{"id":"494f1dc2-cc59-4d2e-9a29-20b32d6f3431","arxiv_id":"2507.12683","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"A hybrid GPR-LSTM model trained on synthetic Holzapfel viscoelastic data reproduces and extrapolates multiaxial cyclic stress response while keeping dissipation non-negative.","lead":"The authors combine Gaussian process regression and LSTM recurrent networks into one physics-informed model that predicts how soft materials stretch, relax, and dissipate energy under repeated multiaxial loads. The system is trained on synthetic data from a known viscoelastic model and tested on loading paths outside the training range.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. 11 assumes the non-equilibrium stress lies in the span of current-C generators {I, C, C^{-1}} without a representation theorem for internal-variable functionals; under non-coaxial triaxial histories this basis is incomplete, so the general multiaxial claim is not yet established.","rationale":"The reader's weakest assumption identifies the same point: Eq. 11 asserts, without proof, that the history-dependent non-equilibrium stress can be written with the same two generators used for equilibrium hyperelasticity. This is the correct place to stress-test the paper. The paper's central claim is that the surrogate generalizes to unseen multiaxial loading states, but the only multiaxial states actually exercised are axisymmetric, with lambda_2 = lambda_3 and fixed principal axes. For such states the two-generator representation is complete for purely diagonal stresses, so the numerical success of the paper does not provide evidence for the general triaxial case. The paper has genuine strengths: the stress decomposition into volumetric, isochoric equilibrium, and isochoric viscoelastic parts is coherent; the use of invariants and response functions is a sensible path to data-efficient learning; the dissipation check is explicit; and the synthetic noise test is a useful robustness probe. However, no representation theorem is supplied for functions of C and internal variables, no code or data are provided, and no quantitative error metrics or baselines are reported. These are secondary concerns; the primary issue is the unproved completeness of Eq. 11. The verdict should remain CONDITIONAL, because the framework may well be correct for the data-generating Holzapfel family under axisymmetric loading, but the general multiaxial claim cannot be assessed until the representation question is settled.","tokens_in":29116,"tokens_out":10266,"duration_ms":129937,"concrete_test":"Generate ground-truth data from the same Holzapfel evolution law (Eq. 25) under a genuinely triaxial, non-proportional path—for example, a two-step loading in which the principal axes rotate by 45 degrees between steps, or a path with lambda_1, lambda_2, lambda_3 all distinct—extract response functions via Eq. 14, train the same surrogate (Eqs. 15c and 16c), and report mean relative Frobenius errors on unseen segments. If the error is comparable to the axisymmetric cases, Eq. 11 may be adequate for this model; if it grows substantially, the missing internal-variable generators are load-bearing. As an analytical counterpart, expand Q(t) = integral exp(-(t-s)/tau) dS_iso^infty(C(s)) for a two-step non-commuting path and test whether the result lies in span{Dev(I), Dev(C^{-1})}; if it does not, the assumption fails before any data are needed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the assertion in Section 2.2, immediately before Eq. 11, that 'the sum of all non-equilibrium branches can also be expressed using the same integrity basis' as the equilibrium stress. For a single current tensor C, the set {I, C, C^{-1}} is a complete isotropic integrity basis, but the non-equilibrium stress in an internal-variable viscoelastic material is a function, or functional, of both C and the internal variables/history. The representation theorem for isotropic functions of two tensors includes additional generators that involve the internal variables, such as H, CH+HC, C^2H+HC^2, and these are absent from Eq. 16c. The LSTM hidden state can encode history, but the output Q is still projected onto the two fixed generators Dev(I) and Dev(C^{-1}), so any non-equilibrium stress not coaxial with the current C cannot be represented. Because every numerical test uses lambda_2 = lambda_3 (axisymmetric, fixed principal axes), the test suite cannot detect this gap: for such diagonal rank-2 states, any deviatoric stress with the same two-eigenvalue structure lies in the span of the chosen generators. Under general triaxial or rotating-principal-axis loading, however, Q(t) is a weighted history of past equilibrium stresses S_iso^infty(C(s)), which are not generally in the span of the current generators. Thus Eq. 11 is not a general representation, and the framework's central generality claim is unsupported by the presented evidence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a hybrid physics-informed machine learning framework for finite-strain, isotropic, compressible visco-hyperelasticity. The equilibrium volumetric and isochoric stresses are represented through Gaussian Process Regression on invariant-based response functions, while the non-equilibrium, history-dependent stress is represented through an LSTM-based RNN whose outputs are projected onto a two-generator integrity basis. Thermodynamic constraints are imposed through