{"id":"b769ae56-45ee-48d6-853a-f469235d294a","arxiv_id":"2412.06304","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Backpropagation through deuteron and NN scattering calculations gives uncertainty estimates that mostly match direct sampling, but can fail for momenta near quadrature nodes.","lead":"This paper uses PyTorch backpropagation to turn uncertainties in nuclear potential values into uncertainties in deuteron binding energy and nucleon-nucleon phase shifts. The method is checked against Monte Carlo sampling, with good agreement in most cases and notable exceptions near quadrature points.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Table 3 shows backpropagation uncertainty estimates that are off by orders of magnitude when p0 nears a quadrature node, so the abstract's unqualified 'very good agreement' claim is not yet supported without a detection criterion.","rationale":"The reader's verdict is CONDITIONAL with moderate confidence, and I agree with the identified weakest assumption. The paper's strongest claim is the abstract's validation statement, and Table 3 directly contradicts it for two of the eight tested momenta. The likely cause is well identified in the text, but the paper does not convert this into a safeguard, so the method is not yet a drop-in tool for arbitrary p0. Independent support includes the released code and data repository, the Kolmogorov-Smirnov normality tests, and good agreement for the deuteron binding energy and most scattering channels. These do not resolve the quadrature-proximity failure. My read does not change the verdict: CONDITIONAL remains appropriate, with the condition being that the authors qualify the abstract and provide a way to detect or avoid the problematic grid points.","tokens_in":14209,"tokens_out":2485,"duration_ms":26438,"concrete_test":"Rerun the 1P1 phase-shift uncertainty calculation for p0 = 0.5 and p0 = 4.0 fm^-1 using the same N = 48 Gaussian grid shifted by half the local grid spacing, and also with N = 96 and N = 192 Gauss-Legendre grids; compare sigma_backprop to sigma_samples. If the backpropagation values move from 3225.6/217.8 degrees toward 0.4/26.0 degrees while the phase shifts remain stable, this confirms that the disagreement is caused by node proximity. Record the minimum distance |p0 - p_k| divided by the local grid spacing for each p0 to derive a quantitative criterion for when the backpropagation estimate is trustworthy.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that backpropagated gradients through the PyTorch Lippmann-Schwinger implementation yield uncertainty estimates validated by direct sampling. The strongest counter-evidence is Table 3: at p0 = 0.5 fm^-1 and p0 = 4.0 fm^-1, sigma_backprop is 3225.6 and 217.8 degrees versus 0.4 and 26.0 degrees from sampling, a discrepancy of roughly a factor of 8000 and 8 respectively. The authors correctly trace this to the proximity of p0 to Gaussian quadrature nodes (0.4999 and 4.0200 fm^-1). This is not a peripheral numerical detail: the discretization in Eqs. (23)-(25) places p0 on the extended grid with zero weight, but the principal-value kernel contains 1/(p0^2 - k^2), so when p0 is close to a quadrature node k, the matrix A becomes nearly singular and the gradient with respect to V is amplified by a large factor. The resulting uncertainty is then dominated by discretization rather than by the physical uncertainty of the observable. The paper's suggested remedy, that a large backpropagation uncertainty can be used as a signal to reconsider the numerical parameters, is only a heuristic; no threshold, no automatic detection, and no modification of the abstract's unqualified validation statement are provided. Since the method is presented as a practical tool, users need to know when the answer is unreliable. The deuteron results and the other Table 3 rows agree well, so the method is conditionally valid, but the central validation claim must be qualified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents a computational method for estimating uncertainties in the deuteron binding energy and in nucleon-nucleon phase shifts induced by uncertainties in the scalar functions that define a two-nucleon potential. The potential is represented on a discrete momentum-angle grid, and the deuteron eigenvalue problem and the Lippmann-Schwinger equation are implemented in PyTorch so that