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REVIEW 5 major objections 5 minor 21 references

State-Space Kolmogorov Arnold Networks for Interpretable Nonlinear System Identification

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read From raw data, SS-KAN recovers a Duffing cubic and a diode saturation.

desk verdict A modest but honest architectural contribution that shows KANs inside a state-space model can recover the expected nonlinearity on two benchmarks; the interpretability claim is real but softer than the abstract suggests. read the letter →

arxiv 2506.16392 v1 pith:AWOBBNWG submitted 2025-06-19 cs.LG cs.SYeess.SY

Gonçalo Granjal Cruz , Balazs Renczes , Mark C Runacres , Jan Decuyper This is my paper · ORCID
classification cs.LGcs.SYeess.SY
keywords nonlinearsystemidentificationstate-spacemodelsKolmogorov-ArnoldnetworksinterpretabilitysparsityregularizationDuffingoscillatorWiener-Hammersteinbenchmarkgrey-boxmodelling
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 proposes State-Space Kolmogorov-Arnold Networks (SS-KAN), a grey-box architecture that places learnable univariate (single-variable) KAN functions inside a discrete-time state-space model instead of a large black-box network. The aim is to identify nonlinear dynamical systems from input-output measurements in a way that lets an engineer read the physics off the trained model. On the Silverbox (Duffing oscillator) benchmark the learned functions show a dominant cubic term matching the hardening stiffness, and on the Wiener-Hammerstein circuit they show a linear response with saturation matching the diode-resistor nonlinearity. These insights come at a quantitative price: on both benchmarks the test RMSE is larger than that of the best polynomial and deep-encoder black-box models, though an order of magnitude smaller than the best linear approximation. The paper positions SS-KAN as a deliberate accuracy-versus-interpretability trade-off rather than as a new accuracy champion.

What carries the argument

The load-bearing object is the state-space representation of Eq. (5): the unknown nonlinearities $f(\cdot)$ and $g(\cdot)$ in a discrete-time state-space model are replaced by KANs, $\operatorname{KAN}_f(x(k),u(k))$ and $\operatorname{KAN}_g(x(k),u(k))$, while $(A,B,C,D)$ keep the linear dynamics. Each KAN univariate function is a sum of a SiLU residual activation and a linear combination of cubic B-splines on a five-point grid, and the training cost adds an $\ell^1$ penalty on the KAN parameters for sparsity and an $\ell^2$ penalty on the linear matrices. The argument works because the KAN takes state variables directly as inputs and only combines them through sums of univariate functions, so a single dominant nonlinearity can be isolated by fixing the other inputs and plotting one curve. In the Wiener-Hammerstein variant, the KAN is instead exposed as a static scalar nonlinearity $w(k)=\operatorname{KAN}(v(k))$ sandwiched between two linear state-space blocks.

What would settle it

Apply an invertible linear change of state coordinates to a trained SS-KAN model, which leaves the input-output behavior identical, and replot the KAN curves: if the cubic shape and its position in a particular state update disappear or migrate, the visual interpretation depends on the coordinate convention. Alternatively, retrain Silverbox from several random near-identity initializations and check whether the same univariate curve appears every time.

Watch

Extended reading notes

Core claim

The central claim is that when a KAN is used to model only the nonlinear part of a state-space model, the trained univariate functions retain a readable relation to the physical nonlinearity. For the two-state Silverbox model, $\operatorname{KAN}_f$ for the velocity update, plotted against position with velocity and input fixed, is nearly $y \approx -996x^3 + 12.8x^2 - 24.6x - 0.115$; the cubic term carries the Duffing stiffness and a small quadratic term reflects a secondary effect known in the physical realization. For the Wiener-Hammerstein system, the learned KAN that sits between the two linear blocks forms a linear ramp that saturates, directly matching the diode-resistor behavior. In both cases the authors conclude that SS-KAN reveals system nonlinearities at the cost of accuracy relative to state-of-the-art black-box models, with test RMSEs of $0.0039\,\mathrm{V}$ and $0.0114\,\mathrm{V}$ on the two benchmarks.

Load-bearing premise

The load-bearing premise is that the trained state variables keep their physical meaning (position and velocity in the Silverbox case), so a univariate plot of the KAN with the other inputs fixed is a plot of the physical nonlinearity rather than of some mixed coordinate.