a stress-free reference state in the GPR and a dissipation penalty in the RNN loss. Training data are generated from the Holzapfel differential viscoelastic model with Neo-Hookean volumetric and Mooney-Rivlin isochoric energies, for two synthetic datasets (short-term and long-term relaxation). The model is tested on multiaxial cyclic protocols with varied strain rates, stretch levels in tension and compression, extrapolation beyond training ranges, dissipation non-negativity, and synthetic noise. The authors conclude that the framework accurately captures nonlinear, rate-dependent, history-dependent behavior and generalizes beyond its training domain.","tokens_in":29546,"tokens_out":3504,"duration_ms":43976,"significance":"If the central claims hold, this is a useful step toward interpretable, physics-constrained surrogates for history-dependent soft materials. The strength of the paper is its architectural design: learning response functions on an integrity basis rather than raw stress-strain pairs, combining GPR (equilibrium) with LSTM (memory), and explicitly checking stress-free states, objectivity, symmetry, and dissipation. The multiaxial cyclic test suite, including extrapolation and noise, is broader than in many comparable studies. However, the evidence presented is largely self-consistency validation: all training and test data derive from the same Holzapfel generator, the dissipation constraint itself imports the Holzapfel-specific form of the internal-variable evolution, and the non-equilibrium stress representation in Eq. (11) is assumed complete without a representation theorem for internal-variable functionals. The generalizability and 'data-driven discovery' claims therefore need either additional evidence or substantial reframing.","major_comments":[{"comment":"The load-bearing assumption that the sum of non-equilibrium branches can be expressed using the same two-generator basis {I, C^{-1}} as the equilibrium stress is not justified by a representation theorem. For an internal-variable viscoelastic material, the non-equilibrium stress is a functional of C and the internal variables/history; isotropic functions of two tensors require additional generators involving the internal variables (e.g., H, CH+HC, C^2H+HC^2). The LSTM hidden state can encode history, but the output is still projected onto Dev(I) and Dev(C^{-1}), so non-equilibrium stresses that are not coaxial with the current C cannot be represented. All numerical tests use lambda_2 = lambda_3 with fixed principal axes, for which any deviatoric stress with the same two-eigenvalue structure lies in the span of these generators; hence the test suite cannot detect this gap. The central generality claim ('general', 'multiaxial') is therefore unsupported. Please either provide a representation theorem for the assumed form, or restrict the claims to axisymmetric coaxial histories and add a non-proportional/rotating-principal-axis test.","section":"Section 2.2, Eq. (11)"},{"comment":"The phrase 'data-driven discovery' is too strong for the present validation. Training data, test data, and the physics penalty all come from the same Holzapfel-Mooney-Rivlin generator: the response functions are extracted from that model, and the dissipation constraint in Eqs. (29)-(30) is imported from the same generating model. Consequently, the extrapolation tests in Cases 1-3 are self-consistency checks of the surrogate's ability to interpolate/extrapolate the generator, not evidence that the framework discovers a general constitutive law. The dissipation verification in Case 4 is likewise a self-consistency check: a model trained on outputs of a dissipative generator and penalized with that generator's own dissipation inequality will naturally produce non-negative dissipation. To support the discovery claim, the framework should be tested on data generated by a different constitutive class or on experimental data, or the claims should be explicitly limited to 'surrogate modeling of a given generator family.'","section":"Sections 3.2, 4.1, and Case 4"},{"comment":"The paper defines a percent relative error metric in Eq. (24) but never reports quantitative error values for any of the five cases. The conclusions rely on visual inspection of figures ('model predictions match the training data closely', 'accurate predictions'), which is not a reproducible quantitative standard. Please report mean and maximum errors for training and testing in each case, including the extrapolation and noise cases, and state which cases use normalized vs. absolute errors. This is needed to support the accuracy and generalization claims.","section":"Section 4.3, Eq. (24)"},{"comment":"The exact relaxation time spectra and branch weights for the short-term and long-term datasets are never specified. The text says the ST dataset has four Maxwell branches with 'a relaxation time spectrum and uniform weighting factors' and the LT dataset has a single 'long relaxation time', but no numerical values are given. Without these values the synthetic datasets cannot be reproduced, and it is impossible to assess whether the tested time ranges (0.1 s to 200 s) actually sample the intended relaxation regimes. Please provide the full generation parameters, including relaxation times, weights, mu_alpha, and the loading/unloading time histories.","section":"Section 4.1.2 and Figures 15"}],"minor_comments":[{"comment":"The GPR kernel description is