backpropagation can compute gradients of the selected eigenvalues and phase shifts with respect to every grid value of the potential. Assuming independent Gaussian errors with relative standard deviations alpha(i) on the scalar-function values, the authors propagate these errors through a first-order formula and compare the resulting uncertainties with direct Monte Carlo sampling of the same error model. Good agreement is reported for the deuteron eigenvalue and for most of the tested scattering channels, but Table 3 shows strong disagreements for p0 = 0.5 and 4.0 fm^-1, which the authors attribute to proximity to Gaussian quadrature nodes.","tokens_in":14576,"tokens_out":9159,"duration_ms":100527,"significance":"If the method is reliable, it is a useful addition to uncertainty quantification in few-nucleon calculations: it avoids repeated solves, works with hundreds of thousands of input parameters, and the code and plot data are publicly available. The approach is not circular, because the alpha(i) error model is an external input and the validation compares two ways of evaluating the same uncertainty model. However, the central validation claim is currently overstated: the discrepancies in Table 3 are not compatible with the abstract's unqualified 'very good agreement', and the deuteron binding-energy uncertainty is not validated by direct sampling of the fitted quantity. These issues are local and fixable, but they are load-bearing for the paper's main message.","major_comments":[{"comment":"The central validation claim is not supported by the full Table 3. At p0 = 0.5 fm^-1 the backpropagated uncertainty is 3225.6 degrees versus 0.4 degrees from sampling, and at p0 = 4.0 fm^-1 it is 217.8 degrees versus 26.0 degrees. The text's explanation (proximity to quadrature nodes 0.4999 and 4.0200 fm^-1, in the context of Eqs. (23)-(25)) is plausible, but the prescription that a large backpropagated uncertainty can be used as a signal is post hoc and provides no threshold or test. Because the paper presents this as a practical tool, please (i) qualify the abstract and the 'very good agreement' statement to exclude near-node momenta, (ii) provide an automatic diagnostic, for example min_k |p0 - p_k| or the condition number of the discretized A matrix in Eq. (25), and (iii) demonstrate that the diagnostic flags the failed rows and that refining or shifting the quadrature grid removes the discrepancy.","section":"Abstract; §3.2, Table 3"},{"comment":"The sampling validation in the deuteron section is performed for the eigenvalue lambda at a single energy, not for the deuteron binding energy E_d reported in Table 1. The uncertainty sigma(E_d) is obtained from a weighted least-squares fit in which the errors of lambda(E_i) are treated as independent, although the lambda(E_i) values share the same random potential perturbation and are therefore correlated. The claimed validation by direct sampling therefore does not currently cover E_d. Please sample whole lambda(E) curves and compare the distribution of fitted E_d with the reported sigma(E_d), or restrict the validation claim to the single-energy eigenvalue uncertainties.","section":"§2.2, Table 1 and Figs. 2, 4, 6"},{"comment":"The p0 = 4.0 fm^-1 row shows that direct sampling is itself sensitive to the numerical grid, with sigma_samples = 26.0 degrees versus 1.1 and 1.6 degrees at the neighboring p0 = 3 and 5 fm^-1. The issue is therefore not only a backpropagation artifact but also a property of the discretized principal-value integral. The paper should include a grid-convergence study for at least one near-node energy and one safe energy, reporting how both sigma_backpropagation and sigma_samples change when the number of quadrature points or the quadrature rule is varied. Such a study would give users a practical way to decide when a computed uncertainty is trustworthy.","section":"§3.2, Table 3"}],"minor_comments":[{"comment":"The phrase 'thePythonpytorchlibrary' should read 'the Python pytorch library'.","section":"Abstract"},{"comment":"Several table captions contain 'T able' instead of 'Table'; this should be corrected throughout.","section":"Tables"},{"comment":"The text following Eq. (11) spells 'Clebsh-Gordan'; the standard spelling is 'Clebsch-Gordan'.","section":"§2.1"},{"comment":"In Table 3, the units of p0 should appear in the column header (for example, 'p0 [fm^-1]') rather than