{"id":"298ea05c-f90d-4ee3-97a7-127575825534","arxiv_id":"2508.19468","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A weighted Levenberg-Marquardt algorithm using a weighted Fisher information metric produces more balanced, physically reasonable fits of the CoH3 nuclear model to multichannel cross section data.","lead":"This paper introduces a weighted version of the Levenberg-Marquardt fitting algorithm that balances datasets with different sizes and uncertainties when fitting nuclear cross section models. It shows on neutron data for samarium-148 that the weighted fits keep model parameters in a physically reasonable range.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The wLM step in Eq. (40) does not match the gradient of the marginalized likelihood in Eq. (8), so the central claim that wLM minimizes the weighted likelihood is not established by the printed derivation.","rationale":"The reader's weakest_assumption focused on the un-derived physical-consistency band in Figs. 5 and 7. That is a genuine weakness, but I find a more fundamental internal inconsistency in the algorithm derivation: Eq. (40) uses w_k(θ)J^T r as the gradient, while the negative log of the marginalized likelihood in Eq. (8) has gradient with weight (n_k+2)/(χ²_k+2), not (n_k+2)χ²_k/(χ²_k+2)^2. This means the printed wLM is not the LM step for the likelihood that the abstract and Section II claim to optimize. The empirical demonstration may still be valid if the implemented code differs from the printed equations, or if the authors intended a different objective, but the manuscript does not say so. The physical-consistency band issue remains secondary and would be worth addressing, but the gradient mismatch is the condition most load-bearing for the central claim. Because the concern is a fixable derivation issue rather than a demonstrated failure of the empirical method, I keep the reader's CONDITIONAL verdict unchanged.","tokens_in":15391,"tokens_out":8632,"duration_ms":77637,"concrete_test":"Independently re-derive the gradient of the negative marginalized log-likelihood from Eq. (8). Then replace the gradient term in Eq. (40) with Σ_k (n_k+2)/(χ²_k+2) Σ_i r_i ∂_ν r_i while keeping the wFIM from Eq. (19), and rerun the 148Sm fits (raw and FBET-smoothed) from the same initialization. If the converged parameters and the log-likelihood trajectories of Figs. 4–8 differ materially, the published wLM is not minimizing the claimed weighted likelihood; if they do not differ, the mismatch is a harmless typo and the central claim survives.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing problem is an internal inconsistency between the stated objective and the wLM update. Eq. (8) gives the marginalized likelihood L(θ|y) ∝ ∏_k (χ²_k(θ)+2)^{-(n_k+2)/2}. The negative log-likelihood has gradient ∂_μ(-ln L) = Σ_k (n_k+2)/(χ²_k+2) Σ_i r_i ∂_μ r_i. Eq. (40), however, uses the surrogate gradient Σ_k w_k(θ) Σ_i r_i ∂_μ r_i with w_k defined in Eq. (21) as (n_k+2)χ²_k/(χ²_k+2)^2. These differ by a factor χ²_k/(χ²_k+2); they coincide only when χ²_k ≫ 2, which is not the regime of the fits at the optimum. Thus the printed wLM step is not the LM/natural-gradient step for the likelihood the paper claims to optimize; it corresponds to a different, unstated objective, and derivative-of-weight terms are also omitted. Since the central claim is that incorporating dataset weights 'through a generalized likelihood framework' improves parameter estimation, this mismatch is decisive: the demonstration must be re-run with either the correct gradient or with the objective that actually corresponds to Eq. (40). The 'physical consistency' band in Figs. 5/7 is additionally un-derived, but the step mismatch is the more fundamental obstacle.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a weighted extension of the Levenberg-Marquardt algorithm (wLM) for fitting multichannel nuclear cross-section data. The weights are derived by marginalizing over per-dataset inverse-variance hyperparameters with several priors (exponential, uniform, cropped exponential), leading to a weighted Fisher information metric. The authors test the method on neutron total and capture cross sections for 148Sm using the CoH3 Hauser-Feshbach code, fitting multiplicative tweaks to the Koning-Delaroche optical potential. They report that weighted schemes balance the total and capture channels better than unweighted LM, keep fitted optical-model tweaks inside an asserted physical