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arxiv: 2606.05940 · v1 · pith:KPYAMZTZnew · submitted 2026-06-04 · ⚛️ nucl-th

Seed-Robust PINN Determination of s-Wave Bound States and Jost-Function-Based vertex constants in _(Λ)²⁰⁸Pb

J.T Tshipi , A.S Cornell This is my paper

Pith reviewed 2026-06-27 23:27 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords physics-informed neural networksbound stateshypernucleiWoods-Saxon potentialJost functionsvertex constantsRayleigh-Ritz methodrandom-seed robustness
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The pith

A normalized residual loss in PINNs produces seed-independent bound-state energies and radii for the Lambda hyperon in lead-208 that match theoretical values.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether physics-informed neural networks can solve the two-body bound-state problem for a hypernucleus by representing the radial wave function with a neural network and extracting energies via the Rayleigh-Ritz variational quotient. Two different residual-loss formulations are run across four random seeds to check stability; the normalized version produces the smallest variation and yields energies and root-mean-square radii in close agreement with independent calculations for a Woods-Saxon potential. The same wave functions are then used to build Jost functions via Wronskians with Riccati-Hankel functions, from which the residue at the bound-state pole, asymptotic normalization constants, and nuclear vertex constants are obtained.

Core claim

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.

What carries the argument

Normalized residual loss applied to the Rayleigh-Ritz variational quotient on a neural-network representation of the radial bound-state wave function.

If this is right

  • Bound-state energies and root-mean-square radii obtained with the normalized loss agree closely with theoretical reference values.
  • Jost functions are constructed from the network wave functions via the Wronskian with incoming and outgoing Riccati-Hankel functions.
  • The residue of the partial-wave S-matrix at the bound-state pole, the asymptotic normalization constant, and the nuclear vertex constant are extracted from those Jost functions.
  • These extracted quantities agree with theoretical values at a level comparable to the eigenenergies, albeit with larger standard deviations across seeds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same normalized-loss strategy could be tested on potentials that include spin-orbit terms or on three-body hypernuclear systems to check whether seed robustness persists.
  • Because the method supplies both the wave function and the Jost function in a single forward pass, it may reduce the computational overhead of separately solving the Schrödinger equation and then performing analytic continuation.
  • If the larger scatter in vertex constants proves systematic, it would indicate that the network representation is less faithful at large radii where the asymptotic normalization is determined.

Load-bearing premise

The radial bound-state wave function can be represented by an artificial neural network such that the Rayleigh-Ritz variational quotient applied to the network output yields accurate eigenenergies for the chosen Woods-Saxon potential, independent of random-seed initialization.

What would settle it

Re-running the identical network architecture and normalized loss on the same Woods-Saxon potential but with a fifth independent seed that produces eigenenergies differing by more than the reported coefficient of variation would falsify the seed-robustness claim.

Figures

Figures reproduced from arXiv: 2606.05940 by A.S Cornell, J.T Tshipi.

Figure 1
Figure 1. Figure 1: Incoming(f in) and outgoing(f out) Jost functions calculated from Eq. (8) At large distances the bound state wave function may be expressed in terms of the Jost functions, u0(Ed, r) −−−→ r→∞ χ (−) 0 (Ed, r)f (in) 0 (Ed) + χ (+) 0 (Ed, r)f (out) 0 (Ed) (10) here, we used k = iκ to represent the momentum of the bound state. Since the reduced s-wave Schrödinger equation contains no first-derivative term, Abel… view at source ↗
Figure 2
Figure 2. Figure 2: CV and the corresponding SNR using Eq. (18) and normalized residual Eq. (22) losses. from the Rayleigh–Ritz quotient converge within fewer than 103 iterations out of the total 105 training iterations for all three s-wave bound states, indicating rapid convergence of the method. Beyond approximately 1000 iterations, the predicted energies remain unchanged, and this regime is therefore not shown. The converg… view at source ↗
Figure 3
Figure 3. Figure 3: Bias-Variance plots for the residual Eq. (18) and normalized residual Eq. (22) losses. where the potential vanishes. In this region, the Jost functions are expected to be constant. Inside the potential region, however, this asymptotic definition is no longer guaranteed to apply in the same way. This behaviour is visible in Figs. 1, 6, and 7, where the Wronskian expressions used to determine the Jost functi… view at source ↗
Figure 4
Figure 4. Figure 4: Training history of the computed s-wave eigenvalues. The dashed horizontal lines denote the theoretical values, while the solid curves represent the ANN predictions [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Woods-Saxon potential together with the three lowest s-wave bound-state wave functions for ℓ = 0 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Incoming (fin) and outgoing (fout) Jost functions calculated from Eq. (8) for the first excited state. non-zero. This observation suggests that the PINN-based Jost-function construction retains sufficient information to reconstruct the bound-state wave functions across the full domain. However, this result should be interpreted as numerical evidence rather than a formal proof of analytic continuation or of… view at source ↗
Figure 7
Figure 7. Figure 7: Incoming (fin) and outgoing (fout) Jost functions calculated from Eq. (8) for the second excited state [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The plot of the SNR (bottom) and CV (top) of the outgoing Jost function for the ground state using a window width of 3. The point of interest is the segment r ∈ [10, 10.5] which produced the highest (lowest) SNR (CV) [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The plot of the SNR (bottom) and CV (top) of the outgoing Jost function for the first excited state using a window width of 3. The point of interest is the segment r ∈ [13, 14] which produced the highest(lowest) SNR (CV). Refs. [17, 15]. The extracted residue of the S-matrix, NVC, and ANC are also in reasonable agreement with the reference values, although their standard deviations are larger than those ob… view at source ↗
Figure 10
Figure 10. Figure 10: The plot of the SNR (bottom) and CV (top) of the outgoing Jost function for the second excited state using a window width of 3. The point of interest is the segment r ∈ [11, 12] which produced the highest(lowest) SNR (CV) [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of the ground (left), first excited (middle) and second excited (right) s-wave bound state [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
read the original abstract