only being explained in the text.","section":"§3.2, Table 3"},{"comment":"The type of Gaussian quadrature and the momentum cutoff used for the scattering results in Tables 2-5 are not stated; Section 3.1 should report the same grid parameters that are given for the deuteron calculation.","section":"§3.1"},{"comment":"The caption of Figure 2 states that 48 momentum points were used in the interval (0, 10) fm^-1, while Figure 1 uses (0, 20) fm^-1; please clarify whether these are different grids and why.","section":"Fig. 2 caption"},{"comment":"Reference [7] lists 'H. Krebs' twice in the author list; the correct list should be 'P. Reinert, H. Krebs, and E. Epelbaum'.","section":"References"},{"comment":"Equation (17) is presented without a derivation or citation; a standard reference for first-order error propagation would help readers assess the approximation.","section":"§2.1"}],"recommendation":"major_revision","confidential_remarks":"To the editor: This is a useful methods paper and the central idea is sound. The main shortcomings are fixable: the validation claim needs qualification and a diagnostic for the quadrature-node failure in Table 3, and the deuteron binding-energy uncertainty should either be validated by sampling the fitted quantity or the claims should be narrowed. I do not see circularity or hidden use of the data. The paper is within the scope of the journal and, after revision, could be a solid contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper is a genuine methodological contribution—reverse-mode automatic differentiation (backpropagation) through PyTorch to propagate uncertainties in deuteron and NN scattering observables with hundreds of thousands of potential lattice values. The validation against direct sampling is convincing for most kinematics. The real caveat is Table 3: at p0 = 0.5 and 4.0 fm^-1 the backpropagated uncertainty is 3225.6 and 217.8 degrees versus 0.4 and 26.0 from sampling. The authors correctly trace this to the on-shell momentum sitting near a Gaussian quadrature node, and they suggest the large uncertainty itself is a signal to reconsider the numerical grid. But that's a heuristic, not a safeguard, and the abstract's unqualified 'very good agreement' is not supported by those rows.\n\nWhat's new: earlier uncertainty work used forward-mode AD (Carlsson et al. 2016) or repeated sampling. Backpropagation is the right tool for many inputs/few outputs, and this is the first few-nucleon implementation that I know of. The code is on GitHub, the error propagation is standard first-order Gaussian propagation, and the non-trivial piece—gradients through the eigensolver and the linear solves—checks out against sampling in the deuteron and in the 1S0, 1P1 (most p0 values), 3S1-3D1, and 3P2-3F2 channels. That's a solid validation.\n\nSoft spots: the quadrature-node sensitivity is the main one. The matrix in Eq. (25) becomes nearly singular when p0 is close to a grid point, and the gradient is amplified by the 1/(p0^2 - k^2) kernel. A user needs a way to detect this automatically, not just a rule of thumb. The assumed alphas are arbitrary, which is fine for a methods paper but means the numerical values carry no physics until applied to a realistic error budget. The uncorrelated-error assumption is acknowledged as future work, and that's fine.\n\nOverall, it's a worthwhile methods paper, honest about its limitations, and I'd send it to review. The referee should ask for a qualified abstract and a more robust treatment of the node problem.","headline":"Genuinely new and mostly validated backpropagation approach to two-nucleon uncertainty propagation, but the Table 3 quadrature-node blowups require qualifying the abstract's 'very good agreement' claim.","tokens_in":15086,"tokens_out":3063,"would_cite":true,"duration_ms":28250,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.65.Ge","24.10.