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A unified framework for equation discovery and dynamic prediction of hysteretic systems

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read 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

desk verdict 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. read the letter →

arxiv 2512.02408 v3 pith:7UZCUUSX submitted 2025-12-02 cs.CE

Siyuan Yang , Wei Liu , Zhilu Lai This is my paper · ORCID
classification cs.CE
keywords hysteresisequationdiscoverysymbolicregressioninternalvariablelearningBouc-Wenmodelstate-spaceidentificationdynamicpredictionfractionalexponent
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Section 3.2/3.3, Eq. (8)] 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.
  2. [Section 4.2, Tables 3-4] 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.
  3. [Section 3.2, Eq. (7)] 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.
minor comments (4)
  1. [Abstract and Section 3.3] 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.
  2. [Equation (8)] 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'.
  3. [Section 4] 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.
  4. [Section 4.2, Figures 8-9] 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.

Circularity Check

1 steps flagged · score 4.0 of 10

Partial circularity in Full Equation Discovery: z is a free learnable parameter fitted by Eq. (8b) to the measured response, and the discovered g_phi is a symbolic regression on that fitted latent trajectory; this mode is tested only under the training excitation.

  1. fitted input called prediction [Sec. 3.1-3.3 (Case 2 definition; Eq. (8b); SR(ż_pred, ẋ_pred, z_pred))]
    "For Case 2: Full Equation Discovery, since the structure of the dynamic motion equation is unknown, it is represented by a neural network N(·), parameterized by learnable parameters θ, i.e., f_θ = N(x, ẋ, u; θ, z). Here, the internal variable z is also treated as an additional learnable parameter within the model. ... For the hysteretic link equation (g_ϕ), ... symbolic regression (SR) is applied to the learned trajectories of ẋ_pred, z_pred, and ż_pred to search for mathematical expressions ż = g_ϕ(ẋ,z), and this can be written as SR(ż_pred, ẋ_pred, z_pred)."

    Eq. (8b) optimizes z (and θ) by back-propagating the mismatch of the solver output against the measured y_m, so z_pred is a fitted latent variable, not observed data. The SR target is then Diff(z_pred), and its regressors are z_pred and ẋ_pred; all three come from the same fitted trajectory. In Case 2, f_θ is an unconstrained neural network and z is a free time series, so for a single excitation many pairs (f_θ,z) reproduce y_m. Hence the reported g_ϕ is a regression on an arbitrary, unidentifiable latent process rather than a law independently recovered from data.

full rationale

The paper is not circular as a whole. Case 1 (Hysteresis Discovery) on the Bouc-Wen benchmark is trained on a sinesweep and tested on a fixed multisine excitation (Testing 2), and the shaking-table experiments use different earthquake records for training and testing; with the dynamics f_θ known, the learned z is constrained by Eq. (3), and these out-of-excitation tests provide real independent support. The circularity burden is specific to Case 2 (Full Equation Discovery): Section 3.2 learns z by fitting Eq. (8b) to the measurements, and Section 3.3 then 'discovers' g_ϕ by regressing a finite-difference derivative of that fitted z. Since f_θ is a neural network and z is free, the decomposition is not identifiable, and the paper gives no identifiability or excitation-richness analysis. Moreover, Section 4.2 validates Full Equation Discovery only by changing the initial condition while keeping u(t)=sin(2t), so the test does not exercise new forcing and cannot distinguish the true equations from a forcing-tuned fit. These limitations make the central general-mode claim partially circular, but the external and out-of-excitation Case 1 and experimental results keep the paper from being wholly circular; score 4.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The framework does not postulate new physical entities; z is inherited from the Bouc-Wen/hysteresis modeling tradition. The central free assumptions are the identifiability of z from the chosen measurements and the sufficiency of the operator set used by PySR. No code, data, or formal certificates accompany the paper, so the ledger entries above are the main load-bearing choices the reader must accept.