band, and remain stable when the capture data are first smoothed with FBET. A geometric scaling of the update step is added to accelerate convergence.","tokens_in":15776,"tokens_out":13804,"duration_ms":122162,"significance":"If the derivation and numerics were consistent, the paper would offer a practically useful recipe for heterogeneous nuclear-data evaluation: a transparent way to down-weight overrepresented or overly precise datasets, a natural-gradient-type update, and a cheap acceleration heuristic for sloppy models. The empirical comparison across four weighting schemes on a real CoH3 problem, with z-score and eigenvalue diagnostics, is a useful contribution to the nuclear-data fitting literature. However, the central theoretical derivation currently contains load-bearing inconsistencies, and the main qualitative conclusion ('physical consistency') rests on an unstated band. The strengths are the clear problem formulation, the explicit acceptance/rejection protocol for the LM step, and the comparative diagnostics in Figs. 8-10; these give the paper a solid empirical core that could support a revised version.","major_comments":[{"comment":"The wLM update is not the natural-gradient or Gauss-Newton step for the marginalized likelihood in Eq. (8). From Eq. (8), the negative log-likelihood has gradient -∂_μ ln L = Σ_k (n_k+2)/(χ_k^2+2) Σ_i r_i ∂_μ r_i, whereas Eq. (40) uses Σ_k w_k(θ) Σ_i r_i ∂_μ r_i with w_k = (n_k+2)χ_k^2/(χ_k^2+2)^2. These coefficients coincide only in the limit χ_k^2 ≫ 2, which is not the regime of the reported fits. Consequently, the printed wLM step does not minimize the likelihood the paper claims to optimize; it corresponds to a different, unstated objective, and if one tries to interpret it as a weighted least-squares step the derivative-of-weight terms are also omitted. The central algorithmic claim therefore needs either a corrected gradient in Eq. (40) or an explicit statement of the objective that Eq. (40) actually minimizes, together with a re-run of the numerical demonstration under that objective.","section":"II.B, Eq. (40)"},{"comment":"The general weight formula in Eq. (15) has the wrong sign on the variance term. For a dataset with marginal likelihood L_k(s) = L_s[α^{n_k/2} f(α)], the Fisher-information coefficient is ℓ'(s)+ℓ''(s) with ℓ = -ln L_k, i.e. L_1/L_0 + (L_1/L_0)^2 - L_2/L_0, not L_1/L_0 + L_2/L_0 - (L_1/L_0)^2 as printed. For the exponential prior this makes Eq. (15) reduce to (n_k+2)(χ_k^2+4)/(χ_k^2+2)^2 rather than the correct Eq. (21), (n_k+2)χ_k^2/(χ_k^2+2)^2. Because Eq. (15) is the basis for the uniform and cropped-exponential weights, all weights derived through it inherit the error; the numerical results for those schemes in Section IV should be re-examined.","section":"II.A, Eq. (15)"},{"comment":"The incomplete-gamma functions appear with an off-by-one index. With the definition γ(s,x)=∫_0^x t^{s-1}e^{-t}dt in Eq. (26), the Laplace transform of α^{n_k/2}f(α) requires γ(n_k/2+1, ·), not γ(n_k/2, ·). Equation (25) and the Δ functions in Eqs. (34)-(37) use n_k/2, which corresponds to the transform of α^{n_k/2-1}. This changes the numerical weights for the uniform and cropped-exponential priors and must be corrected before the corresponding fits can be trusted.","section":"II.A, Eqs. (25), (34)-(37)"},{"comment":"The central qualitative claim that unweighted fits are 'physically inconsistent' while weighted fits are not rests entirely on the dashed horizontal lines at approximately 0.8-1.2 times the Koning-Delaroche values. No derivation, citation, or quantitative criterion is given for this band. If the unweighted tOW value is in fact physically admissible, the main conclusion of the paper is not established. Please justify the band from the adopted optical-model uncertainties or templates, or replace the physical-consistency argument with an explicit quantitative metric.","section":"IV, Figs. 5 and 7"}],"minor_comments":[{"comment":"The normalization in Eq. (8) appears incorrect: marginalizing the likelihood in Eq. (4) over α_k with prior e^{-α_k} gives a coefficient 2Γ(n_k/2+1)/π^{n_k/2}, not 2Γ(n_k/2+2)/π^{n_k/2}. The constant does not affect the minimization, but the formula as printed is misleading.","section":"II.A, Eq. (8)"},{"comment":"The expression for E[α] for the cropped exponential distribution mixes α with the integration limits and is hard to parse; please write it explicitly in terms of α_L and α_U and verify the η = 1, symmetric-support convention