We investigate physics-informed neural networks (PINNs) for computing the $s$-wave bound states of the hypernucleus $_{\Lambda}^{208}$Pb, modeled as a two-body system composed of a $\Lambda$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates physics-informed neural networks (PINNs) to compute s-wave bound states of the hypernucleus 208ΛPb modeled as a Λ hyperon plus 207Pb core with a Woods-Saxon potential (no spin-orbit term). The radial wave function is represented by a neural network, eigenenergies are obtained via the Rayleigh-Ritz variational quotient, and two residual-loss formulations are compared across four random seeds using accuracy, coefficient of variation, SNR, bias-variance, and spectral-ordering metrics. The normalized residual loss is identified as most stable; resulting energies and rms radii agree well with theory. Wave functions are then used to build Jost functions via Wronskians with Riccati-Hankel functions, from which the S-matrix residue, ANC, and nuclear vertex constants are extracted and compared to theoretical values.

Significance. If the seed-robustness claim can be substantiated, the work supplies a concrete demonstration that a standard variational PINN formulation, paired with a normalized residual loss, can deliver stable bound-state solutions and downstream asymptotic quantities (Jost functions, ANC, vertex constants) for a realistic hypernuclear two-body problem. This is a modest but useful addition to the growing literature on PINNs in nuclear physics, as it directly connects the network output to established quantities (Wronskian Jost function, residue at bound-state pole) without introducing circular definitions. The explicit comparison of loss formulations and the use of multiple stability diagnostics are positive features.

major comments (1)
  1. [Abstract and stability-analysis paragraph] Abstract and stability-analysis paragraph: the central claim that the normalized residual loss yields a 'seed-robust' spectrum (and therefore reliable Jost-function/vertex-constant extractions) rests on results from exactly four random seeds. Neural-network optimization landscapes for eigenvalue problems are known to contain narrow basins that produce large errors for some initializations; four samples provide no statistics on failure rates or coefficient-of-variation behavior under a larger ensemble (e.g., 20–50 seeds). This sample size is insufficient to support the 'seed-robust' qualifier in the title and the subsequent use of the wave functions for asymptotic quantities.
minor comments (2)
  1. [Method section] The manuscript does not supply the explicit numerical values of the Woods-Saxon parameters or the precise training-set construction details; these omissions hinder independent reproduction even though the method itself is standard.
  2. [Results section] Quantitative error bars or confidence intervals on the extracted ANC and vertex constants are not reported, only standard deviations across the four seeds; adding these would strengthen the comparison with theory.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive assessment and for highlighting both the strengths and the limitations of our stability analysis. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and stability-analysis paragraph] Abstract and stability-analysis paragraph: the central claim that the normalized residual loss yields a 'seed-robust' spectrum (and therefore reliable Jost-function/vertex-constant extractions) rests on results from exactly four random seeds. Neural-network optimization landscapes for eigenvalue problems are known to contain narrow basins that produce large errors for some initializations; four samples provide no statistics on failure rates or coefficient-of-variation behavior under a larger ensemble (e.g., 20–50 seeds). This sample size is insufficient to support the 'seed-robust' qualifier in the title and the subsequent use of the wave functions for asymptotic quantities.

    Authors: We agree that four independent seeds provide only a limited statistical basis and cannot rule out the existence of narrow failure basins that might appear in a larger ensemble. The four seeds were selected to demonstrate practical reproducibility under standard random initializations, and the normalized residual loss produced low coefficients of variation together with consistent spectral ordering in every case. Nevertheless, the referee’s point is valid: the qualifier “seed-robust” in the title and the downstream use of the wave functions for Jost-function quantities are not fully substantiated by the present sample size. We will therefore revise the title, abstract, and stability-analysis section to replace “seed-robust” with the more precise phrasing “stable across the four seeds examined,” add an explicit statement on the sample-size limitation, and include a brief discussion of the need for larger ensembles in future work. The numerical results themselves remain unchanged. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation applies the standard Rayleigh-Ritz variational quotient to obtain eigenenergies from a neural-network representation of the radial wave function, then extracts Jost functions via the Wronskian definition with Riccati-Hankel functions and computes residues/ANC/NVC from those Jost functions. None of these steps reduces by construction to a fitted parameter being re-derived as a prediction, nor relies on a self-citation chain or imported uniqueness theorem. The seed comparison is an empirical stability check using four initializations and standard metrics; it does not create a definitional loop. The paper is self-contained against external benchmarks (theoretical Woods-Saxon values) with no load-bearing self-referential elements.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit listing of potential parameters, convergence criteria, or network architecture details is available.

free parameters (1)
  • Woods-Saxon potential parameters
    The interaction is described by a Woods-Saxon potential; depth, radius, and diffuseness values are required but not stated in the abstract.
axioms (1)
  • domain assumption The hypernucleus can be modeled as a two-body system with a local Woods-Saxon potential and no spin-orbit coupling.
    Explicitly stated in the abstract as the physical model employed.

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Reference graph

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