{"id":"b445d1ec-d961-443a-9980-3b1bc55a4b80","arxiv_id":"2505.24578","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A two-stage framework uses neural operator predictions as training data for sparse regression, recovering interpretable hysteresis ODEs that generalize to new voltage fields.","lead":"This paper combines a Fourier neural operator with sparse regression to discover interpretable differential equations describing piezoelectric hysteresis. The method is tested on synthetic voltage-displacement data, where it recovers known equations and is reported to generalize better than black-box neural operators to unseen voltage signals.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Butterfly experiments are unreproducible as described: the library in §IV-B/§V-A has no latent variable y, yet Exp 3–4 report discovered two-state ODEs in y, so the central OOD-generalization claim rests on an undeclared step.","rationale":"The reader's weakest assumption points to the same keystone: the candidate library must contain the true functional terms, and in the butterfly experiments a latent state must be discoverable and representable. The text provides no procedure for recovering y, and the specified library cannot represent the reported terms, so the strongest claim cannot be verified from the paper as written. I considered the other concerns raised by the reader — the structurally unfair generalization comparison and the internally inconsistent Table I baseline column. These are real and should be fixed in revision, but they are secondary to the missing latent-state mechanism: if Exp 3 and 4 cannot be reproduced from Algorithm 1, the paper's broadest claim about butterfly hysteresis and model-agnostic symbolic discovery falls. The proposed concrete test is minimal: run the pipeline with the stated library and no hidden y column, or require the authors to disclose the y-reconstruction step. Either outcome settles whether the concern lands. Because this concern reinforces rather than redirects the reader's REJECT verdict, the recommended verdict stays unchanged.","tokens_in":16723,"tokens_out":8163,"duration_ms":102548,"concrete_test":"Reproduce Experiment 3 using only the exact library construction stated in §IV-B/§V-A — columns from v, v̇, |v|, |v̇|, d and their degree-2 products — with no y column. If Algorithm 1 neither outputs the reported ˙y equation nor any equivalent two-state model, the butterfly result is unsupported. If the authors instead reconstruct y, they must publish the exact reconstruction formula and input it as an additional column; then re-run Exp 3/4 and verify on a different latent-dynamics generator (e.g., ẏ = 2v̇ − 1.25|v̇|y − 0.1v̇ y²).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing condition for the central claim is that Algorithm 1 can actually produce the reported white-box ODEs from the data it is given. For Experiments 3 and 4 this condition is not met by the text. Section IV-B defines the candidate library as Θ(D,V,f(V)), and Section V-A specifies that STLSQ uses \"voltage fields, their derivatives, absolute valued fields and their derivatives, and the displacement profiles as the basis functions\" with polynomial features up to degree 2. The two reported butterfly models, however, are coupled ODEs in d and y, e.g., ˙d=2|v̇|vy−4.70|v̇|dy+3v̇y and ˙y=|v̇|v−2.35|v̇|y+1.5v̇. A library containing only v, v̇, |v|, |v̇|, d and degree-2 products cannot represent terms such as v y, d y, or |v̇| y, and in any case y is not measured, not an output of the FNO (Stage I returns only displacement), and no reconstruction procedure for y is given anywhere. Without either an explicit y-recovery step or a different library construction, the Exp 3–4 \"identified ODEs\" cannot be outputs of Algorithm 1. The paper's claim of model agnosticism (\"does not require explicitly including candidate terms governing the dynamics\") is also contradicted: the discovered Exp-1/2 equations are exactly the monomials the library was built to contain. Thus the headline generalization result is either circular or relies on an undeclared step; this is the most load-bearing weakness.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a neuro-symbolic operator (NSO) framework for hysteresis modeling. Stage I trains a Fourier neural operator on voltage-displacement pairs sampled from one input family (Sine kernel). Stage II uses the FNO predictions on Sine test inputs to build a candidate library of polynomial and absolute-value terms and runs sequential threshold least-squares to extract a sparse ODE for the displacement. The identified ODE is then simulated on held-out voltage fields from RBF and Matérn kernels. The paper reports seven experiments: two first-order hysteresis laws, two butterfly-shaped two-state hysteresis laws, noise and low-fidelity ablations, and a sparsity-threshold ablation.","tokens_in":17130,"tokens_out":10225,"duration_ms":131292,"significance":"If the results were fully supported, NSO would be a useful bridge between neural operators and symbolic model discovery: Experiments 1-2 recover coefficients close to the generating equations, and the noise/fidelity ablations show a practical advantage of using an operator surrogate as a denoiser before sparse regression. However, the butterfly experiments and the model-agnosticism claim depend on an undeclared latent variable and on a library that is constructed