-i"],"model":"deepseek-v4-flash","headline":"This paper claims that backpropagation can turn uncertainties in a two-nucleon potential into error bars on observables, and that those error bars match direct sampling.","keywords":["backpropagation","automatic differentiation","theoretical uncertainties","nucleon-nucleon scattering","deuteron bound state","Lippmann-Schwinger equation","error propagation","nuclear potential"],"falsifier":"Sweep the on-shell momentum \\(p_0\\) continuously between two adjacent quadrature nodes of the same grid and compare the backpropagated uncertainty with the Monte Carlo sampled uncertainty at each point; a sharp spike in the ratio at any node would confirm that the method's accuracy depends on grid placement rather than on the physics alone.","tokens_in":1679,"feed_emoji":"⚛️","tokens_out":2177,"duration_ms":78164,"temperature":0.7,"pith_summary":"This paper argues that backpropagation, the automatic-differentiation technique behind machine learning, can compute gradients of two-nucleon observables with respect to every value in the lattice that defines the nuclear potential, and that those gradients turn potential uncertainties into observable uncertainties through first-order error propagation. The authors implement the deuteron bound-state problem and the nucleon-nucleon scattering problem as differentiable tensor computations, then propagate uncorrelated Gaussian errors in the scalar potential functions to the binding energy and phase shifts. They validate the result by sampling many perturbed potentials and find good agreement with the backpropagated standard deviations. This makes uncertainty estimates that would otherwise require many repeated calculations available from a single backward pass, even when there are hundreds of thousands of input parameters.","feed_headline":"Backpropagation puts error bars on nuclear observables","feed_subtitle":"One backward pass reproduces Monte Carlo uncertainties for deuteron binding and NN phase shifts, at a fraction of the cost.","key_machinery":"The carrying object is the differentiable implementation of the two-nucleon calculation. The potential is written as a linear combination of spin–momentum operators with scalar coefficient functions \\(v_i(p',p,x)\\) stored as a discrete array \\(V_{ijkl}\\); partial-wave matrix elements are built by tensor contractions with precomputed angular-integration arrays, and the resulting eigenproblem or linear system is solved with differentiable library routines. Because every operation is recorded in a computation graph, backpropagation computes the gradient of an observable with respect to all entries of \\(V\\) in one pass, and the uncorrelated-error formula \\(\\$\\sigma$(O) = \\sqrt{\\sum_{ijkl} \\$\\sigma$(V_{ijkl})^2 (\\partial O/\\partial V_{ijkl})^2}\\) turns those gradients into uncertainties.","core_discovery":"The central claim is that error propagation from the scalar functions defining a two-nucleon potential to observables can be done through the calculation itself: once the eigenvalue equation for the deuteron and the Lippmann–Schwinger equation for scattering are written as differentiable tensor operations, the backward pass yields \\(\\partial O/\\partial V_{ijkl}\\) for each lattice value of each scalar function, and the variance formula with \\(\\$\\sigma$(V_{ijkl}) = \\$\\alpha$(i)|V_{ijkl}|\\) gives the observable uncertainty. Direct Monte Carlo sampling of the potential values confirms the result: for the \\(^1S_0\\) and \\(^1P_1\\) channels and the \\(^3S_1\\)-\\(^3D_1\\) and \\(^3P_2\\)-\\(^3F_2\\) coupled channels, the backpropagated and sampled standard deviations agree closely in most cases, and the sampled eigenvalue distributions are approximately Gaussian. The method is presented as especially efficient for many inputs and few outputs, and as a tool for identifying which parts of a potential drive the uncertainty of a given observable.","pith_inferences":["The same gradient machinery could propagate correlated errors among lattice values, which the paper notes as future work, making realistic parameter covariances from chiral potentials accessible without large sample ensembles.","Because the method only requires the calculation to be an eigenproblem or a linear solve, it could be carried over to Faddeev three-nucleon equations and to nuclear structure calculations implemented in other languages, as the authors suggest.","Gradients from one backward pass could be reused to build linear emulators of observables around a reference potential, making uncertainty estimates for many energies or angles almost free once the reference calculation is done.","The near-node blow-up could be turned into a grid-adaptation criterion: choose quadrature points so that the backpropagated uncertainty for a test observable is stable under small grid shifts."],"forward_implications":["Deuteron binding-energy