free parameters (4)
  • Internal hysteretic variable z(t) (trainable vector) = reported only via plots; no numeric values in text
    Section 3.2: z is defined as a learnable parameter optimized to match measured x, ẋ, ẍ through the solver loss (Eq. 8). This is the main hidden degree of freedom.
  • Linear structural parameters (m, c, k, α) = e.g., m=1.9998, c=10.0231, k=49893.86, α=1.0110 (Table 1, noise-free)
    Case 1: Hysteresis Discovery fits these coefficients to data; they appear in the discovered equations.
  • Neural network parameters θ for f_θ in Full Equation Discovery = not reported (intermediate representation)
    Section 3.1: f_θ=N(x,ẋ,u;θ,z) is trained in Case 2, then SR is fit to its output; θ values are not given.
  • SR hyperparameters (operator set, complexity penalty λ, search budget) = not reported
    Section 3.3 mentions operators (e.g., |·|^n with free n) and complexity control but gives no concrete values; results are sensitive to these choices.
assumptions (5)
  • domain assumption The unmeasured internal hysteretic variable z(t) is sufficient to close the state-space description of the system (Eq. 7).
    Section 3.1: the framework models hysteretic dynamics with a single internal variable z; systems where one scalar internal variable is insufficient would break the method.
  • domain assumption The measurement data and excitation are sufficiently rich that optimizing z as a free parameter converges to a unique/meaningful latent trajectory.
    Section 3.2: z is treated as a learnable parameter without an identifiability or excitation-conditions analysis; no guarantee the fitted z is the physical z.
  • standard math RK4 time-stepping with finite-difference Diff(z) accurately approximates the true coupled ODE solution.
    Section 2.2: RK4 is embedded in the solver; this is standard, but with z free the internal consistency of the z dynamics is not enforced during training.
  • domain assumption The Bouc-Wen differential form (Eq. 4/11) is the correct generative model for the synthetic benchmarks.
    Section 4.1/4.2: ground truth is generated from Bouc-Wen equations; evaluation is only meaningful within that model class.
  • domain assumption Shake-table test data from [26] are accurate and the SDOF equation of motion mẍ+cẋ+kx+αz=-mu is applicable.
    Section 4.3: experimental validation reuses [26] data and assumes the stated structural form.

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Cite this review

Pith. "Pith review of A unified framework for equation discovery and dynamic prediction of hysteretic systems." pith.science (2026). https://pith.science/paper/7UZCUUSX

@misc{pith2026251202408,
  author       = {Pith},
  title        = {Pith review of: A unified framework for equation discovery and dynamic prediction of hysteretic systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7UZCUUSX}},
  note         = {Machine review of arXiv:2512.02408}
}
read the original abstract

Hysteresis is a nonlinear phenomenon with memory effects, where a system's output depends on both its current state and past states. It is prevalent in various physical and mechanical systems, such as yielding structures under seismic excitation, ferromagnetic materials, and piezoelectric actuators. Analytical models like the Bouc-Wen model are often employed but rely on idealized assumptions and careful parameter calibration, limiting their applicability to diverse or mechanism-unknown behaviors. Existing equation discovery approaches for hysteresis are often system-specific or rely on predefined model libraries, which limit their flexibility and ability to capture the hidden mechanisms. To address these challenges, this research classifies equation discovery problems for hysteretic systems and develops a unified framework in which the state-space form is reformulated, and hysteretic variables are treated as trainable parameters from data. The framework further employs symbolic regression (SR) to automatically recover explicit governing equations without relying on predefined libraries, unlike methods such as sparse identification of nonlinear dynamics (SINDy). Experimental results demonstrate that the proposed method is effective in recovering governing equations for hysteretic systems, even in a challenging Full Equation Discovery setting, where prior information is extremely limited, and solving the equations naturally enables the dynamic prediction of hysteretic systems.

Figures

Figures reproduced from arXiv: 2512.02408 by the authors.

Figure 1
Figure 1. Classification of equation discovery approaches for hysteretic systems according [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The proposed unified framework for equation discovery and dynamic prediction [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Different types of external excitation of benchmark data. The sinesweep signal [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Hysteresis loops of benchmark data. (a) x−x˙ hysteresis loop. (b) x−z hysteresis loop. (c) x − F(F = kx + αz) hysteresis loop. We define two different types of testing cases. In Testing 1, the external excitation is the same as in Training, but the data are taken from …
Figure 5
Figure 5. Figure 5: Displacement results of benchmark data. (a) Training results. (b) Testing 1 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Velocity results of benchmark data. (a) Training results. (b) Testing 1 (same [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Hysteresis loops of complex structure data. (a) [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: Training results of complex structure data. (a) Displacement results. (b) Veloc [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: Testing results of complex structure data. (a) Displacement results. (b) Velocity [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: Shaking table test of a braced 3-story structure (equivalent to a SDOF system), [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]
Figure 11
Figure 11. Figure 11: x − x˙ hysteresis loops of yielding structure. (a) Without NSD. (b) With NSD [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]
Figure 12
Figure 12. Figure 12: Training results of yielding structure data without NSD. (a) Displacement [PITH_FULL_IMAGE:figures/full_fig_p024_12.png]
Figure 13
Figure 13. Figure 13: Testing results of yielding structure data without NSD. (a) Displacement re [PITH_FULL_IMAGE:figures/full_fig_p025_13.png]
Figure 14
Figure 14. Figure 14: Training results of yielding structure data with NSD. (a) Displacement results. [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]
Figure 15
Figure 15. Figure 15: Testing results of yielding structure data with NSD. (a) Displacement results. [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.