{"id":"6cf7790a-3ebd-439d-9bf8-5ee9d780c319","arxiv_id":"2512.02408","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A solver-based latent-variable learning and symbolic regression pipeline recovers explicit equations for hysteretic dynamics, including cases where both the motion law and the hysteresis law are unknown.","lead":"Hysteretic systems have a hidden 'memory' that makes their response depend on the past. This paper trains that hidden memory as a curve, then uses symbolic regression to print out the governing equations from data, without choosing a model library in advance.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Full Equation Discovery is tested only under the same excitation used for training; with a free latent z and neural-network fθ, the discovered equations may be a non-unique data fit rather than the true governing laws.","rationale":"The reader's weakest assumption is that the internal variable z is not identifiable from the measurements, and the Full Equation Discovery section's same-excitation validation is the concrete place where this assumption is load-bearing. In Case 2, both fθ and z are flexible (neural network plus free time series), so infinitely many decompositions can produce the same observed displacement under u(t)=sin(2t). The symbolic-regression step then operates on the learned z; if z is an artifact, gφ is an artifact. Because the testing set uses the same excitation as training, the reported test NRMSEs cannot falsify this. This is not a claim of internal inconsistency; the Case 1 benchmark's multisine Testing 2 demonstrates that the solver-based learning can work when fθ is structurally constrained and the excitation is varied. The unresolved gap is specifically the Full Equation Discovery claim, which is the paper's main novelty ('no prior work has demonstrated a fully symbolic equation discovery framework capable of addressing this most general setting'). A single new-excitation test would settle whether the discovered equations are the true laws or just a fit. I therefore agree with the reader's conditional assessment and see no reason to change the verdict, but the concern is specific and testable.","tokens_in":16796,"tokens_out":6685,"duration_ms":64380,"concrete_test":"Re-run the Full Equation Discovery procedure of Section 4.2 with the same training setup (u1(t)=sin(2t), five initial conditions), but generate a second test set from the same true system using an unseen excitation such as u2(t)=sin(3t) (or a 5-150 Hz multisine) with the unseen initial condition. Solve the discovered fθ and gφ under u2 and compute NRMSE (Eq. 10) for x, ẋ, and ẍ. If the NRMSEs are comparable to the existing in-excitation test values in Table 4 (e.g., ≤~5%), the discovered equations generalize and the identifiability concern is resolved; if errors degrade markedly (e.g., displacement NRMSE >20%), Full Equation Discovery has identified a non-unique decomposition rather than the true governing laws.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central novelty claim hinges on Full Equation Discovery (Section 3.3, Case 2), but its only validation (Section 4.2) uses u(t)=sin(2t) for both training and testing; only initial conditions are varied. In Case 2, fθ is a neural network N(x,ẋ,u;θ,z) and z is a free learnable time series (Eq. 8b). For any chosen z(t), an NN fθ can be defined to reproduce the measured x(t) under this single excitation, so the pair (fθ,z) is not identifiable. SR then regresses ż=Diff(z) onto (ẋ,z) and reports the result as the hysteresis law gφ. If z is one of many latent processes consistent with the observed response, the discovered gφ is an artifact; solving the discovered equations can fit the training excitation without recovering the true dynamics. Testing on the same excitation with new initial conditions checks interpolation over initial states, not generalization to new forcing, and therefore cannot distinguish the true equations from an arbitrary decomposition. The Case 1 Bouc-Wen experiment (Testing 2 with multisine) provides genuine out-of-excitation support for Hysteresis Discovery, but the most general and novel mode lacks any such test.