inconsistent: Section 3.4 states a Matérn 3/2 kernel is used, while Section 4.2 says the kernel is a constant kernel combined with a radial basis function kernel. Please reconcile these statements and specify which kernel was actually used.","section":"Section 4.2"},{"comment":"The equation numbering is disordered: Eq. (25) is introduced in Section 4.1.2, then Eqs. (26)-(27), then Eq. (28) in Section 3.5.2, and Eq. (24) is defined later in Section 4.3. This makes cross-referencing difficult; please renumber sequentially.","section":"Equation numbering"},{"comment":"In Eqs. (29)-(31) the notation is unclear: Gamma_alpha is both an internal deviatoric history variable and appears with a finite-difference update, while Q_alpha is called both a deviatoric internal variable and a non-equilibrium stress. Please define the relation between Gamma_alpha and Q_alpha explicitly and state which quantity the RNN actually outputs.","section":"Section 3.5.2"},{"comment":"The caption contains a typo: 'Isochoric viscoelastic tress-stretch behavior' should read 'stress-stretch behavior'.","section":"Figure 14 caption"},{"comment":"The GPR evaluation in Case 1 only shows qualitative curves. Since the GPR is a core component of the framework, please report its training and test errors separately from the RNN errors, particularly for the extrapolation cases.","section":"Section 4.3.1"},{"comment":"No code or data availability statement is provided. Given the synthetic-data setup, releasing the generation scripts and trained model configurations would substantially strengthen reproducibility.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely to be of interest to readers of a computational mechanics journal, but the gap between the claims ('data-driven discovery', 'general' framework) and the evidence (single-generator self-consistency validation) is significant. The most important fixes are: (i) justify or restrict the two-generator representation for non-equilibrium stress, (ii) test on at least one genuinely different constitutive model or experimental dataset, (iii) report quantitative errors, and (iv) provide the missing relaxation parameters. With these changes the contribution could become a solid surrogate-modeling paper, though the 'discovery' framing may still need to be softened."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid engineering integration, not a discovery. Combining GPR on integrity-basis response functions for equilibrium stress with an LSTM for the history-dependent branch is new and worth a serious look. The authors know the relevant literature, and the extrapolation test set is more ambitious than most. If they had shipped code and actual error tables, I would be more enthusiastic.\n\nWhat is good: the way objectivity, isotropy, and symmetry are enforced through the tensor-basis decomposition is clean. The distinction between the identity-based and C-based non-equilibrium terms is helpful. The short-term/long-term relaxation datasets and the out-of-range tests are sensible experimental designs. The derivation from the visco-hyperelastic framework to Eq. 16 is readable. This is a competent execution of a reasonable architecture.\n\nSoft spots:\n\n1. The representation claim around Eq. 11 is the load-bearing weakness. The paper asserts that the sum of non-equilibrium branches can be expressed using the same integrity basis as the equilibrium stress. For a function of one tensor, {I, C, C^-1} is complete, but the non-equilibrium stress is a functional of C and internal variables/history. A proper representation for functions of two tensors would add generators involving the internal variables. The LSTM hidden state can encode history, but the output is still projected onto Dev(I) and Dev(C^-1). Since every numerical test uses lambda2 = lambda3, the two-eigenvalue structure hides the gap. General non-coaxial triaxial loading would expose it. This directly weakens the paper's central generality claim.\n\n2. The closed-loop validation. Training data come from a Holzapfel generator, and the dissipation constraint in Eq. 29 is imported from the same Holzapfel model. So Case 4 checks self-consistency with the generating model, not thermodynamic consistency of a learned constitutive law. That is acceptable for a methods paper, but it does not support the phrase \"data-driven discovery.\"\n\n3. Missing numbers and baselines. The paper defines a percent relative error but never reports the resulting values. There are no comparisons to a plain LSTM, GPR-only, or a classical fitted Holzapfel model. No code or data are provided. The \"accurate\" claims rest on figures and selected qualitative descriptions.\n\n4. Minor: the title promises multiaxial loading, but all experiments are axisymmetric with fixed principal axes. That is a subset, not the full claim.