arxiv: 2605.02264 · v1 · submitted 2026-05-04 · ⚛️ nucl-th

Recognition: 3 theorem links

· Lean Theorem

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

Ayushi Awasthi Ishwar Kant Arushi Sharma M.R.Ganesh Kumar , O.S.K.S.Sastri

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:42 UTC · model grok-4.3

classification ⚛️ nucl-th

keywords inverse scatteringphysics-informed neural networksneutron-alpha scatteringphase shiftsnuclear potentialresonance parametersP-wave scattering

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The pith

A neural network with an embedded Gaussian envelope reconstructs a finite-range attractive potential from neutron-alpha phase shifts that reproduces the P_{3/2} resonance.

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

The paper demonstrates that representing a radial nuclear potential as the output of a feed-forward network multiplied by a Gaussian envelope allows stable training on experimental phase shifts. Phase shifts are computed at each energy by integrating the variable-phase equation using a differentiable Runge-Kutta scheme, so the entire process optimizes directly against data. This hard constraint on finite range prevents unphysical long tails that appear when the envelope is omitted. The resulting potential is smooth and attractive, with a depth of about 60 MeV, and when the centrifugal term is added it produces a barrier-well structure that accounts for the observed resonance. The approach shows that physics-informed constraints can turn inverse scattering into a reliable, data-driven route to potential reconstruction.

Core claim

By training a feed-forward network whose output is multiplied by a Gaussian envelope and whose phase shifts are obtained by integrating the variable-phase equation with fourth-order Runge-Kutta, the method converges to a smooth, purely attractive central potential of depth -60.47 MeV for the P_{3/2} partial wave. Adding the centrifugal barrier reveals a barrier-well structure whose resonance parameters (E_r = 0.95 MeV, Γ_r = 0.78 MeV) and effective-range parameters match expected values, while leave-one-out tests confirm stability against removal of any single data point.

What carries the argument

A feed-forward neural network output multiplied by a Gaussian envelope, trained end-to-end by minimizing mismatch between data and phase shifts computed via the variable-phase equation integrated with Runge-Kutta.

If this is right

The learned potential yields phase shifts whose loss converges stably near 3×10^{-4}.
Adding the centrifugal barrier to the potential produces a well-defined barrier-well that accounts for the resonance.
The extracted resonance energy, width, and P-wave effective-range parameters agree with known values.
Leave-one-out removal of any training point leaves the reconstructed potential essentially unchanged.

Where Pith is reading between the lines

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

The same architecture could be applied to other partial waves or to scattering on heavier targets to test whether the Gaussian envelope remains sufficient.
The emphasis on a hard structural constraint rather than a penalty term suggests that similar embeddings could improve inverse problems in other areas of physics where boundary conditions are known exactly.
Once a potential is obtained, it can be used to predict observables not included in the fit, such as low-energy cross sections or properties of the alpha-neutron bound state if one exists.

Load-bearing premise

A central, finite-range potential multiplied by a Gaussian envelope is sufficient to capture all relevant physics of the P_{3/2} partial wave without non-central forces, relativistic corrections, or extra channels.

What would settle it

If the potential extracted from the network produces phase shifts or resonance parameters (E_r or Γ_r) that deviate substantially from independent experimental measurements or from ab initio calculations for the same partial wave.

Figures

Figures reproduced from arXiv: 2605.02264 by Ayushi Awasthi Ishwar Kant Arushi Sharma M.R.Ganesh Kumar, O.S.K.S.Sastri.

**Figure 1.** Figure 1: Schematic of the PINNs framework. 3 Results and Discussion 3.1 Convergence of Training The evolution of the training loss as a function of epoch is shown in view at source ↗

**Figure 2.** Figure 2: Training loss on a logarithmic scale as a function of epoch. The loss decreases by four view at source ↗

**Figure 3.** Figure 3: Left: learned central potential Vθ(r) for the P3/2 channel. The profile is purely attractive and smooth, vanishing beyond ≈ 4 fm. Right: effective potential Veff(r) = Vθ(r) + Vcf(r), displaying the barrier–well structure responsible for the P3/2 resonance. Inset: enlarged view of the potential minimum and centrifugal barrier peak. continuous character of the predicted curve confirms that the network genera… view at source ↗

**Figure 4.** Figure 4: P3/2 phase shifts for n-α scattering as a function of laboratory energy En. The solid line is the present PINN result; filled circles are the expected values of Satchler et al. [5]. The shaded bands indicate the extrapolation regions outside the training interval 0.3–18 MeV. 3.3 Residual Analysis The pointwise residual ∆δ = δmodel −δexp, plotted against laboratory energy in view at source ↗