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a unified framework for equation discovery in hysteretic systems. It reformulates the system in state-space form with an internal hysteretic variable z and treats z as a learnable parameter optimized through a differentiable ODE solver against measured responses (Eq. 8). Symbolic regression is then applied to the learned trajectories to obtain explicit equations for the hysteretic link g_phi and, in the Full Equation Discovery case, the dynamics f_theta. The framework is tested on the Bouc-Wen benchmark (Hysteresis Discovery), a synthetic system with cubic stiffness and fractional hysteresis exponent (both Hysteresis and Full Equation Discovery), and a shake-table SDOF yielding structure. The Bouc-Wen benchmark includes a genuinely different excitation for testing; the Full Equation Discovery validation uses the same excitation with different initial conditions.","tokens_in":17136,"tokens_out":7437,"duration_ms":67841,"significance":"If the Full Equation Discovery claims are fully substantiated, the paper would fill a real gap: a model-free, library-free method that simultaneously discovers both the primary dynamics and the hysteretic link equation. The paper makes a useful contribution by classifying equation-discovery settings for hysteretic systems and by showing that solver-based latent-variable learning combined with symbolic regression can recover known Bouc-Wen structures, including fractional exponents, more flexibly than SINDy. The Hysteresis Discovery results on the public Bouc-Wen benchmark are credible, with low NRMSE under a different testing excitation, and the experimental yielding-structure case adds real-data evidence. However, the most general and novel mode, Full Equation Discovery, is currently validated under conditions that cannot rule out overfitting of the free latent variable; the central claim therefore needs additional support before the contribution can be accepted.","major_comments":[{"comment":"The central loop is circular: z is optimized as a free learnable vector to fit the measured response (Eq. 8a/8b), and then the symbolic regression target is dot(z)=Diff(z) regressed onto (dot(x),z) (Section 3.3). Because z is never constrained by the discovered g_phi during training, a wide family of latent trajectories can be consistent with the same measured x when f_theta is sufficiently flexible. No identifiability or excitation-conditions analysis is given. The paper should provide identifiability conditions, show sensitivity of the discovered g_phi to initialization/regularization, or constrain z through the dynamical model.","section":"Section 3.2/3.3, Eq. (8)"},{"comment":"Full Equation Discovery—the article's most general and novel mode—is validated only under the same excitation u(t)=sin(2t) for training and testing; only the initial condition changes. Under a single excitation, the pair (f_theta,z) is not identifiable: a neural-network f_theta can absorb any chosen z and still reproduce the measured x. The new-initial-condition test therefore checks interpolation over initial states, not generalization to unseen forcing. The Hysteresis Discovery mode in Section 4.1 has a genuine out-of-excitation test (Testing 2, multisine), but Full Equation Discovery does not. Add a cross-excitation test for Case 2.","section":"Section 4.2, Tables 3-4"},{"comment":"The state-space formulation in Eq. (7) implies that z evolves under g_phi and that the solver integrates [x,dot(x),z]. The learning procedure, however, treats z as a learnable parameter vector whose derivative is computed by finite differences (the text in Section 3.1 says 'only the value of z is required in this step'). Thus g_phi is not part of the forward dynamics during training, and the training loss never validates the discovered g_phi. This decoupling means that after SR replaces Diff(z) with an analytic g_phi, the closed-loop system may have no reason to reproduce the training response. Please clarify, and ideally integrate g_phi (or a provisional surrogate) into the solver loop.","section":"Section 3.2, Eq. (7)"}],"minor_comments":[{"comment":"The phrase 'without predefined libraries' overstates the case, since symbolic regression itself uses a predefined operator set and complexity measure. The contrast with SINDy should be phrased as 'without a fixed library of candidate terms' to avoid overclaiming.","section":"Abstract and Section 3.3"},{"comment":"Typo: 'selction matrix' should be 'selection matrix'. Also, the notation is slightly confusing because the symbol S is used for the selection matrix while the signal-to-noise ratio uses 'SNR'.","section":"Equation (8)"},{"comment":"Reproducibility details are missing: PySR hyperparameters (population size, iterations, operator set, complexity penalty lambda), the RK4 time step, the optimizer and learning rate, and how z(0) is initialized for test-time solution of the discovered equations. Please report these.","section":"Section 4"},{"comment":"The text says five initial conditions are used for training and one for testing, but the figures appear to show a single continuous time trace. Clarify whether the traces are concatenated and whether z is learned as a single vector across the concatenated data or per trace.","section":"Section 4.2, Figures 8-9"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within scope for a computational engineering / signal processing venue. The major issue is that the most novel mode (Full Equation Discovery) lacks an out-of-excitation validation, and the free latent variable raises an identifiability concern. These are fixable with additional experiments and analysis, so I recommend major revision rather than rejection. No concerns about citation behavior or novelty disclosure."