{"id":"143f872b-4725-44c5-8e34-b0daaa4580c2","arxiv_id":"2412.03970","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A DNN-based pipeline with Gauss-Jacobi quadrature and alternating sparse/global optimization discovers fractional differential equations from sparse, noisy data.","lead":"This paper builds a machine-learning pipeline that reads noisy measurements and outputs a fractional differential equation, a type of equation with memory and nonlocal effects. It tests the pipeline on simulated diffusion, frozen-soil creep data, and Lévy motion trajectories, recovering known equation forms in each case.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3.3 Lévy validation omits the drift term implied by its own generating stable law, so structure recovery in that headline case is not established.","rationale":"The reader's weakest_assumption focused on the closed candidate library and on the accuracy of DNN-based derivative estimates, while also listing the Lévy drift omission as one of several issues. I agree that the closed-library and derivative-accuracy concerns are real, but the drift omission in Section 3.3 is more load-bearing because it identifies a concrete internal inconsistency in one of the three validation cases used to support the central claim. The paper itself, in Appendix C, derives that the stated stable law corresponds to a space-fractional advection-diffusion equation with a drift term, yet Table 3 reports a learned equation and a 'ground truth' equation with no advection term. That is not a matter of tuning, error bars, or comparison methodology; it is a question of whether the reported experiment actually recovered the structure of the true governing equation. If the drift is genuinely present in the generating process and the learned model omits it, the framework's structure-recovery capability is called into question in a headline demonstration. If the drift was intentionally zero, the text's stable-law parameters are misleading, and the experimental validation does not match the stated problem. In either case, the claim that the method robustly identifies FDE structure across diverse scenarios is not fully supported by Section 3.3. The FADE recovery in Section 3.2 and the frozen-soil example in Section 3.1 provide independent evidence that the method can work, which is why I do not recommend rejection. The verdict remains CONDITIONAL, as the reader concluded, but with the emphasis shifted to the Lévy drift inconsistency as the decisive item to resolve before acceptance.","tokens_in":20715,"tokens_out":6723,"duration_ms":68895,"concrete_test":"Run the released code for Section 3.3 with the stated generating parameters S_{1.8}(1, 0.66, -0.32) and, independently, with a version where the location parameter is set to zero. Check whether the sparse regression output contains a c_x term in the first case and no c_x term in the second. If the nonzero-location simulation still yields a learned equation without advection, the structure-recovery claim in the Lévy case fails; if it yields an advection term, Table 3's ground-truth equation must be corrected to include drift.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of the paper is robust discovery of explicit FDE structure from data, and Section 3.3 is one of the three main demonstrations. The generating process is stated as a stable time series S_{1.8}(1, 0.66, -0.32). Under the paper's own parameterization in Eq. (24) and Appendix C, a stable law with beta = 1 and location mu = v t corresponds to the space-fractional advection-diffusion equation ct = -v cx + D cxx^alpha. The nonzero location parameter -0.32 is therefore a drift, and the corresponding PDE should contain an advection term. However, Table 3 reports the learned equation c_t^(0.99) = 0.48 c_xx^(1.84) and labels the ground truth as ct = 0.5 c_xx^(1.8), with no first-order spatial term. Either the reported ground truth is incomplete, or the simulation did not actually include the stated drift. In both cases, the Lévy experiment does not demonstrate recovery of the full FDE structure; if the drift was present and the learned model omitted it, the sparse regression incorrectly discarded a real term, and if the drift was absent, the experimental setup contradicts the stated stable-law parameters. This is an internal inconsistency in a headline validation case, not merely a missing comparison or a stylistic issue. The FADE and frozen-soil results remain supportive, but the central 'robustness across diverse examples' claim is weakened until the Lévy case is clarified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a stepwise data-driven framework for discovering explicit fractional differential equations (FDEs) from sparse and noisy data. The pipeline consists of (i) a deep neural network that denoises and reconstructs the field and supplies automatic differentiation, (ii) Gauss-Jacobi quadrature for evaluating fractional derivatives with few auxiliary nodes, and (iii) an alternating optimization scheme in which STRidge estimates sparse linear coefficients and a global optimizer (Differential Evolution or Powell) adjusts the fractional orders. The method is validated on three cases: experimental creep data of frozen soils (fractional Kelvin model), synthetic space-time fractional advection-diffusion data under clean, 5% noise, and 25% noise, and single-particle trajectories modeled by an alpha-stable Lévy motion. The authors also compare with DL-PDE and analyze the effect of the regularization parameter lambda.","tokens_in":20985,"tokens_out":5297,"duration_ms":50488,"significance":"If the claims hold, the framework is a useful contribution to data-driven discovery of fractional PDEs, especially the combination of