Editorial extensions

If this is right

  • On the Silverbox benchmark, the learned velocity-state update is dominated by a cubic term, so a practitioner can read the Duffing stiffness off the model without knowing the equations in advance.
  • On the Wiener-Hammerstein benchmark, the recovered intermediate function is linear with saturation, so the diode-resistor nonlinearity is visible even though the internal states are never measured.
  • SS-KAN reaches test RMSEs of $0.0039\,\mathrm{V}$ and $0.0114\,\mathrm{V}$ on the two benchmarks, roughly an order of magnitude better than the best linear approximation but worse than the polynomial and deep-encoder nonlinear baselines.
  • Training is practical, taking about an hour on the Silverbox case and six to eight hours on the Wiener-Hammerstein case on laptop-class hardware, versus multi-day deep-encoder training.
  • The sparsity penalty plus one-at-a-time plotting identifies which input state drives the nonlinearity and shows the other directions contribute negligibly.

Reading between the lines

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

  • The interpretability reading is coordinate-dependent in a way the paper does not formally settle: a similarity transformation of the state-space model leaves input-output behavior identical but would mix the KAN inputs, so a visible cubic curve could be tied to the chosen coordinate convention rather than to the physical displacement.
  • A robustness test would be to retrain SS-KAN from several different near-identity linear initializations on the same Silverbox data; if the cubic curve consistently lands in the same state-update slot, the physical reading is much stronger.
  • The same construction should transfer to other identifiable structural nonlinearities, such as Coulomb friction, backlash, or piecewise-linear stiffness, and the recovered univariate shapes could be compared against those known nonlinear maps.
  • The paper sets $\operatorname{KAN}_g$ to zero for Silverbox; keeping it active would let SS-KAN expose sensor or output-mapping nonlinearities as well, with a second set of visualizable curves.
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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

5 major / 5 minor

Summary. The paper proposes State-Space Kolmogorov-Arnold Networks (SS-KAN), which embed KANs as additive residual functions in a discrete-time state-space model, jointly training the linear matrices (A,B,C,D) and the KAN weights with L1/L2 regularization. The method is evaluated on two benchmark problems: the Silverbox Duffing oscillator and the Wiener-Hammerstein circuit. The paper reports RMSE values against BLA, PNLSS, and Deep Encoder baselines, and interprets visualized KAN functions as revealing a cubic stiffness nonlinearity in Silverbox and a diode-resistor saturation nonlinearity in Wiener-Hammerstein. The central claim is that SS-KAN trades a modest amount of accuracy for enhanced interpretability via sparsity and direct visualization of univariate functions.

Significance. If the interpretability claim is established, SS-KAN is a useful grey-box architecture that combines a linear state-space core with flexible univariate nonlinearities, and the two public benchmark demonstrations are relevant to the nonlinear system identification community. The paper is honest about the possibility that states may not remain physical and explicitly flags the assumption rather than hiding it. Its strengths include the use of widely used benchmark datasets, a clear formulation of the cost function, and a training cost that is competitive with existing nonlinear state-space methods. However, the central scientific claim that the visualized functions correspond to physical nonlinearities is currently supported mainly by a post-hoc reading of two fitted surfaces, one of which is supported by an analysis that is explicitly omitted from the manuscript. The evidence therefore falls short of the paper's stated claim that SS-KAN 'reveals' the physical nonlinearities, rather than merely being consistent with them under an unverified coordinate assumption.