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, read this one if you work on data-driven structural dynamics. The genuinely new piece is treating the internal hysteretic variable z(t) as a trainable parameter inside an RK4 solver, then running symbolic regression on the learned trajectory to recover an explicit z-dot = g(x-dot, z). That is a sensible move, and it handles fractional exponents naturally, which library-based SINDy cannot. The classification in Figure 1 is also useful and worth borrowing.\n\nCredit where it is due: the Bouc-Wen benchmark results are solid. Training on a sinesweep and testing on a different multisine, with NRMSE under a few percent even at 20 dB noise, is real evidence that the discovered z-evolution law generalizes. The shake-table experiments are informative too; the discovered equations are interpretable, and the larger testing error in the NSD case is reported honestly rather than hidden. The SINDy comparison on the fractional-exponent case makes the library-free point convincingly.\n\nThe soft spot is exactly in Full Equation Discovery, which is also the most advertised novelty. In Case 2, z is a free learnable time series and the motion equation is a neural network N(x, x-dot, u; theta, z). For any chosen z(t), an NN can reproduce the observed x(t) under one excitation, so the pair (f_theta, z) is not identifiable. SR then regresses z-dot = Diff(z) on (x-dot, z) and reports the result as the hysteresis law. The test in Section 4.2 uses u(t) = sin(2t) for both training and testing; only initial conditions change. That checks interpolation over initial states, not generalization to unseen forcing. So the most novel mode lacks the out-of-excitation validation that the Case 1 benchmark has. The paper provides no identifiability or excitation-conditions analysis. This is a load-bearing gap for the general claim, though it does not invalidate the Case 1 results.\n\nMinor points: no code or data, no error bars or repeated runs, and the novelty claim against refs [30] and [34] is asserted more than demonstrated. These are fixable.\n\nWho is this for: people doing structural identification, equation discovery, or hysteresis modeling will get value. It deserves a serious referee, but the revision either needs an identifiability analysis, a richer-excitation test for Full Equation Discovery, or a downgraded claim about that mode. My recommendation: send it to review, and ask for those additions explicitly.","headline":"A genuinely useful framework for hysteresis equation discovery, with strong Case 1 validation, but the flagship Full Equation Discovery mode is tested only under the training excitation, leaving the identifiability of the latent z unresolved.","tokens_in":744,"tokens_out":931,"would_cite":true,"duration_ms":36859,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper aims to establish that the hidden memory state of a hysteretic system can be learned as a free parameter inside a numerical solver, after which symbolic regression writes out the explicit governing equations—including the hysteres","keywords":["hysteresis","equation discovery","symbolic regression","internal variable learning","Bouc-Wen model","state-space identification","dynamic prediction","fractional exponent"],"falsifier":"Run the method on a simulated Bouc-Wen system driven by a single low-amplitude harmonic with no noise: if the discovered ˙z equation changes when the optimizer is re-initialized or when the excitation is swapped while the true law stays fixed, then z is not identifiable and the claimed direct discovery is falsified.","tokens_in":16683,"feed_emoji":"🔄","tokens_out":4199,"duration_ms":41359,"temperature":0.7,"pith_summary":"The paper tries to establish a unified, library-free pipeline for discovering explicit governing equations of hysteretic systems from data, where the unmeasured internal variable z(t) is learned jointly with system parameters by embedding it in a differentiable ODE solver. It claims that once z and its time derivative are recovered, symbolic regression returns explicit equations for both the dynamic motion equation and the hysteretic link equation, even when neither structure is known in advance. The authors argue this removes the reliance on predefined candidate function libraries, which is especially important for non-integer exponents and absolute-value or sign nonlinearities that are hard to enumerate. If correct, this shifts hysteresis modeling from selecting and calibrating a fixed model toward deriving the model itself from measurements, with dynamic prediction following directly from solving the discovered equations.","feed_headline":"Hysteresis laws recovered from data with no preset library","feed_subtitle":"A