from the true equation terms. The central generalization claim is therefore not established in the current version.","major_comments":[{"comment":"Experiments 3 and 4 report discovered ODEs in the state (d, y), for example ˙d = 2|v̇|vy − 4.70|v̇|dy + 3v̇y and ˙y = |v̇|v − 2.35|v̇|y + 1.5v̇, but the candidate library Θ(D,V,f(V)) defined in Section IV-B and the STLSQ basis described in Section V-A contain only voltage fields, their derivatives, absolute valued fields and their derivatives, and displacement profiles with polynomial features up to degree two. Terms such as v y, d y, and |v̇| y cannot be formed from these basis functions, y is not an output of the FNO, and Algorithm 1 contains no latent-state reconstruction step. The butterfly results therefore cannot be produced by the described method. If y is available from the synthetic data generator, that use must be disclosed and a deployment-relevant procedure for obtaining y must be supplied; as written, the claim that NSO discovers these two-state models is unsupported.","section":"§IV-B, §V-A, §V-D"},{"comment":"The discussion in Section VI claims that NSO 'does not require explicitly including candidate terms governing the dynamics in the library' and is model-agnostic, but this is contradicted by the experimental setup. The candidate library is built from exactly the voltage, derivative, absolute-value, and displacement terms appearing in the generating equations, and the recovered ODEs in Experiments 1-2 are precisely the monomials contained in the library. The library composition is a free design choice, and no misspecification experiment is reported. The paper should either demonstrate discovery with an incomplete or perturbed library or substantially weaken the model-agnosticism claim.","section":"§VI and §V-A"},{"comment":"Table I contains an internally inconsistent DON column. For each experiment the DON RMSE is identical across Sine, RBF, and Matern rows (8.63e-02 in Exp 1, 6.14e-02 in Exp 2, 4.02e-02 in Exp 3, and 3.11e-01 in Exp 4), while the corresponding R and MAE values differ across those rows. Such a pattern cannot arise from independent evaluations on different test sets. The table needs to be corrected or an explanation provided before the claim that all baseline operators fail out-of-distribution can be accepted.","section":"Table I"}],"minor_comments":[{"comment":"The description of the second stage should state explicitly that NSO's RBF and Matern predictions are obtained by simulating the discovered ODE with the test voltage fields, not by evaluating the FNO on those fields; Table I and Table IV as presented make it difficult to distinguish the two predictors.","section":"§V-A"},{"comment":"Experiment 6 is said to use the Experiment 4 data with the Matern32 kernel, but Table III labels the third kernel as Matern52; unify the notation.","section":"§V-G/Table III"},{"comment":"In the last paragraph of Section V-G, 'NNO' should be 'NSO'.","section":"§V-G"},{"comment":"The notation f(V) is never defined; the paper should list the exact basis functions included in Θ, including whether |d| and |y| are included, since the reported Exp 2 and Exp 4 equations require terms such as v̇|d|.","section":"§IV-B"},{"comment":"The sentence about Stage II using 500 functions based on FNO predictions is ambiguous; clarify whether these 500 are disjoint from the 1000 test functions used for the other baselines and whether the same functions are also used in the error metrics of Table I.","section":"§V-A"}],"recommendation":"reject","confidential_remarks":"The manuscript is clearly written and the first-order examples are clean, but the current text does not make the butterfly experiments reproducible. I would be willing to reconsider a revised version that adds an explicit latent-variable recovery procedure and a misspecification study; in the present form I cannot recommend acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The two-stage idea—train an FNO, then run STLSQ on its predictions—is not a bad one, and the paper has one clean result: in the noise and low-fidelity ablations (Exp 5a/5b), the FNO's denoising/upsampling step genuinely helps symbolic regression recover the true coefficients, while SINDy and Lasso drift. That is the part I would take seriously.\n\nThe rest of the paper does not support the claims. The butterfly experiments (Exp 3–4) are not reproducible from the text. Section IV-B/V-A define the library as functions of V, its derivatives, absolute values, and the displacement D. There is no latent variable y anywhere in the algorithm. The FNO outputs displacement only. Yet the paper reports discovered coupled ODEs in d and y. No reconstruction procedure for y is given, and the library as described cannot represent terms like v y or |v̇| d y. Either there is an undeclared step or the reported equations did not come out of Algorithm 1. That is load-bearing, because the butterfly cases are the supposed demonstration of generality.\n\nThe generalization comparison is also not apples-to-apples. For RBF and Matern test fields, NSO's predictions are generated by simulating the ODE fitted in Stage II, while the neural operators are evaluated directly on those fields. Since the fitted ODE is effectively the data-generating equation (with correct coefficients), of course it generalizes. That shows symbolic models generalize, not that the NSO framework adds anything to the FNO.