{"id":"0f3142bd-bcc9-471e-afe9-b99a3b044134","arxiv_id":"2505.03021","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Neural ODE models trained on 19 experimental creep-fatigue tests of Alloy 617 reproduce stress-strain loops better than the standard Chaboche model, and symbolic regression converts one into an interpretable polynomial.","lead":"This paper trains two neural-network-based ODE models on experimental creep-fatigue stress-strain data from Alloy 617 at 950 degrees Celsius, and compares them to the standard Chaboche model. The neural models fit the data more closely, and a symbolic regression step converts the best one into a simple polynomial equation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Per-curve normalization in Sec. 3.1 Eq. (12) leaks each validation curve's future stress extrema into the evaluation, so the claimed accuracy advantage over Chaboche is not yet established as a predictive result; a training-only normalization or leave-one-out evaluation is needed.","rationale":"I considered whether missing SINDy equations, absent calibrated parameters, or lack of code and data are more load-bearing; those are serious reproducibility gaps, but they do not by themselves contradict the central empirical claim. The normalization protocol, by contrast, directly undermines the validity of the accuracy comparison: the validation target for each held-out curve is scaled using that curve's own future stress extrema. The paper's own Section 6 limitation statement, admitting that out-of-sample testing is not feasible, supports the same reading: the reported advantage over Chaboche is a curve-fitting result, not a demonstrated predictive result. The reader's weakest assumption identifies the same issue, and my analysis confirms it as the most load-bearing concern. A conditional verdict remains appropriate because the flaw is fixable with a clean leave-one-out evaluation using training-only normalization; the secondary reproducibility gaps should also be addressed in revision.","tokens_in":18121,"tokens_out":9657,"duration_ms":109793,"concrete_test":"Recompute the full comparison with leave-one-curve-out cross-validation, using a normalization transform fit only on the training fold (or no stress normalization at all), and report validation MSE in physical stress units (MPa) for the black-box, neural flow-rule, SINDy, and Chaboche models, recalibrating Chaboche on each training fold. If the black-box model's average validation MSE is not consistently below Chaboche, the per-curve normalization in Eq. (12) is responsible for the claimed advantage.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the Neural ODE models 'precisely capture' and 'exceed the standard Chaboche model's accuracy' rests on a validation comparison made in a normalized space. Section 3.1 (Eq. (12)) normalizes each stress-strain curve using that curve's own global minimum and maximum before training and validation. For strain, the extrema are known from the controlled command, but for stress they are measured outputs of the very validation curve being predicted. The evaluation therefore requires knowing the future stress range of each held-out experiment to define the target; at deployment that information is unavailable, so the trained model cannot produce physical stress from strain history without the missing extrema. The paper's Conclusion explicitly concedes that 'out-of-sample testing is not feasible for this paper.' With only one 15/4 split and no cross-validation, the Fig. 14 MSE comparison is a curve-fitting comparison, not a predictive evaluation. The normalization is load-bearing: it is the mechanism by which the abstract's superiority claim could survive even if the model had no generalization ability.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two Neural ODE architectures, a fully black-box model and a hybrid 'Neural flow rule' model, for uniaxial creep-fatigue stress-strain response of Alloy 617 at 950°C. The models are trained on 15 experimental stress-strain curves and evaluated on 4, and compared against a calibrated three-backstress Chaboche model. The authors report that the black-box Neural ODE exceeds Chaboche accuracy, that the Neural flow rule model is comparable or better than Chaboche, and that a SINDy-derived polynomial model extracted from the black-box network achieves accuracy comparable to Chaboche while being interpretable. The paper also presents a state-space stability analysis of the SINDy model.","tokens_in":18429,"tokens_out":3319,"duration_ms":35039,"significance":"If the central claims are valid, the paper would make a useful contribution: it applies Neural ODEs directly to experimental creep-fatigue data, uses a small dataset (a realistic constraint), provides an open-source implicit ODE solver (pyoptmat), and attempts to interpret the learned dynamics via SINDy. The explicit modeling of the elastic part in the Neural flow rule model is