used in the fits.","section":"II.A, Eq. (31)"},{"comment":"The quantity s in Eq. (46) is called an 'angle', but it is an unnormalized metric inner product and is not bounded in [-1,1]. Scaling δθ by this quantity therefore changes the step magnitude as well as its direction; please clarify the intended geometric meaning and state whether normalization by the relevant norms is required.","section":"II.C, Eq. (46)"},{"comment":"The notation g^{(W,λ)}_{μν} is used both for the damped matrix and for its inverse; using upper and lower indices consistently, or a different symbol for the inverse, would remove ambiguity.","section":"II.B, Eqs. (40)-(43)"},{"comment":"The indices in the acceptance criterion appear shifted (θ_n vs θ_{n-1}); please make the iteration index consistent throughout the recursion.","section":"II.B, Eqs. (44)-(45)"}],"recommendation":"major_revision","confidential_remarks":"The paper does not include code, data files, or explicit EXFOR dataset identifiers in machine-readable form, which limits reproducibility of the numerical claims. Given the number of formulas that need correction (Eqs. 15, 25, 34-37, 40), the refereeing process would benefit from a version that supplies the derived weights in a form that can be checked symbolically, and a statement of which objective function the implemented algorithm actually minimizes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe one-sentence take: the paper has a real idea and a real bug. The wLM update in Eq. (40) does not minimize the marginalized likelihood in Eq. (8), so the theoretical foundation as written is wrong. The empirical demonstration is suggestive, but the central claim needs to be reworked.\n\nWhat's actually new: the paper adapts Hobson et al.'s dataset-weighting marginalization to multichannel nuclear-data fitting, constructs a weighted Fisher Information Metric, and adds a geometric scaling heuristic. That is a reasonable extension of existing ideas, not a paradigm shift. The single-nucleus example (148Sm) shows that weighting changes the fit in a plausible direction, and the paper is honest that the best prior choice depends on the dataset. The FIM spectrum analysis is a nice touch, and the geometric scaling is a pragmatic accelerator worth testing elsewhere.\n\nSoft spots, in proportion:\n\n- The gradient mismatch is the big one. The negative log of Eq. (8) has gradient Σ_k (n_k+2)/(χ_k²+2) Σ_i r_i ∂_μ r_i. Eq. (40) uses Σ_k w_k Σ_i r_i ∂_μ r_i with w_k=(n_k+2)χ_k²/(χ_k²+2)^2. The factor χ_k²/(χ_k²+2) is missing, and the derivative of the weights is ignored. So the update is not a natural-gradient step for the stated objective. This is not a typo; it invalidates the derivation of the algorithm. The authors need to re-derive the update from Eq. (8) or state the actual objective being minimized.\n\n- There are off-by-one incomplete-gamma indices in Eqs. (25), (34), and (37), which change the numerical weights.\n\n- The \"physical consistency\" band in Figs. 5 and 7 is asserted without derivation. It is a reasonable prior, but not evidence by itself.\n\n- No code is provided. For a methods paper, that is a serious omission; the empirical results cannot be checked.\n\nBottom line: this deserves a serious referee, but not acceptance in current form. The idea is worth engaging; the math needs fixing and the implementation should be released. I would be skeptical of the conclusions until the corrected derivation reproduces the same behavior.","headline":"A genuinely useful empirical idea, but the wLM update as printed does not minimize the stated marginalized likelihood; the paper needs major revision before the central claim can be trusted.","tokens_in":16239,"tokens_out":5899,"would_cite":false,"duration_ms":54603,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Replacing the Fisher information metric with a dataset-weighted version keeps Levenberg-Marquardt fits of multichannel nuclear cross sections inside physically plausible parameter regions and improves convergence, and a geometric…","keywords":["weighted Levenberg-Marquardt","weighted Fisher information metric","multichannel nuclear cross section fitting","Hauser-Feshbach statistical model","optical model parameters","sloppy models","dataset weighting","uncertainty quantification"],"falsifier":"Repeat the same CoH3 fitting pipeline on another nucleus with similarly heterogeneous total and capture data, and check whether the unweighted fit again leaves the 0.8–1.2 multiplier band