DNN-based reconstruction with G-J quadrature and alternating optimization of linear coefficients and nonlinear fractional orders. The paper ships code and data, provides a clear algorithmic description, and includes a comparison with DL-PDE and a regularization sensitivity study. The FADE recoveries in Table 2 are clean and match the generating equation at all three noise levels, and the frozen-soil case recovers the Kelvin-model structure from experimental data. However, the Lévy validation has an internal inconsistency that undermines one of the three headline demonstrations, and the 'robustness' claim is supported only by single runs with model selection on the same test data. These issues need to be resolved before the central claim is fully established.","major_comments":[{"comment":"The Lévy validation is internally inconsistent. The text states that the stable time series is S_{1.8}(1, 0.66, -0.32), and Appendix C establishes that a stable law S_alpha(1, (Dt|cos(pi alpha/2)|)^{1/alpha}, vt) corresponds to the solution of the space-fractional advection-diffusion equation (25), which contains the advection term -v c_x. The nonzero location parameter -0.32 therefore implies a drift. Nevertheless, Table 3 reports the ground truth as c_t = 0.5 c_xx^{1.8} and the learned equation as c_t^(0.99) = 0.48 c_xx^(1.84), with no first-order spatial term. If the simulation included the stated drift, the sparse regression discarded a real term; if the drift was absent, the experimental setup contradicts the stated stable-law parameters. In either case, Section 3.3 does not demonstrate recovery of the full FDE structure, and the claim of robustness across the three demonstrations is weakened. Please clarify the generator, correct the ground truth, and rerun the experiment accordingly.","section":"Section 3.3, Table 3, Eq. (25), Appendix C"},{"comment":"The claim of robustness across noise levels is not statistically supported because each configuration is run once. For each noise level, a single realization is generated, a single DNN is trained, and a single sparse-regression path is reported, with no standard errors, random seeds, or repeated trials. In addition, lambda = 10^-3 is selected on the same noisy test cases in Table 6 and then used to produce the results in Table 2, which is a form of test-set selection. The consistent recoveries are encouraging, but the robustness claim needs repeated trials (or at least a sensitivity analysis over seeds and data realizations) and an independent model-selection procedure to rule out favorable chance outcomes.","section":"Section 3.2, Tables 2 and 6"},{"comment":"The method can only discover equations whose terms lie in the predefined candidate library, yet the abstract and introduction describe the framework as discovering FDEs 'without prior knowledge' of structure. In all three validations, the true terms are placed inside the library by construction. The authors acknowledge the closed-library limitation in Section 5, but the wording of the central claim overstates the degree to which the method is free of prior structural assumptions. Please qualify the claim explicitly, for example by stating that discovery is relative to the chosen library and that the library is a form of prior knowledge about which terms may appear.","section":"Section 2.2, Eq. (6); Section 5"}],"minor_comments":[{"comment":"The caption says 'Comparison of FADE results between our method and DL-PDE', but the benchmark in that table is the classical advection-diffusion equation with integer-order diffusion. Retitle the caption to avoid confusion.","section":"Table 5 caption"},{"comment":"The 'Error' and 'Error^2' columns are not defined. State whether the reported values are L2 errors, relative errors, or mean squared errors, and specify the data split over which they are computed.","section":"Tables 1-3"},{"comment":"There are several wording issues: 'lost function' should be 'loss function', and 'Appendix Appendix B' and 'Appendix Appendix C' should be 'Appendix B' and 'Appendix C'.","section":"Introduction and appendices"},{"comment":"The choice of five G-J quadrature nodes is stated without a convergence or sensitivity study. A brief numerical test in Appendix B showing how accuracy varies with the number of nodes would support this choice.","section":"Section 2.3 and Appendix B"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for a nonlinear dynamics journal and has a clear algorithmic contribution with code availability. The main blocker is the Section 3.3 Lévy inconsistency: the stated stable-law parameters imply a drift term that is absent from both the reported ground truth and the learned equation. The single-run nature of the experiments and the test-set-based selection of lambda also need attention. If the authors can clarify and correct the Lévy experiment and add repeated trials or error bars, the paper would be suitable for publication. I do not see grounds for rejection if these issues are fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"One-line take: useful pipeline, clean FADE recovery, but the Lévy validation case has a drift inconsistency that undercuts the general robustness claim.\n\nThe paper's real contribution is combining DNN-based reconstruction with Gauss-Jacobi quadrature to build fractional-derivative libraries, then alternating STRidge and global optimization for fractional orders. That combination is new, the code is public, and the quadrature appendix is genuinely useful. The FADE recovery (Table 2) is clean at all three noise levels, and the frozen-soil demonstration is a nice real-data case. Credit where due.