a sensible hybrid design. However, the evaluation methodology currently does not support the abstract's claim of 'precisely capture' and 'exceeding the standard Chaboche model's accuracy.' The per-curve normalization leaks future information into validation, the Chaboche baseline parameters are not actually reported, and the SINDy model is fit to the network's own trajectories rather than to independent experimental data. These issues are load-bearing for the paper's main conclusions.","major_comments":[{"comment":"The normalization described in Eq. (12) uses each stress-strain curve's own global minimum and maximum before the training/validation split. For the stress variable, these extrema are measured outputs of the very validation curve being predicted, so the validation procedure assumes knowledge of future stress values that would not be available at deployment. The MSE comparison in Fig. 14 and the abstract's superiority claim are therefore not a genuine out-of-sample evaluation. The authors should redo the evaluation with normalization constants computed only from training data (or, better, with leave-one-out cross-validation where normalization is refit on each training fold) and report results in physical units. The current Conclusion statement that 'out-of-sample testing is not feasible' is in tension with the paper's earlier validation claims.","section":"Section 3.1, Eq. (12)"},{"comment":"Table 2 is captioned 'Calibrated Parameters for Chaboche Model' but its content is a list of experimental conditions, not the calibrated Chaboche parameters (E, sigma0, K, eta, n, C_i, gamma_i, delta, R). Without these numbers, the Chaboche baseline is not reproducible, and the reader cannot judge whether the calibration is of comparable quality to the neural models or whether the comparison is fair. The authors should either report the calibrated parameters or clearly indicate where they can be found, and should also state which of the 19 experiments were used for calibration (e.g., all 19, or only the 15 training curves, and with what loss).","section":"Section 4, Table 2"},{"comment":"The SINDy model is fit to the black-box neural network's own trajectories (X and Xdot extracted from the trained network), not directly to the experimental data. As a result, the accuracy comparison of the 'interpretable model' against experimental data in Fig. 16 is not an independent test of the discovered equations; it measures how well a cubic polynomial approximates the network, compounded by the network's own fit quality and the normalization leakage. The claim that a 'purely data-driven approach achieved accuracy close to the Chaboche model' is therefore overstated. Additionally, the detailed polynomial equations promised after Eq. (18) are not actually printed, so the interpretability and stability analysis (Eqs. (19)-(20)) cannot be verified by the reader. The authors should report the discovered equations and, ideally, re-fit or at least evaluate the SINDy model directly on experimental data.","section":"Section 5.1"},{"comment":"The evaluation rests on a single 15/4 split with no cross-validation, no repeated splits, and no error bars, as shown in Fig. 14. With only four validation curves and a per-curve normalization that leaks information, the claimed 'superior predictive capability' of the black-box model is not statistically supported. The authors should provide either a leave-one-out evaluation, repeated random splits with reported means and standard deviations, or at minimum a clear statement that the 'validation' here is not predictive but an assessment of interpolation within the same experimental database. The Section 6 statement that 'out-of-sample testing is not feasible for this paper' should be reflected in a more guarded version of the abstract's claim.","section":"Section 2.2 / 4.2, Validation methodology"}],"minor_comments":[{"comment":"The internal state h is introduced as 'arbitrary' and its dimension is never specified. Since h evolves through a neural network, its initialization and the identifiability of the combined (sigma, h) system should be discussed; otherwise the SINDy extraction and the physical meaning of the derived internal variables x1 and x2 remain ambiguous.","section":"Section 2.2, Eq. (9)"},{"comment":"The Monte Carlo initialization procedure selects the initial point with the lowest training loss among 30 He-initialized candidates. This selection is made on the same training loss that is later optimized, so it may bias the reported training performance; using a validation-based selection or reporting the variance across initializations would be more transparent.","section":"Section 2.4.2"},{"comment":"Tables 1 and 2 appear to contain the same experimental-condition data; the Table 2 caption is incorrect and should be fixed. The paper should also justify the selection of 19 of the original 33 datasets and state the data-quality criteria