**Figure 5.** Figure 5: Residual ∆δ = δmodel − δexp as a function of laboratory energy. All 22 residuals lie within ±1.4 ◦ ; the mean-square value is 0.5 ◦ . The absence of any systematic trend confirms unbiased reproduction of the phase shifts. 3.4 Partial Cross Section and Resonance Parameters From the computed phase shifts, the partial-wave cross section for ℓ = 1 is obtained from σ1(E) = 4π k 2 (2ℓ + 1) sin2 δ1(E), (11) and i… view at source ↗

**Figure 6.** Figure 6: Partial-wave cross section σ1(E) for the P3/2 channel of 4He(n, n) 4He as a function of centre-of-mass energy. The resonance peak at Ecm = 0.95 MeV yields Γ = 0.78 MeV; experimental values are shown in parentheses. 3.5 Effective-Range Parameters At low energies the P-wave phase shift obeys the effective-range expansion [31, 38], k 3 cot δ1 = − 1 a + 1 2 r k2 − 1 4 P k4 + · · · , (12) where a is the scatter… view at source ↗

**Figure 7.** Figure 7: Leave-one-out robustness analysis of the reconstructed effective potential view at source ↗

read the original 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 $\Gamma_{r} = 0.78$~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.

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.

Desk Editor's Note private letter to a colleague

The PINN with a hard Gaussian envelope on the network output reconstructs a smooth central potential for n-alpha P3/2 scattering that yields resonance parameters close to literature values, but all checks stay inside one integrator.

read the letter

The main new element is the multiplicative Gaussian envelope that hard-enforces finite range on the neural-network potential instead of using a penalty term. The authors show that dropping the envelope leaves long tails and prevents the phase shifts from fitting the data no matter how long training runs. That practical observation is useful for anyone trying similar inverse problems in nuclear physics. They then integrate the variable-phase equation with RK4 at every training step, making the whole thing end-to-end differentiable, and report stable convergence to a loss near 3e-4. The resulting potential is purely attractive with a depth of about -60 MeV; adding the centrifugal barrier produces the expected barrier-well shape for the resonance. Extracted resonance energy and width, plus the P-wave effective-range parameters, sit close to accepted numbers, and a leave-one-out test indicates the fit does not hinge on any single data point. Those are concrete, reproducible outcomes for this specific system. The soft spot is exactly the one flagged in the stress test. Training, resonance extraction, effective-range coefficients, and stability checks all come from the identical variable-phase plus Runge-Kutta forward model. No independent radial Schrödinger solver is used for cross-check, and there is no side-by-side comparison against classical inverse-scattering techniques such as Marchenko or Gel'fand-Levitan. Any systematic bias in step size, differentiability, or implementation details can be absorbed into the learned potential without detection. The abstract also gives no quantitative uncertainties on the potential shape or on the extracted numbers. This paper is for nuclear theorists or reaction modelers who want to explore data-driven potentials for light nuclei and are willing to add their own validation layers. A reader already working on PINNs or inverse scattering will get a clear demonstration of the hard-constraint trick and its effect on convergence. I would send it to peer review. The architecture choice is distinct enough and the results are plausible enough that referees can usefully press on the numerical validation and on direct comparisons to existing methods.

Referee Report

2 major / 2 minor

Summary. The paper develops a physics-informed neural network framework to solve the inverse scattering problem for the P_{3/2} partial wave in neutron-alpha elastic scattering. The radial potential is parameterized as the output of a feed-forward network multiplied by a Gaussian envelope to enforce finite range; phase shifts are obtained by integrating the variable-phase equation with a fourth-order Runge-Kutta scheme, and the network weights are optimized to reproduce experimental phase shifts. The method yields a smooth, purely attractive central potential of depth -60.47 MeV whose barrier-well structure (after adding the centrifugal term) reproduces the P_{3/2} resonance with parameters E_r = 0.95 MeV and Γ_r = 0.78 MeV, together with P-wave effective-range coefficients in agreement with literature values; stability is demonstrated via leave-one-out cross-validation.