while the weighted fit stays inside; a case where both stay inside would weaken the claimed advantage, and an independent microscopic optical-model calculation of the out-of-band parameters would settle whether they are truly unphysical.","tokens_in":15230,"feed_emoji":"⚛️","tokens_out":7874,"duration_ms":79938,"temperature":0.7,"pith_summary":"This paper introduces a weighted Levenberg-Marquardt (wLM) algorithm for fitting nuclear reaction models to heterogeneous experimental cross-section data, targeting a known weakness of the classical method when models have many weakly constrained, 'sloppy' parameters. The algorithm builds a weighted Fisher information metric by marginalizing over per-dataset weight hyperparameters, so that a channel with many data points or very small reported errors does not silently dominate the fit. Applied to neutron cross sections on 148Sm, the paper shows that weighted fits keep optical-model tweaks near physically expected values, while the unweighted fit drives them outside that band. It also shows that a geometric scaling of the update step, based on the angle between successive steps measured with the weighted metric, speeds convergence without moving the final parameters. The paper argues this is a practical alternative to trust-region methods for nuclear data evaluation.","feed_headline":"Weighted Levenberg-Marquardt keeps nuclear fits physical","feed_subtitle":"For 148Sm, unweighted fitting leaves the allowed parameter band; per-dataset weights pull it back.","key_machinery":"The central object is the weighted Fisher information metric (wFIM), constructed by treating each dataset's reported uncertainties as scaled by a per-dataset inverse-variance weight $\\alpha_k$ and marginalizing the likelihood over $\\alpha_k$ under a chosen prior (exponential, bounded uniform, or cropped exponential). This yields an effective per-dataset weight $w_k(\\theta)$, for example $w_k = (n_k+2)\\chi_k^2/(\\chi_k^2+2)^2$ for the exponential prior, multiplying each group's unweighted FIM contribution $g^{(0,k)}_{\\mu\\nu}$. The wLM update replaces the FIM in the classical LM step with this weighted metric, $\\delta\\theta^\\mu = -(g^{(W,\\lambda)})^{\\mu\\nu} \\sum_k \\sum_{i_k} w_k(\\theta)\\,\\partial_\\nu r_{k i_k}\\, r_{k i_k}$, so that datasets with extreme chi-squared values are automatically rebalanced. A geometric scaling factor $s = g^{(W,\\lambda)}_{\\mu\\nu}\\,\\delta\\theta^\\mu_{n-1}\\,\\delta\\theta^\\nu_n$ rescales each update by the wFIM-angle between successive steps, lengthening aligned steps and shortening abrupt directional changes.","core_discovery":"The paper claims that classical Levenberg-Marquardt on a multichannel, sloppy model such as the coupled-channels reaction code CoH3 tends to concentrate on the best-populated or most precise channel, producing fits whose parameters leave a physically reasonable band around the standard optical-model baseline. By contrast, the weighted Levenberg-Marquardt algorithm, which incorporates a dataset-specific weight obtained by marginalizing an inverse-variance hyperparameter under a chosen prior, balances the channels and keeps the fitted multiplicative tweaks within that band for both raw and FBET-smoothed data. On the 148Sm benchmark, the exponential weighting prior gives the lowest overall chi-squared but the broadest posterior uncertainties, while the cropped-exponential prior performs best after FBET smoothing; the choice of optimal weighting is data-dependent. The paper also demonstrates that geometric scaling of the update steps consistently reduces the number of iterations to convergence without altering the converged parameter estimates.","pith_inferences":["Editorial inference: the weighting construction naturally acts as a data-balancing regularizer, so it could also be tested as a general robustness device for fitting any multichannel dataset where one experimental group is overrepresented or assigned overconfident uncertainties.","Editorial inference: a direct way to separate the paper's physical-consistency claim from its choice of the 0.8–1.2 multiplier band would be a synthetic-data test with known true parameters and deliberately imbalanced channels, checking whether weighted estimates stay closer to the truth than unweighted ones.","Editorial inference: the geometric scaling factor uses the wFIM inner product of successive steps, so it should behave like a local preconditioner; comparing it directly with standard adaptive