\n\nThe soft spot that matters: Section 3.3. The generating stable law is S_{1.8}(1, 0.66, -0.32). Under the paper's own parameterization in Appendix C, the location parameter -0.32 is a drift, so the corresponding FDE should contain an advection term. The reported ground truth is c_t = 0.5 c_xx^1.8, with no drift, and the learned equation also omits it. That is an internal inconsistency in a headline validation. If the drift was present and missed, sparse regression discarded a real term; if it was absent, the stated stable-law parameters don't match the experiment. Either way, the Lévy case doesn't demonstrate full structure recovery.\n\nOther issues are smaller. Single runs with no error bars or seeds, so 'robustness across noise levels' is not statistically supported. The regularization lambda is tuned on the same FADE benchmark that is later judged. And Vats et al., the closest prior FDE-discovery work, is discussed but not benchmarked. The closed-library limitation is acknowledged in the conclusions, which is honest.\n\nBottom line: the method is plausible and the FADE result gives real evidence, but the Lévy inconsistency needs to be resolved before I'd trust the central claim. I'd send it to peer review—a good referee can ask for a corrected Section 3.3, error bars, and a Vats comparison.","headline":"Clean FADE recovery and a useful DNN+quadrature pipeline, but the Lévy validation case has a drift inconsistency that undercuts the general robustness claim.","tokens_in":21574,"tokens_out":4134,"would_cite":true,"duration_ms":36978,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34A08","35R11","65D30","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"A stepwise data-driven method discovers explicit fractional differential equations directly from sparse, noisy data.","keywords":["fractional differential equations","knowledge discovery","sparse regression","Gauss-Jacobi quadrature","machine learning","anomalous diffusion","Lévy motion"],"falsifier":"Generate data from a known FDE whose correct term is deliberately absent from the candidate library, run the full pipeline, and check whether the recovered equation is structurally wrong; additionally, on the clean FADE benchmark, compare the DNN-plus-quadrature fractional derivative against a high-resolution spectral derivative of the true solution at the same points, since large pointwise error with a correct recovered equation would indicate accidental cancellation rather than reliable estimation.","tokens_in":20475,"feed_emoji":"📐","tokens_out":10160,"duration_ms":85591,"temperature":0.7,"pith_summary":"The paper sets out to show that explicit fractional differential equations (FDEs) can be discovered directly from sparse, noisy measurements, without knowing the structure of the equation in advance. Its proposed pipeline uses a deep neural network to denoise and interpolate the data, Gauss-Jacobi quadrature to evaluate the singular convolution integrals that define fractional derivatives, and an alternating routine that combines sparse regression for the linear coefficients with global optimization for the fractional orders. The authors validate the pipeline on synthetic anomalous-diffusion data, experimental creep data from frozen soils, and single-particle Lévy trajectories, recovering equation structures that match the known models in all three cases. If the claim holds, fractional models of memory and nonlocal effects can be obtained from observations in fields where first-principles derivation is impractical.","feed_headline":"New pipeline extracts fractional equations from noisy data","feed_subtitle":"Neural reconstruction, singular-integral quadrature, and sparse regression recover explicit memory models without prior knowledge.","key_machinery":"The load-bearing machinery is a candidate library $\\Theta(\\beta)$ built from fractional derivatives, combined with Gauss-Jacobi quadrature and an alternating optimization scheme. Gauss-Jacobi quadrature is a numerical integration rule for integrands with endpoint singularities; here it evaluates the convolution integral in the Caputo or Riemann-Liouville fractional derivative using a small fixed set of weighted function values from the DNN-reconstructed field, replacing dense meshes. The alternating scheme is necessary because fractional orders enter the library nonlinearly: STRidge (sequential threshold ridge regression) determines the sparse linear coefficient vector $\\xi$, while a global optimizer (differential evolution for multi-parameter problems and a derivative-free optimizer for single-parameter problems) adjusts the fractional orders $\\alpha$ and $\\beta$. The loss surface is non-convex and discontinuous because the number of nonzero coefficients changes with the fractional orders, which is why the authors use global rather than gradient-based optimization.","core_discovery":"The paper's central claim is that the combination of a DNN surrogate, Gauss-Jacobi quadrature, and alternating sparse/global optimization recovers explicit FDEs from data. Concretely, the reported discoveries include the fractional Kelvin model $\\varepsilon_t^{(0.374)} = -2.402\\varepsilon + 8.125$ for clay creep, the fractional advection-diffusion equation $c_t^{(0.790)} = -1.006 c_x + 0.501 c_x^{(1.720)}$ from clean synthetic transport data, and the space-fractional diffusion equation $c_t^{(0.99)} = 0.48 c_{xx}^{(1.84)}$ from $\\alpha$-stable single-particle trajectories. The method also learns an integer-order time derivative as a limit case, recovering the order as 0.99 when the true order is 1. These results are offered as evidence that interpretable, closed-form FDEs can be identified without prior structural knowledge.","pith_inferences":["Beyond the paper's tests, the same alternating scheme could be pointed at variable-order fractional equations if