major comments (5)
  1. [Sec. IV-B, Eq. (5)] The cubic interpretation of Fig. 2 assumes that the trained state vector retains the physical meaning of position and velocity. The state-space realization in Eq. (5) is invariant under any invertible similarity transform x' = T x, with A' = T A T^{-1}, B' = T B, C' = C T^{-1}, and KAN_f'(x',u) = T KAN_f(T^{-1}x',u). Nothing in the cost function (7), the L1 penalty, or the close-to-identity initialization proves or tests that training selects the original coordinate system. The text in Sec. IV-B that 'the KAN preserves the function inputs without mixing' addresses mixing inside the KAN layer, not the state-space realization ambiguity. The authors should add a concrete test, for example comparing the identified state trajectories with measured physical states, checking the learned A against a discretized physical model, or showing that the cubic shape is preserved under allowable similarity transforms. Without such evidence, the apparent cubic function in Fig. 2 could be a coordinate artifact.
  2. [Sec. IV-B, Silverbox results] The sentence in Sec. IV-B describing the 'complementary analysis, omitted here' is a load-bearing piece of evidence: it is used to claim that varying the velocity state and the input produces negligible univariate functions, thereby supporting the conclusion that sparsity isolated the position state as the dominant nonlinear input. Since this analysis is not shown, and since it directly supports the central interpretability claim, it should be included as a figure or at least as a quantitative summary table. The current text asks the reader to accept the main conclusion on the basis of an unverifiable result.
  3. [Eqs. (5) and (7), Fig. 2] There is a non-identifiability between the linear part A and the linear components of KAN_f: any linear function of x can be represented either in A or in the linear terms of the B-splines, so the decomposition into 'residual linear dynamics' and 'nonlinear terms' is not unique. The polynomial fit y ≈ -996x^3 + 12.8x^2 - 24.6x - 0.115 includes a linear term and a quadratic term alongside the cubic term, and the interpretation of these extra terms as 'subtle dynamic effects' is post-hoc. The authors should demonstrate, for instance on simulated data where the true nonlinearity is known, that the optimized decomposition recovers the true cubic coefficient and does not merely fit an arbitrary polynomial in the chosen coordinate system.
  4. [Sec. V-A, Eq. (10)] The Wiener-Hammerstein model is strongly structured: the cascade form (10) is imposed a priori, and the linear matrices are initialized using Chebyshev filter information from the benchmark reference. Consequently, the learned saturation curve in Fig. 4 is partly a consequence of the imposed structure and initialization rather than a discovery from input-output data alone. The paper should state explicitly which components are assumed and which are learned, and it should include a sensitivity test, such as random or less informed linear initializations, to show that the saturation shape is robust rather than a side effect of the chosen initialization.
  5. [Tables I and II] The RMSE comparisons combine numbers from different publications with different test protocols, model classes, and training setups, and no variance or repeated-seed statistics are reported for SS-KAN. The 'at the cost of accuracy' claim in the abstract and conclusion is therefore less precise than it should be. The authors should either add same-protocol baseline runs for at least one standard method or explicitly state the comparability limitations of the literature-derived RMSE values.
minor comments (5)
  1. [Table I paragraph] There is a typo in 'Deep Enconder'; it should read 'Deep Encoder'.
  2. [Abstract] The first sentence 'While accurate, black-box system identification models lack interpretability of the underlying system dynamics' is a fragment; it should be rephrased, for example 'Although they are accurate, black-box system identification models lack interpretability...'.
  3. [Eq. (5)] The notation uses KAN_f and KAN_g as both the function names and their vector-valued outputs; using explicit composition notation, such as f(x(k),u(k)) ≈ KAN_f(x(k),u(k)), would make the equations clearer.
  4. [Sec. IV-A and V-A] Hyperparameters such as λ_L1 = λ_L2 = 10^{-4}, the 5-point grid, and the hidden layer sizes are stated without sensitivity analysis; since the interpretability conclusion depends on the sparsity level, a short discussion or small ablation of λ_L1 would strengthen the claim.
  5. [Fig. 2 caption] The y-axis label 'KANf(x(k), ˙x, u) Nonlinear Output' is ambiguous because KAN_f is vector-valued; the caption should state explicitly which component of the KAN output is being plotted for the position update and which for the velocity update.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: SS-KAN fits public benchmarks and its interpretability evidence is a post-hoc reading of fitted functions, not a prediction forced by construction or by self-citation.

full rationale

The paper's derivation chain is not circular. The model equations (5) and (10) are direct architectural definitions, the cost function (7) is a standard data-fit objective, and the reported RMSE values are validated against independent public benchmarks (Silverbox, Wiener-Hammerstein) and external baselines. No equation is defined in terms of the claimed result: the cubic shape in Fig. 2 and the saturation shape in Fig. 4 are outputs of the trained KANs, not constraints or fitted parameters renamed as predictions. The paper explicitly frames these as learned functions that 'reveal system nonlinearities', which is a post-hoc interpretation of fitted model components rather than a prediction derived from the model's own assumptions; that distinction weakens the interpretability claim's force but is not circularity. The self-citations in Section III-C ([4], [5], [15]) provide background on decoupled state-space models and are not load-bearing: the paper argues from the architecture that KANs take state variables as direct inputs 'without mixing', and it hedges the physical-interpretability conclusion with 'under the assumption of a well-approximated nonlinearity'. That hedge is an unproven correctness assumption, not a circular step. Two passages deserve explicit flagging but do not raise the circularity score. First, Section IV-B describes a 'complementary analysis ... omitted here' that allegedly confirms negligible contributions from velocity and input; this is unverifiable, unreported evidence and weakens empirical support. Second, the central interpretability claim depends on learned states retaining physical coordinates (position and velocity), which state-space similarity invariance makes non-obvious; the paper does not prove this, citing it as an assumption. Both are correctness/evidence limitations, not cases where the result reduces to its own inputs by construction.