step-size or momentum schedules on the same CoH3 problem would clarify whether the acceleration comes from the angle alignment or from the damping schedule.","Editorial inference: because the paper finds that the best weighting prior shifts between raw and smoothed data and across nuclei, a practical extension would be to choose the prior width and shape by cross-validation or predictive holdout rather than by prior guesswork."],"forward_implications":["On the 148Sm benchmark, the weighted schemes fit both total and capture channels in a single run without letting one channel dominate, whereas the unweighted LM fit visibly sacrifices one channel for the other.","The exponential weighting prior gives the lowest overall chi-squared but the broadest posterior parameter uncertainties, so comparing fits by chi-squared alone is misleading for sloppy models.","Geometric scaling reduces the number of iterations needed to reach convergence for every weighting scheme tested, while leaving the final parameter estimates essentially unchanged.","When the capture data are smoothed with FBET, the cropped-exponential prior performs best, and the optimum parameters remain consistent with the raw-data fits.","The unweighted LM Fisher information for this CoH3 problem is nearly degenerate, with only three nonzero eigenvalues, explaining its failure to converge unless initialized at a good point; the weighted variants produce a broader eigenvalue spectrum."],"supporting_citations":[{"why":"Supplies the classical Levenberg-Marquardt update that the weighting scheme modifies.","marker":"[1]"},{"why":"Provides the marginalization-over-weights construction from which the weighted Fisher information metric is derived.","marker":"[6]"},{"why":"Defines the standard optical-model baseline whose multiplicative tweaks are fitted.","marker":"[22]"},{"why":"Supports the CoH3 reaction code used for coupled-channels Hauser-Feshbach cross-section calculations.","marker":"[20, 21]"},{"why":"Provides the FBET linearized Bayesian smoothing procedure applied to the capture data.","marker":"[11]"},{"why":"Supplies the sensitivity-matrix construction used in the FBET smoothing step.","marker":"[33]"},{"why":"Connects the inverse Fisher information metric to parameter covariance via the Cramér-Rao bound, motivating the use of the wFIM for uncertainties.","marker":"[5]"},{"why":"Establishes the sloppy-model context that motivates weighting and explains the wide eigenvalue spectrum.","marker":"[3]"},{"why":"Aggregates the experimental total and capture cross-section data used in the fits.","marker":"[10]"}],"fun_headline_variants":["Weighted LM balances nuclear fits","Per-channel weights fix sloppy nuclear fits","Weighted LM: robust nuclear data fitting","Geometric scaling speeds nuclear fitting","Better nuclear fits with reweighted LM"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's case that weighted fits are more physically consistent rests on the unstated assumption that each fitted multiplier should stay between roughly 0.8 and 1.2 of the baseline optical-model value; if the unweighted fit's out-of-band parameters are actually admissible, the central comparison loses its force.","fun_headline_variants_meta":{"raw":{"variants":["Weighted LM balances nuclear fits","Per-channel weights fix sloppy nuclear fits","Weighted LM: robust nuclear data fitting","Geometric scaling speeds nuclear fitting","Better nuclear fits with reweighted LM"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000188,"raw_usage":{"total_tokens":1312,"prompt_tokens":904,"completion_tokens":408,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":520,"completion_tokens_details":{"reasoning_tokens":346}},"tokens_in":520,"tokens_out":408,"duration_ms":4614,"temperature":1.0,"reasoning_tokens":346,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:54:16.408721+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the same CoH3 fitting pipeline on another nucleus with similarly heterogeneous total and capture data, and check whether the unweighted fit again leaves the 0.8–1.2 multiplier band while the weighted fit stays inside; a case where both stay inside would weaken the claimed advantage, and an independent microscopic optical-model calculation of the out-of-band parameters would settle whether they are truly unphysical.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Aggregates the experimental total and capture cross-section data used in the fits."}],"review_version":1}