solver learns the hidden memory state, then symbolic regression derives the laws and predicts unseen responses.","key_machinery":"The mechanism is a two-stage pipeline built on the state-space triple (x, ẋ, z). First, z is treated as a free learnable time series and integrated through a differentiable RK4 solver, which enforces temporal consistency while updating system parameters. Second, the learned trajectories are fed to symbolic regression, which searches over expression trees (including terms like |·|^n with n free) to produce the explicit equations for the motion law and the hysteresis law, replacing the predefined function library used by sparse identification methods.","core_discovery":"The central claim is that reformulating a hysteretic system in state-space form, with the internal hysteretic variable z treated as a trainable parameter inside an embedded Runge-Kutta solver, makes the unobservable memory state learnable from measured responses. After convergence, the predicted trajectories provide the input-output data for symbolic regression, which discovers an explicit expression for the hysteresis evolution law ż = g(ẋ, z) and, in the full setting, for the motion equation ẍ = f(x, ẋ, z, u). The paper reports successful recovery of the Bouc-Wen-like law including a fractional exponent n = 1.5, and on experimental shake-table data recovers a rate-dominated linear law when","pith_inferences":["The same solver-based learnable-state idea could be applied to other hidden-state dynamics—plasticity variables, battery state-of-charge, or ferromagnetic domains—where only a subset of states is measured and the evolution law is unknown.","The method's reliance on optimizing z without an identifiability guarantee suggests that excitation richness and noise level directly control whether the recovered g_phi is the true physical law or merely one of many functions that reproduce the training data.","The reported 14% testing displacement error in the negative-stiffness-device case hints that the framework's generalization gap widens as hysteresis becomes stronger and more history-dependent, so a diagnostic for when z is reliably identified would sharpen its practical use."],"forward_implications":["For the Bouc-Wen benchmark, the framework recovers the structure and coefficients of both governing equations with displacement prediction errors around 1–3% on test excitations, even under 20 dB measurement noise.","For a system with cubic stiffness and a fractional hysteretic exponent of 1.5, both Hysteresis Discovery and Full Equation Discovery recover the fractional exponent with small error, whereas library-based regression cannot represent non-integer powers and introduces spurious terms.","On experimental shake-table data, the framework automatically returns a rate-dominated linear law when no negative-stiffness device is present, and a richer nonlinear law when the device is installed, showing that the discovered equation adapts to the actual hysteresis level.","Because the discovered equations are explicit, solving them yields forward predictions on unseen excitations, so equation discovery and dynamic prediction are handled in one unified workflow."],"fun_headline_variants":["Hysteresis equations discovered from data alone","Hidden memory state learned, then laws recovered","Symbolic regression unearths hysteresis laws without priors","Trainable internal state enables equation discovery","From data to explicit hysteresis laws, no library needed"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the hidden internal variable z(t) is identifiable from the measured responses, so the optimized z is the true hysteresis state and not an arbitrary function that happens to reproduce the training data.","fun_headline_variants_meta":{"raw":{"variants":["Hysteresis equations discovered from data alone","Hidden memory state learned, then laws recovered","Symbolic regression unearths hysteresis laws without priors","Trainable internal state enables equation discovery","From data to explicit hysteresis laws, no library needed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000166,"raw_usage":{"total_tokens":1090,"prompt_tokens":744,"completion_tokens":346,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":488,"completion_tokens_details":{"reasoning_tokens":276}},"tokens_in":488,"tokens_out":346,"duration_ms":5071,"temperature":1.0,"reasoning_tokens":276,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T06:40:00.468708+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the method on a simulated Bouc-Wen system driven by a single low-amplitude harmonic with no noise: if the discovered ˙z equation changes when the optimizer is re-initialized or when the excitation is swapped while the true law stays fixed, then z is not identifiable and the claimed direct discovery is falsified.","supporting_citations":[],"review_version":1}