\n\nThe candidate library is also tailored to the known generators; claiming model-agnosticism is a stretch when the library contains the exact monomials of the target ODEs. And Table I has a tell: DON's RMSE is identical across Sine/RBF/Matern for every experiment, which can't be right.\n\nSo: the noise-robustness experiment is a useful data point, and Exp 1–2 show the pipeline works when the library is right. But the central claim—interpretable and generalizable characterization, including butterfly hysteresis—rests on an unreproducible step and an unfair baseline. This is not ready for review; I would desk-reject. If the authors fix the butterfly issue and rerun the comparison properly, the paper could become a solid contribution to operator distillation.","headline":"The noise-robustness experiment is clean, but the butterfly results are unreproducible as written and the OOD comparison is structurally unfair.","tokens_in":17627,"tokens_out":3828,"would_cite":false,"duration_ms":44384,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proposes a neuro-symbolic operator (NSO) that distills a Fourier neural operator into a sparse, interpretable ODE governing piezoelectric hysteresis, and shows this discovered ODE predicts displacement for voltage fields…","keywords":["neuro-symbolic operator","neural operators","piezoelectric hysteresis","sparse model discovery","Fourier neural operator","out-of-distribution generalization","butterfly hysteresis","interpretable modeling"],"falsifier":"Generate synthetic voltage-displacement data from a hysteresis ODE whose library is deliberately missing a term, e.g. $\\dot{d}=0.4|\\dot{v}|v-0.85|\\dot{v}|d+0.2\\dot{v}+0.5\\sqrt{|\\dot{v}|}$, run NSO with the paper's library, and check whether the recovered ODE either omits the $\\sqrt{|\\dot{v}|}$ term or fails to predict a held-out RBF voltage field; either outcome would refute the claimed generalizability.","tokens_in":16536,"feed_emoji":"⚡","tokens_out":6076,"duration_ms":74302,"temperature":0.7,"pith_summary":"The paper proposes a two-stage neuro-symbolic operator (NSO) for piezoelectric hysteresis. It first trains a Fourier neural operator mapping voltage fields to displacement profiles, then uses sequential threshold least-squares regression on the operator's predictions to discover a sparse, interpretable ordinary differential equation that governs the hysteresis. The central claim is that this white-box ODE not only matches the training data but predicts displacement profiles for voltage fields drawn from different stochastic-process kernels than anything seen in training, including butterfly-shaped hysteresis loops. The motivation is that standard neural operators are black boxes that fail outside their training input distribution, whereas an explicit ODE can be inspected, simulated, and applied to new voltage inputs.","feed_headline":"White-box ODE predicts piezoelectric displacement beyond training","feed_subtitle":"A Fourier neural operator plus sparse regression recovers the hysteresis law from voltage data alone.","key_machinery":"The load-bearing mechanism is the two-stage distillation: Stage I uses a Fourier neural operator with $K$ spectral convolution layers to learn a smooth surrogate $\\hat{P}=N_\\theta: \\mathcal{V}\\to\\mathcal{D}$, which denoises and upsamples the data; Stage II assembles a library $\\Theta(D,V,f(V))$ of candidate terms built from voltage, displacement, derivatives, absolute values, and degree-2 polynomial combinations, then solves the sparse regression $\\dot{D}=\\Theta(D,V,f(V))\\xi$ via sequential threshold least squares. The resulting sparse coefficient vector $\\xi^*$ selects the few terms that define the discovered ODE. What makes the argument work is that the neural operator's predictions are clean enough for derivative estimation, so the sparse regression can pick out the true structure from a library that already contains it.","core_discovery":"On the paper's own terms, the central discovery is that a neural operator's learned mapping can be distilled into a parsimonious differential model without losing predictive accuracy. Concretely, training an FNO on Sine-kernel voltage inputs and then applying STLSQ to the FNO's predicted displacement profiles recovers equations such as $\\dot{d} = 0.39|\\dot{v}|v - 0.83|\\dot{v}|d + 0.2\\dot{v}$ for a standard hysteresis law and two-equation systems with a latent variable $y$ for butterfly-shaped hysteresis. These recovered ODEs, when solved forward, give accurate displacement predictions for RBF and Matern voltage fields, whereas the neural operators alone degrade by an order of magnitude. The paper takes this as evidence that the discovered symbolic model, not the neural network, carries the generalizable content of the hysteresis operator.","pith_inferences":["The method's core assumption is that the hysteresis law is sparse in a human-chosen library; if that is accepted, NSO is effectively an equation-discovery pipeline that uses a neural operator as a data-cleaning front end.","A natural extension is to apply NSO to measured experimental voltage-displacement data where no closed-form law is known and validate the discovered ODE on a completely different voltage class, something the paper demonstrates only on synthetic ODE-generated data.","The butterfly experiments suggest the latent state $y$ is identifiable from trajectories, but the paper