\n\nBottom line: this deserves a serious referee. The core architecture is worth developing, and the representation gap is identifiable and addressable with non-proportional loading tests. I would not cite it in my own work until code, data, and error metrics are available, but I would engage with it.","headline":"A coherent hybrid GPR/LSTM constitutive surrogate with real architectural merit, but the 'discovery' claim is overreaching: all evidence is synthetic Holzapfel data and the non-equilibrium basis has a representation gap the axisymmetric tests cannot detect.","tokens_in":30008,"tokens_out":2018,"would_cite":false,"duration_ms":28072,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["74D10","74B20","68T07"],"pacs":["83.60.Bc","83.80.Va","07.05.Mh"],"model":"deepseek-v4-flash","headline":"A hybrid GPR-LSTM surrogate trained on tensor-invariant response functions reproduces a nonlinear visco-hyperelastic model under multiaxial cyclic loading, including extrapolated stretches, strain rates, and tension/compression.","keywords":["physics-informed machine learning","visco-hyperelasticity","Gaussian process regression","LSTM","integrity basis","multiaxial cyclic loading","thermodynamic consistency","constitutive modeling"],"falsifier":"Train the same GPR-LSTM surrogate on data generated by a visco-hyperelastic model whose dissipative stress requires additional generators, for example a model with a transversely isotropic or orthotropic viscous branch, or a two-network theory with a non-isotropic internal tensor; if the surrogate cannot reproduce the unseen multiaxial stress components despite good training fits, the completeness of the two-generator basis for the non-equilibrium stress is falsified.","tokens_in":28934,"feed_emoji":"🔄","tokens_out":3294,"duration_ms":31452,"temperature":0.7,"pith_summary":"The paper tries to establish that a physics-informed machine learning surrogate can replace an internal-state-variable visco-hyperelastic constitutive model without losing physical fidelity. It claims that by training a Gaussian Process Regression (GPR) model on the equilibrium stress response functions and a Long Short-Term Memory (LSTM) recurrent network on the non-equilibrium response functions, the surrogate accurately captures nonlinear, rate-dependent, history-dependent stress under multiaxial cyclic loading. The trained model is shown to predict stress for deformation states outside the training range, including different stretch levels, strain rates, tension and compression, while satisfying objectivity, material symmetry, angular momentum balance, and non-negative dissipation. A sympathetic reader would care because this offers a path to data-driven constitutive modeling that generalizes beyond observed loading conditions without enforcing an explicit closed-form strain energy function.","feed_headline":"Hybrid GPR-LSTM surrogate extrapolates viscoelastic stress beyond training data","feed_subtitle":"Physics constraints and tensor invariants let the model predict multiaxial cyclic loading in tension and compression it never saw.","key_machinery":"The key machinery is the integrity basis representation of stress, Eqns. 10-11, where stress components are written as weighted linear combinations of the isotropic generators I, C, and $C^{{-1}}$, with the weights being response functions of the invariants I1, I2 (and J). The equilibrium response functions are learned by GPR with a Matérn 3/2 kernel, while the history-dependent response functions are learned by a three-layer LSTM RNN with a physics-informed loss that adds a penalty when the computed dissipation (from the Clausius-Duhem inequality) is negative. This representation enforces objectivity and material symmetry by construction and converts constitutive modeling into supervised regression of scalar invariant-dependent coefficients.","core_discovery":"The central claim is that the total stress of a generalized internal-state-variable visco-hyperelastic material can be decomposed into equilibrium volumetric, equilibrium isochoric, and non-equilibrium isochoric parts, and that each part can be learned separately as a function of tensor invariants using a hybrid GPR-LSTM architecture. The GPR learns the volumetric response function delta(J) and the isochoric response functions chi1(I1,I2) and chi2(I1,I2), while the LSTM learns the time-dependent response functions xi1(t,I1,I2) and xi2(t,I1,I2) that capture the summed Maxwell branches. Because stresses are expressed as linear combinations of a fixed integrity basis (identity, C, $C^{{-1}}$) multiplied by these learned response functions, objectivity, isotropy, and symmetry are built in, and the second law is enforced as a dissipation penalty in the LSTM loss. Using the Holzapfel differential viscoelastic model to generate training data, the paper claims the surrogate generalizes beyond training domains in stretch, strain rate, and tension/compression, with dissipation staying non-negative and predictions robust to about 6% synthetic noise.","pith_inferences":["The paper's success on extrapolation in stretch and time suggests, but does not prove, that the learned invariant-response functions have locally smooth extrapolation behavior; a testable extension would be to query the surrogate on random multiaxial paths that are not one-cyclic or proportional.","The framework is demonstrated only on data generated by the Holzapfel model with isotropic symmetry and two isotropic generators; a natural extension is to anisotropic soft tissues or fiber-reinforced elastomers, where the integrity basis must be enlarged and the claim of complete representation would need to be revisited.","The stress-decomposition methodology, which fits asymptotic equilibrium stress and then subtracts it, would fail if a material has a very slow relaxation component that never equilibrates within the test window; a practical extension would require a systematic protocol for choosing test durations such that the asymptotic fit is reliable."],"forward_implications":["If the surrogate