used.","section":"Section 3.1, Table 1 / Table 2"},{"comment":"The notation uses X for the state vector, the trajectory matrix, and the library in Eq. (16), which is confusing. Please use distinct symbols, e.g., x, Y, and Theta, for these quantities.","section":"Section 5.1, Eq. (16)"},{"comment":"References [1], [15], and [25] refer to the same work (Chen and Messner) and are formatted inconsistently; there are also other minor citation formatting issues that should be cleaned up.","section":"References"},{"comment":"The statement that the SINDy model 'explicitly describes the dependence of the stress rate on the stress, strain, and (arbitrary) internal state' is unverifiable because the equations are not shown. Please print the full polynomial system.","section":"Section 5.2"}],"recommendation":"major_revision","confidential_remarks":"The paper has a solid core idea but the evaluation methodology, specifically the per-curve normalization leaking validation stress extrema, undermines the headline claim. I would like the editor to ensure that the revision addresses the normalization issue directly and reports the missing Chaboche parameters and SINDy equations. If the authors are unable to provide a clean out-of-sample evaluation, the paper could still be publishable as a methods demonstration, but the abstract and conclusions must be reworded accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper applies Neural ODEs and SINDy to experimental creep-fatigue data for Alloy 617 and shows a plausible accuracy improvement over a calibrated Chaboche model, but the evaluation has a load-bearing normalization leak and a few reproducibility gaps that keep me from trusting the headline claim as written.\n\nWhat is genuinely useful here: they use 19 experimental curves at 950°C, not synthetic data; they compare two architectures (black-box and neural flow rule) against a standard Chaboche model; the neural flow rule is a reasonable hybrid; and using SINDy to distill the black-box model into an explicit polynomial is a nice touch. The MSE comparison in Fig. 14 is the right kind of head-to-head.\n\nThe main problem is the per-curve normalization in Eq. (12). Every curve is scaled by its own min and max before training and validation. For held-out curves, that means the target is defined using the future stress extrema of that same curve. The reported MSE therefore measures fit in a space where the hardest part of prediction—the absolute level of stress—has been scaled out. The abstract claims the models \"precisely capture\" and \"exceed\" Chaboche; that claim is not established for actual physical stress. At minimum the authors should normalize with training-set statistics only, or report results in original units with a leave-one-experiment-out scheme. The stress-test note is right about this.\n\nAlso, the calibrated Chaboche parameters and the SINDy equations are not actually in the paper. Table 2 appears to list experimental conditions rather than the calibrated parameters, and the polynomial system is missing from Section 5.1—there is a blank where the equations should be. That makes the \"interpretable model\" unreproducible. There is also a single 15/4 split with no error bars or repeated seeds, and the SINDy model is fitted to the black-box network's own trajectories, so its good MSE mainly shows the distillation worked, not that the discovered law is independently valid.\n\nThe normalization issue affects both models: Chaboche is also fitted in normalized space. But that does not make the deployment story easier; it means the physical-stress claim is simply not tested. The conclusion's admission that \"out-of-sample testing is not feasible\" reads as an understatement of this.\n\nFor a journal like IJP or CMAME, I would send this to peer review with major revisions. The dataset and neural-ODE formulation are worth publishing if the evaluation is redone. Tell the authors to redo normalization, add leave-one-out, report the Chaboche parameters and SINDy polynomials, and make code/data available. As it stands, the headline accuracy claim is premature.","headline":"A useful application of Neural ODEs to real creep-fatigue data, but the per-curve normalization leaks validation information and the Chaboche/SINDy outputs are missing, so the headline accuracy claim isn't yet supported.","tokens_in":18901,"tokens_out":3126,"would_cite":false,"duration_ms":32354,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A neural ODE trained directly on 19 creep-fatigue experiments reproduces Alloy 617's measured stress-strain behavior more accurately than the standard Chaboche model, and a symbolic-regression version recovers an interpretable polynomial…","keywords":["creep-fatigue behavior","Neural ODE","constitutive modeling","Alloy 617","Chaboche model","symbolic regression","SINDy","cyclic plasticity"],"falsifier":"Retrain the black-box and Chaboche models with a fixed normalization, for example