:mn> </mml:mmultiscripts> </mml:math> Pb core, with a Woods–Saxon potential without spin-orbit coupling. The radial bound-state wave function is represented by a neural network ansatz; eigenenergies are obtained from the Rayleigh–Ritz variational quotient. Because PINN eigenvalue calculations depend on the residual-loss formulation and random-seed initialization, two residual losses are compared over four independent random seeds. Performance is assessed via eigenvalue accuracy, coefficient of variation, signal-to-noise ratio, bias-variance decomposition, and Hermitian spectral-ordering consistency. The normalized residual loss yields the most stable and physically consistent bound-state spectrum across the seeds. With this loss, computed bound-state energies and root-mean-square radii agree very well with the corresponding theoretical values. The resulting wave functions are then used to construct Jost functions through the Wronskian of 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 Constants (ANC), and the Nuclear Vertex Constants (NVC) are extracted. These quantities show reasonable agreement with theoretical values, albeit with larger standard deviations than 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. To our knowledge, this is the first PINN-based study to construct Jost functions from neural-network bound-state wave functions and to extract associated ANC, NVC, and S -matrix residue.