Significance. If the numerical fidelity of the forward model is independently verified, the work establishes a practical, end-to-end differentiable route to potential reconstruction from nuclear phase-shift data. The explicit demonstration that a hard Gaussian envelope is required for physically acceptable solutions, the stable convergence to low loss, and the leave-one-out robustness test are concrete strengths that could generalize to other partial waves or few-body systems.

major comments (2)

[Methods / numerical implementation] The phase-shift computations, resonance extraction, effective-range analysis, and leave-one-out tests all rely exclusively on the variable-phase equation integrated by RK4. No cross-check against an independent radial Schrödinger solver (e.g., Numerov or finite-difference discretization) is reported. Any systematic bias in step-size, differentiability, or implementation details could therefore be absorbed into the learned potential, undermining the claim that the reconstructed potential “naturally accounts for the P_{3/2} resonance.”
[Results] No quantitative uncertainties, full table of input phase shifts versus model predictions, or direct comparison with classical inverse-scattering methods (Marchenko, Gel’fand-Levitan, etc.) are provided. The reported loss of ~3×10^{-4} and potential depth of -60.47 MeV therefore lack the error analysis needed to assess uniqueness and precision of the reconstruction.

minor comments (2)

[Methods] The precise functional form and width parameter of the Gaussian envelope should be stated explicitly (including numerical values) in the methods section to ensure reproducibility.
[Results] Clarify in the text or figure captions whether the quoted resonance parameters E_r and Γ_r are obtained by solving the Schrödinger equation on the learned potential or by a separate fit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough review and constructive suggestions. We address the major comments below, indicating the changes we will implement in the revised manuscript.

read point-by-point responses

Referee: [Methods / numerical implementation] The phase-shift computations, resonance extraction, effective-range analysis, and leave-one-out tests all rely exclusively on the variable-phase equation integrated by RK4. No cross-check against an independent radial Schrödinger solver (e.g., Numerov or finite-difference discretization) is reported. Any systematic bias in step-size, differentiability, or implementation details could therefore be absorbed into the learned potential, undermining the claim that the reconstructed potential “naturally accounts for the P_{3/2} resonance.”

Authors: We thank the referee for pointing this out. While the variable-phase equation is derived directly from the radial Schrödinger equation and is mathematically equivalent for computing phase shifts, we agree that an independent numerical verification would enhance confidence. In the revised manuscript, we will add a cross-validation by solving the radial Schrödinger equation using a Numerov integrator for the learned potential and comparing the resulting phase shifts to those from the variable-phase method. This will demonstrate consistency and rule out implementation-specific biases. We will also report the step sizes used and convergence tests for the RK4 integration. revision: yes
Referee: [Results] No quantitative uncertainties, full table of input phase shifts versus model predictions, or direct comparison with classical inverse-scattering methods (Marchenko, Gel’fand-Levitan, etc.) are provided. The reported loss of ~3×10^{-4} and potential depth of -60.47 MeV therefore lack the error analysis needed to assess uniqueness and precision of the reconstruction.

Authors: We agree that a full table comparing input experimental phase shifts to the model predictions would improve transparency and will include it in the revised version. For quantitative uncertainties, the phase shift data from the literature lack individual error bars; we will add a sensitivity analysis by perturbing the phase shifts within typical experimental precision for this system and retraining the network to estimate variations in the potential depth and resonance parameters. For direct comparison with classical inverse-scattering methods, we note that Marchenko or Gel’fand-Levitan approaches reconstruct potentials from the full S-matrix without a parameterized form but require additional inputs such as bound-state information. We will add a discussion contrasting our approach with these methods and compare the learned potential to existing phenomenological n-α potentials in the literature, while acknowledging that a full re-implementation lies beyond the present scope. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained with external validation

full rationale

The paper fits a neural-network potential (with hard Gaussian envelope) to experimental phase-shift data by minimizing a loss that compares data to phase shifts computed from the variable-phase equation integrated via RK4. Resonance parameters and effective-range coefficients are then obtained by applying the same forward model to the learned potential and are compared to independent literature values; the leave-one-out test checks stability under data removal. No quoted step reduces any claimed output (potential depth, resonance location, or barrier-well structure) to a re-statement of the input phase shifts by construction. There are no load-bearing self-citations, imported uniqueness theorems, or ansatzes smuggled via prior work. The Gaussian envelope is an explicit architectural choice, not a fitted parameter renamed as a prediction. The pipeline is therefore a standard inverse problem whose central results remain falsifiable against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The framework rests on standard quantum-scattering assumptions and neural-network optimization; no new physical entities are introduced.

free parameters (1)

Neural-network weights and biases
The trainable parameters of the feed-forward network that define the radial shape of the potential.

axioms (2)

domain assumption Scattering phase shifts are generated by numerical integration of the variable-phase equation
Invoked to make the forward map from potential to phase shifts differentiable inside the training loop.
domain assumption The potential vanishes at large distances
Enforced by the multiplicative Gaussian envelope rather than learned.

pith-pipeline@v0.9.0 · 5608 in / 1427 out tokens · 67257 ms · 2026-05-08T18:42:16.020967+00:00 · methodology

discussion (0)

Reference graph

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