a global strain range and a stress range capped from the training set, hold out the same four curves, and compare validation MSE; if the neural ODE no longer beats Chaboche, the reported superiority is an artifact of per-curve extrema leakage. A second test would be to run one new creep-fatigue test outside the training grid and check whether either model's prediction error grows faster than Chaboche's.","tokens_in":17937,"feed_emoji":"🔄","tokens_out":10615,"duration_ms":91849,"temperature":0.7,"pith_summary":"This paper claims that a generic machine-learning architecture, a neural network embedded inside an ordinary differential equation, can learn a creep-fatigue constitutive law directly from experimental stress-strain curves, without an expert-supplied model form. Trained on 19 uniaxial Alloy 617 tests at 950°C spanning strain ranges and hold times, the black-box Neural ODE reproduces the measured hysteresis loops with lower mean-squared error than the standard three-backstress Chaboche model, and it captures the first-cycle overshoot that Chaboche cannot. A second hybrid model, which keeps Hooke's law for elasticity and learns only the inelastic flow with a neural network, performs comparably to or better than Chaboche. Applying sparse symbolic regression to the trained black-box model yields an explicit polynomial ODE whose accuracy is comparable to Chaboche's, trading some accuracy for interpretability. If the claims hold, constitutive modeling for new materials could move from hand-derived equations to data-fitted ODEs using modest experimental datasets.","feed_headline":"Neural ODEs beat Chaboche model on creep-fatigue data","feed_subtitle":"Trained on 19 Alloy 617 tests, a black-box neural ODE bests the standard constitutive model.","key_machinery":"The load-bearing object is the Neural ODE used as a constitutive law: instead of postulating an inelastic flow rule and hardening equations, one takes the time derivative of the state $\\boldsymbol{x}=(\\sigma,h)$ to be a neural network $f(\\boldsymbol{x},\\theta,u)$ and integrates it forward in time. In the black-box model the network maps $(\\sigma,h,\\varepsilon)$ to $\\dot\\sigma$; in the neural flow rule model the network supplies the inelastic strain-rate-like history term $\\dot h$ inside Hooke's law. Training uses an implicit, adjoint-based solver so that the stiff material ODEs can be fit on GPUs, and robust initialization is obtained by repeated He initialization with Monte Carlo selection of the lowest-loss start. Interpretability is recovered with SINDy, which takes trajectories of the trained network and sparse-regresses them against a third-order polynomial library to produce an explicit polynomial ODE system $\\dot{\\boldsymbol{x}}=\\boldsymbol{\\Theta}(\\boldsymbol{x})\\boldsymbol{\\Phi}$.","core_discovery":"On its own terms, the paper's central discovery is that a Neural ODE can act as a material model: the stress rate (and, in the black-box variant, the stress rate plus auxiliary internal variables) is written as a learned function of current stress, an internal state, and the applied strain, and the system is integrated with an implicit solver. In the black-box model the right-hand side is a residual network plus a learned linear driving term $P\\varepsilon$, so the network must represent the entire constitutive response. In the neural flow rule model the equation is $\\dot\\sigma = E(\\dot\\varepsilon - \\dot h)$ with $\\dot h$ learned by a residual network, embedding the elastic/plastic decomposition of classical flow rules. Against 19 experiments the black-box model attains the lowest mean-squared error of all models, the flow-rule model is comparable to or better than the standard Chaboche model, and the polynomial model recovered by SINDy reaches roughly Chaboche-level accuracy. The paper presents this as evidence that data-driven ODE models can replace the need to specify a material-specific constitutive form.","pith_inferences":["My inference: the headline comparison rests on per-curve min/max normalization, so the reported MSE advantage is for reconstructing curves whose range is already known; a fairer deployment test would normalize with a fixed range learned from training tests only, and the ranking could change.","My inference: the four validation curves are held out from parameter fitting but not from normalization, so true extrapolation to unseen strain ranges, hold times, or temperatures is not established by the paper, which acknowledges that out-of-sample testing is not feasible.","My inference: the explicit SINDy polynomial could be inspected term by term to see whether any discovered terms resemble known backstress or recovery terms; if so, the pipeline becomes a hypothesis generator for mechanistic constitutive laws rather than only a predictor.","My inference: the proposed 3D extension via J2 flow theory is assumed rather than validated; confirming it would require multiaxial or