the candidate library were expanded to include order-varying terms; the paper does not attempt this.","The five-node Gauss-Jacobi quadrature is a fixed choice, so an explicit node-count sensitivity study would separate quadrature error from data noise as the limit on recovered fractional orders.","The core reliance on a closed candidate library could be lifted by coupling the pipeline with open-form symbolic search, a direction the conclusion sketches but does not implement here."],"forward_implications":["On the synthetic FADE benchmark, the method recovers the true structure with fractional orders within about 0.02 to 0.16 of the ground truth under clean, 5% noise, and 25% noise conditions.","On experimental frozen-soil creep data, it learns a fractional Kelvin model without being given the equation form, with lower parameter error for clay than for silt.","On $\\alpha$-stable Lévy trajectories, it identifies a space-fractional diffusion equation with a nearly integer time order, showing that memoryless processes fall within its scope.","Compared with an integer-order-only deep-learning PDE discovery method, the fractional library appears necessary: the integer-only method captures only local advection and dispersion and omits the fractional term.","A regularization sweep indicates that $\\lambda = 10^{-3}$ yields the correct FADE structure across all tested noise levels, while too small $\\lambda$ adds redundant terms and too large $\\lambda$ drops the fractional derivative term."],"supporting_citations":[{"why":"Introduces STRidge, the sequential threshold ridge regression used here for the sparse linear coefficients.","marker":"[10]"},{"why":"Supplies the DNN-plus-STRidge strategy for PDE discovery that this framework extends to fractional derivatives.","marker":"[16]"},{"why":"Presents the closest prior white-box FDE-discovery method, which this work improves on via mesh-free quadrature and DNN reconstruction.","marker":"[54]"},{"why":"Provides the Gauss-Jacobi quadrature rule used to approximate fractional integrals with endpoint singularities.","marker":"[55]"},{"why":"Contributes the experimental frozen-soil creep dataset and the fractional Kelvin model used as ground truth.","marker":"[59]"},{"why":"Establishes the stable-law learning problem by recovering fractional diffusion equations from alpha-stable time series with known structure; this paper removes the structural prior.","marker":"[50]"},{"why":"Introduces sparse identification of governing equations, the parent approach for candidate-library sparse regression.","marker":"[9]"}],"fun_headline_variants":["AI uncovers fractional laws from noisy observations","Neural denoising plus sparse regression yields FDEs","Fractional model discovery from single-particle tracks","Hidden memory equations emerge from messy experiments"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the true equation is a linear combination of terms from a hand-chosen candidate library, and that the DNN plus five-node Gauss-Jacobi quadrature produces accurate enough fractional derivatives from the available sparse, noisy data.","fun_headline_variants_meta":{"raw":{"variants":["AI uncovers fractional laws from noisy observations","Neural denoising plus sparse regression yields FDEs","Fractional model discovery from single-particle tracks","Hidden memory equations emerge from messy experiments"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000197,"raw_usage":{"total_tokens":1361,"prompt_tokens":937,"completion_tokens":424,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":553,"completion_tokens_details":{"reasoning_tokens":366}},"tokens_in":553,"tokens_out":424,"duration_ms":4861,"temperature":1.0,"reasoning_tokens":366,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:53:12.609796+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate data from a known FDE whose correct term is deliberately absent from the candidate library, run the full pipeline, and check whether the recovered equation is structurally wrong; additionally, on the clean FADE benchmark, compare the DNN-plus-quadrature fractional derivative against a high-resolution spectral derivative of the true solution at the same points, since large pointwise error with a correct recovered equation would indicate accidental cancellation rather than reliable estimation.","supporting_citations":[{"cited_title":"Data-driven discovery of partial di fferential equations","cited_arxiv_id":null,"evidence_quote":"Introduces STRidge, the sequential threshold ridge regression used here for the sparse linear coefficients."},{"cited_title":"A new perspective for scientific modelling: Sparse reconstruction- based approach for learning time-space fractional di fferential equations","cited_arxiv_id":null,"evidence_quote":"Presents the closest prior white-box FDE-discovery method, which this work improves on via mesh-free quadrature and DNN reconstruction."},{"cited_title":"Gauss-Jacobi-type quadrature rules for fractional directional integrals","cited_arxiv_id":null,"evidence_quote":"Provides the Gauss-Jacobi quadrature rule used to approximate fractional integrals with endpoint singularities."},{"cited_title":"Particle swarm fractional order derivative model of artificial frozen soil creep properties","cited_arxiv_id":null,"evidence_quote":"Contributes the experimental frozen-soil creep dataset and the fractional Kelvin model used as ground truth."},{"cited_title":"Machine learning of space-fractional di fferential equations","cited_arxiv_id":null,"evidence_quote":"Establishes the stable-law learning problem by recovering fractional diffusion equations from alpha-stable time series with known structure; this paper removes the structural prior."}],"review_version":1}