indeed matches the Holzapfel model on unseen multiaxial paths, then data-driven constitutive models can be calibrated from a modest set of multiaxial cyclic experiments and still extrapolate to untrained stretch levels and strain rates.","The decomposition into volumetric, isochoric equilibrium, and isochoric non-equilibrium surrogates means each component can be trained separately, making the learning problem lower-dimensional and more data-efficient than raw stress-strain sequence fitting.","Because the learned response functions are scalar functions of invariants, the resulting model can be checked against known theoretical forms, connected to classical constitutive theory, and reused in finite element codes via the stress expression of Eq. 16.","The non-negative dissipation constraint, enforced as a soft penalty during training, provides a route to thermodynamically consistent RNN-based constitutive models without explicitly enforcing a full evolution equation.","Robustness to about 6% synthetic noise suggests the framework could tolerate realistic experimental noise, which matters if the decomposition procedure (asymptote extraction and basis-coefficient inversion via least squares) amplifies measurement error."],"supporting_citations":[{"why":"Establishes the tensor integrity basis and response-function representation for stress in strain-rate-sensitive soft materials, which the paper extends to internal-state-variable viscoelasticity.","marker":"[106]"},{"why":"Provides the nonlinear Holzapfel differential viscoelastic model used to generate training data and the form of the deviatoric history variables used to enforce non-negative dissipation.","marker":"[16]"},{"why":"Supplies the specific form of the internal deviatoric history variables and the generalized Maxwell-type evolution adopted for the non-equilibrium branches.","marker":"[120]"},{"why":"Provides the background nonlinear solid mechanics, including the projection tensor/Dev operator and finite viscoelasticity framework used in the derivations.","marker":"[17]"},{"why":"Original LSTM reference used to justify the recurrent architecture's ability to capture history-dependent behavior.","marker":"[112]"},{"why":"Demonstrates that RNNs learn viscoelastic constitutive laws, motivating the use of LSTM for the history-dependent stress component.","marker":"[111]"}],"fun_headline_variants":["GPR-LSTM hybrid predicts viscoelastic stress beyond training data","Physics-informed AI learns viscoelastic responses under multiaxial cyclic loads","Tensor invariants plus GPR-LSTM model unseen cyclic loading paths","Hybrid model with physics constraints predicts viscoelastic behavior outside training"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The framework assumes that the complete non-equilibrium stress of any history-dependent isotropic viscoelastic material can be represented using only the two isotropic generators C and $C^{{-1}}$, so that two invariant-based response functions suffice for all dissipative branches.","fun_headline_variants_meta":{"raw":{"variants":["GPR-LSTM hybrid predicts viscoelastic stress beyond training data","Physics-informed AI learns viscoelastic responses under multiaxial cyclic loads","Tensor invariants plus GPR-LSTM model unseen cyclic loading paths","Hybrid model with physics constraints predicts viscoelastic behavior outside training"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00089,"raw_usage":{"total_tokens":3903,"prompt_tokens":1069,"completion_tokens":2834,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":685,"completion_tokens_details":{"reasoning_tokens":2759}},"tokens_in":685,"tokens_out":2834,"duration_ms":21258,"temperature":1.0,"reasoning_tokens":2759,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:41:34.587748+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train the same GPR-LSTM surrogate on data generated by a visco-hyperelastic model whose dissipative stress requires additional generators, for example a model with a transversely isotropic or orthotropic viscous branch, or a two-network theory with a non-isotropic internal tensor; if the surrogate cannot reproduce the unseen multiaxial stress components despite good training fits, the completeness of the two-generator basis for the non-equilibrium stress is falsified.","supporting_citations":[{"cited_title":"Physics-informed Data-driven Discovery of Constitutive Models with Application to Strain-Rate-sensitive Soft Materials","cited_arxiv_id":"2304.13897","evidence_quote":"Establishes the tensor integrity basis and response-function representation for stress in strain-rate-sensitive soft materials, which the paper extends to internal-state-variable viscoelasticity."},{"cited_title":"On large strain viscoelasticity: Continuum formulation and finite element applications to elastomeric structures,","cited_arxiv_id":null,"evidence_quote":"Provides the nonlinear Holzapfel differential viscoelastic model used to generate training data and the form of the deviatoric history variables used to enforce non-negative dissipation."},{"cited_title":"A recurrent neural network-accelerated multi- scale model for elasto -plastic heterogeneous materials subjected to random cyclic and non - proportional loading paths,","cited_arxiv_id":null,"evidence_quote":"Original LSTM reference used to justify the recurrent architecture's ability to capture history-dependent behavior."}],"review_version":1}