component-level experiments, since uniaxial data cannot identify the tensor flow direction."],"forward_implications":["If the claims hold, a black-box Neural ODE can serve as a drop-in constitutive model for strain-controlled cyclic loading, matching or beating the standard Chaboche model without any assumed hardening or flow-rule form.","Because it captures the first-cycle overstress, the black-box model describes transient cyclic behavior that the standard Chaboche form structurally cannot represent.","The SINDy-recovered polynomial ODE gives a compact, transparent equation that could be tabulated or inserted into a finite-element material routine with accuracy close to the empirical baseline.","The neural flow rule model, by preserving Hooke's law, offers a middle path: less flexible than the black box but more robust with limited data and closer to classical mechanical structure.","The framework, if it generalizes, would reduce the cost of developing constitutive models for new or less-studied materials, since only uniaxial experimental data and no material-specific derivations are needed."],"supporting_citations":[{"why":"Supplies the implicit, adjoint-based training machinery used to fit the stiff Neural ODEs.","marker":"[1]"},{"why":"Introduces Neural ODEs and the adjoint sensitivity method on which both model architectures are built.","marker":"[11]"},{"why":"Provides the L-BFGS optimizer used to calibrate the standard Chaboche model that serves as the comparison baseline.","marker":"[24]"},{"why":"Supplies the experimental Alloy 617 creep-fatigue data and the prior calibration context for the comparison.","marker":"[28]"},{"why":"Introduces SINDy, the sparse regression method used to turn the trained black-box network into an explicit polynomial ODE.","marker":"[31]"},{"why":"Provides the SINDy implementation used for the symbolic-regression interpretation.","marker":"[32]"}],"fun_headline_variants":["Neural ODEs beat Chaboche on creep-fatigue","Black-box neural ODEs outperform Chaboche","Neural ODEs model Alloy 617 creep-fatigue accurately","Data-driven ODEs rival Chaboche in creep-fatigue"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that each stress-strain curve is normalized by its own measured global minimum and maximum before training and validation, so the model is effectively told the full range of every test in advance; in a real deployment the future extrema would not be known, and the claimed accuracy may not transfer.","fun_headline_variants_meta":{"raw":{"variants":["Neural ODEs beat Chaboche on creep-fatigue","Black-box neural ODEs outperform Chaboche","Neural ODEs model Alloy 617 creep-fatigue accurately","Data-driven ODEs rival Chaboche in creep-fatigue"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000173,"raw_usage":{"total_tokens":1329,"prompt_tokens":1043,"completion_tokens":286,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":659,"completion_tokens_details":{"reasoning_tokens":213}},"tokens_in":659,"tokens_out":286,"duration_ms":2578,"temperature":1.0,"reasoning_tokens":213,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:01:24.425473+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Retrain the black-box and Chaboche models with a fixed normalization, for example a global strain range and a stress range capped from the training set, hold out the same four curves, and compare validation MSE; if the neural ODE no longer beats Chaboche, the reported superiority is an artifact of per-curve extrema leakage. A second test would be to run one new creep-fatigue test outside the training grid and check whether either model's prediction error grows faster than Chaboche's.","supporting_citations":[{"cited_title":"Training material models using gradient descent algorithms","cited_arxiv_id":null,"evidence_quote":"Supplies the implicit, adjoint-based training machinery used to fit the stiff Neural ODEs."},{"cited_title":"Neural ordinary differential equations,","cited_arxiv_id":null,"evidence_quote":"Introduces Neural ODEs and the adjoint sensitivity method on which both model architectures are built."},{"cited_title":"On the limited memory BFGS method for large scale optimization,","cited_arxiv_id":null,"evidence_quote":"Provides the L-BFGS optimizer used to calibrate the standard Chaboche model that serves as the comparison baseline."},{"cited_title":"Reference constitutive model for Alloy 617 and 316H stainless steel for use with the ASME Division 5 design by inelastic analysis rules,","cited_arxiv_id":null,"evidence_quote":"Supplies the experimental Alloy 617 creep-fatigue data and the prior calibration context for the comparison."},{"cited_title":"Discovering governing equations from data by sparse identification of nonlinear dynamical systems,","cited_arxiv_id":null,"evidence_quote":"Introduces SINDy, the sparse regression method used to turn the trained black-box network into an explicit polynomial ODE."}],"review_version":1}