uncertainty scales roughly linearly with the assumed relative error \\(\\alpha\\) for the values tested, so the backpropagated error bars can be rescaled when the input uncertainty changes.","For the uncoupled \\(^1P_1\\) channel at \\(p_0 = 2\\) fm\\(^{-1}\\), backpropagated and sampled phase-shift uncertainties agree across \\(\\alpha\\) from 1% to 10% (e.g., 4.8 versus 4.6 degrees at \\(\\alpha = 0.1\\)).","For the coupled \\(^3S_1\\)-\\(^3D_1\\) and \\(^3P_2\\)-\\(^3F_2\\) channels, backpropagation reproduces the sampled uncertainties for phase shifts and mixing parameters at \\(p_0 = 2\\) fm\\(^{-1}\\).","The gradient map (e.g., \\(\\partial\\lambda/\\partial v_1\\) over momentum values) shows which regions of a scalar function most influence the binding energy, enabling targeted checks of potential models.","When the on-shell momentum sits close to a quadrature node, the backpropagated uncertainty blows up (3225.6 versus 0.4 degrees at \\(p_0 = 0.5\\) fm\\(^{-1}\\)), so the method doubles as a numerical-quality diagnostic."],"supporting_citations":[{"why":"Supplies the automatic-differentiation library whose backward pass computes gradients of observables with respect to potential lattice values.","marker":"[23]"},{"why":"Supplies the backpropagation algorithm that the method uses to propagate errors through the computation graph.","marker":"[24]"},{"why":"Supplies the partial-wave decomposition procedure used to build potential matrix elements from the scalar functions.","marker":"[30]"},{"why":"Supplies the two-nucleon potential whose scalar functions carry the assumed uncertainties in the numerical tests.","marker":"[27]"}],"fun_headline_variants":["One backward pass estimates nuclear uncertainties","Backpropagation makes nuclear error bars cheap","Fast nuclear uncertainty via automatic differentiation","Backprop validates Monte Carlo for nuclear observables","Error propagation through backprop for few-nucleon systems"],"cache_read_input_tokens":17152,"weakest_assumption_plain":"The method assumes the momentum grid used in the scattering calculation keeps the on-shell momentum away from every quadrature node; when that fails, the backpropagated uncertainty becomes unreliable, as Table 3 shows at \\(p_0 = 0.5\\) and \\(4.0\\) fm\\(^{-1}\\).","fun_headline_variants_meta":{"raw":{"variants":["One backward pass estimates nuclear uncertainties","Backpropagation makes nuclear error bars cheap","Fast nuclear uncertainty via automatic differentiation","Backprop validates Monte Carlo for nuclear observables","Error propagation through backprop for few-nucleon systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000229,"raw_usage":{"total_tokens":1462,"prompt_tokens":911,"completion_tokens":551,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":485}},"tokens_in":527,"tokens_out":551,"duration_ms":5614,"temperature":1.0,"reasoning_tokens":485,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T19:47:38.981801+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Sweep the on-shell momentum \\(p_0\\) continuously between two adjacent quadrature nodes of the same grid and compare the backpropagated uncertainty with the Monte Carlo sampled uncertainty at each point; a sharp spike in the ratio at any node would confirm that the method's accuracy depends on grid placement rather than on the physics alone.","supporting_citations":[{"cited_title":"Pytorch: An imperative style, high- performance deep learning library,","cited_arxiv_id":null,"evidence_quote":"Supplies the automatic-differentiation library whose backward pass computes gradients of observables with respect to potential lattice values."},{"cited_title":"Automatic differentiation in machine learning: a survey,","cited_arxiv_id":null,"evidence_quote":"Supplies the backpropagation algorithm that the method uses to propagate errors through the computation graph."},{"cited_title":"A new way to per- form partial-wave decompositions of few-nucleon forces,","cited_arxiv_id":null,"evidence_quote":"Supplies the partial-wave decomposition procedure used to build potential matrix elements from the scalar functions."},{"cited_title":"Machleidt, The Meson Theory of Nuclear Forces and Nuclear Structure, pp","cited_arxiv_id":null,"evidence_quote":"Supplies the two-nucleon potential whose scalar functions carry the assumed uncertainties in the numerical tests."}],"review_version":1}