does not provide a recovery procedure; whether this generalizes to hysteresis with multiple or unobservable internal states is open.","If the discovered ODE equals the true law, then NSO would make neural operators unnecessary after training: deployment could use the ODE directly, a consequence the paper leaves implicit."],"forward_implications":["NSO's discovered ODEs generalize to RBF and Matern voltage fields even though training used only Sine fields, with errors one to two orders of magnitude lower than standalone neural operators.","The white-box ODE form lets engineers inspect which terms govern the hysteresis instead of treating the model as a black box.","Because Stage I denoises and upsamples, NSO recovers nearly exact coefficients even with 20% Gaussian noise and with only 20 time samples per function.","The framework extends from simple hysteresis loops to butterfly-shaped hysteresis by including a latent variable $y$ in the discovered ODE system.","The sparsity threshold $\\lambda$ controls a trade-off between accuracy and interpretability, with the paper reporting that $\\lambda=0.01$ gives a good balance."],"supporting_citations":[{"why":"Supplies the Fourier neural operator architecture used as the Stage I black-box operator.","marker":"[22]"},{"why":"Supplies the STLSQ sparse regression algorithm used in Stage II to discover the ODE coefficients.","marker":"[29]"},{"why":"Provides the sparse hysteresis model forms and butterfly hysteresis systems that the experiments are built on.","marker":"[24]"},{"why":"Baseline deep operator network compared against NSO on the same voltage-displacement tasks.","marker":"[21]"},{"why":"Baseline convolutional neural operator compared against NSO on the same voltage-displacement tasks.","marker":"[23]"},{"why":"Establishes the neural operator problem of learning maps between function spaces that NSO extends.","marker":"[19]"},{"why":"Explains butterfly-shaped hysteresis loops, the target behavior in Experiments 3 and 4.","marker":"[31]"}],"fun_headline_variants":["Neuro-symbolic operator recovers white-box ODE for hysteresis","Symbolic distillation beats black-box neural operators on unseen data","Interpretable ODE from neural operator predicts piezoelectric behavior","FNO plus sparse regression yields generalizable hysteresis law","White-box model outpredicts neural operators on novel voltages"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole claim rests on the candidate library containing the exact functional forms of the true hysteresis law, and on any latent internal variable $y$ being present or recoverable in that library; if a real system has terms the library does not include, the discovered ODE will not be the true law and out-of-distribution generalization will fail.","fun_headline_variants_meta":{"raw":{"variants":["Neuro-symbolic operator recovers white-box ODE for hysteresis","Symbolic distillation beats black-box neural operators on unseen data","Interpretable ODE from neural operator predicts piezoelectric behavior","FNO plus sparse regression yields generalizable hysteresis law","White-box model outpredicts neural operators on novel voltages"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000236,"raw_usage":{"total_tokens":1531,"prompt_tokens":999,"completion_tokens":532,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":615,"completion_tokens_details":{"reasoning_tokens":451}},"tokens_in":615,"tokens_out":532,"duration_ms":6563,"temperature":1.0,"reasoning_tokens":451,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:18:01.611012+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate synthetic voltage-displacement data from a hysteresis ODE whose library is deliberately missing a term, e.g. $\\dot{d}=0.4|\\dot{v}|v-0.85|\\dot{v}|d+0.2\\dot{v}+0.5\\sqrt{|\\dot{v}|}$, run NSO with the paper's library, and check whether the recovered ODE either omits the $\\sqrt{|\\dot{v}|}$ term or fails to predict a held-out RBF voltage field; either outcome would refute the claimed generalizability.","supporting_citations":[{"cited_title":"Fourier neural operator for parametric partial differential equations,","cited_arxiv_id":null,"evidence_quote":"Supplies the Fourier neural operator architecture used as the Stage I black-box operator."},{"cited_title":"Discovery of sparse hysteresis models for piezoelectric materials,","cited_arxiv_id":null,"evidence_quote":"Provides the sparse hysteresis model forms and butterfly hysteresis systems that the experiments are built on."},{"cited_title":"Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators,","cited_arxiv_id":null,"evidence_quote":"Baseline deep operator network compared against NSO on the same voltage-displacement tasks."},{"cited_title":"Convolutional neural operators for robust and accurate learning of PDEs,","cited_arxiv_id":null,"evidence_quote":"Baseline convolutional neural operator compared against NSO on the same voltage-displacement tasks."},{"cited_title":"Neural operator: Learning maps be- tween function spaces with applications to PDEs,","cited_arxiv_id":null,"evidence_quote":"Establishes the neural operator problem of learning maps between function spaces that NSO extends."},{"cited_title":"Why are some hysteresis loops shaped like a butterfly?","cited_arxiv_id":null,"evidence_quote":"Explains butterfly-shaped hysteresis loops, the target behavior in Experiments 3 and 4."}],"review_version":1}