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

The central claim relies on a short list of chosen hyperparameters and on the physical-interpretability assumption. No invented physical entities appear. The most important unpaid premise is that state coordinates keep their physical meaning after training.

free parameters (5)
  • Regularization weights lambda_L1 and lambda_L2 = 1e-4 (both)
    Chosen by hand, not tuned; they control sparsity and hence the interpretability-accuracy trade-off that is the central claim.
  • Silverbox KAN grid size and architecture = 5-point grid, 2 layers, hidden size 2
    Architecture choices set model capacity and directly affect whether the cubic nonlinearity is visible and accurate.
  • Wiener-Hammerstein KAN architecture = 2 layers, hidden size 15, dynamic grid update
    Capacity choice needed to represent the saturation curve; affects the reported RMSE.
  • Learning rates and batch sizes = 1e-3, batch 64 (Silverbox); 1e-4 decaying, batch 2048 (Wiener-Hammerstein)
    Optimizer hyperparameters chosen empirically, not justified; results may depend on them.
  • Polynomial fit to learned KAN_f velocity update = -996 x^3 + 12.8 x^2 - 24.6 x - 0.115
    Coefficients fitted to the learned curve, not to data; used to support the claim of a cubic stiffness term, so the cubic coefficient is a post-hoc fit.
assumptions (5)
  • standard math Kolmogorov-Arnold representation theorem
    Basis for KAN representation; guarantees arbitrary continuous multivariate functions can be expressed as sums of univariate functions, but says nothing about learnability or physical meaning.
  • domain assumption KAN architecture with chosen layers and grid can approximate residual nonlinearities f and g well enough
    The success of the method depends on this approximation, and it is only validated empirically on two benchmarks.
  • ad hoc to paper State variables remain physically interpretable after training
    Sections III-C and IV-A assume the identified state coordinates correspond to physical position and velocity; no invariance test or similarity-transform analysis is given.
  • domain assumption Fixing other variables at their mean values and varying one input isolates that input's effect
    Used to produce Figs. 2 and 4; valid only if the univariate functions are additively separable and inputs are not mixed, which the trained KAN may only approximately satisfy.
  • domain assumption Wiener-Hammerstein benchmark has a static nonlinearity between two linear blocks
    The model structure in Eq. (10) is assumed a priori, so the identified KAN is constrained to represent the static nonlinearity.

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Cite this review

Pith. "Pith review of State-Space Kolmogorov Arnold Networks for Interpretable Nonlinear System Identification." pith.science (2026). https://pith.science/paper/AWOBBNWG

@misc{pith2026250616392,
  author       = {Pith},
  title        = {Pith review of: State-Space Kolmogorov Arnold Networks for Interpretable Nonlinear System Identification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AWOBBNWG}},
  note         = {Machine review of arXiv:2506.16392}
}
read the original abstract

While accurate, black-box system identification models lack interpretability of the underlying system dynamics. This paper proposes State-Space Kolmogorov-Arnold Networks (SS-KAN) to address this challenge by integrating Kolmogorov-Arnold Networks within a state-space framework. The proposed model is validated on two benchmark systems: the Silverbox and the Wiener-Hammerstein benchmarks. Results show that SS-KAN provides enhanced interpretability due to sparsity-promoting regularization and the direct visualization of its learned univariate functions, which reveal system nonlinearities at the cost of accuracy when compared to state-of-the-art black-box models, highlighting SS-KAN as a promising approach for interpretable nonlinear system identification, balancing accuracy and interpretability of nonlinear system dynamics.

Figures

Figures reproduced from arXiv: 2506.16392 by the authors.

Figure 1
Figure 1. Time-Domain Performance of SS-KAN Model on Silverbox Benchmark. The blue and orange lines show the true output signal for the (partial) training and test sets, respectively. The red line represents the simulation error (y − ySS−KAN ) across both datasets. The vertical dashed line indicates the split between the training and testing data. The horizontal dotted lines indicate the amplitude extrapolation bound where th… view at source ↗
Figure 2
Figure 2. Learned KANf (x(k), x, u ˙ ) nonlinear functions by varying the position state variable (x) for both position x (blue) and velocity x˙ (orange) state updates. The velocity x˙ state update is well-approximated by y ≈ −996x 3 + 12.8x 2 − 24.6x − 0.115 (dotted black). The dominant cubic term captures the Duffing oscillator’s stiffness, while the smaller quadratic and linear terms reveal the KAN’s ability to identify mo… view at source ↗
Figure 4
Figure 4. Learned KAN function for Wiener-Hammerstein nonlinearity. The x-axis represents the input to the KAN, the intermediate signal v(k), and the y-axis represents the output of the KAN, w(k). It exhibits a linear trend with saturation, directly reflecting the behavior of the diode￾resistor nonlinearity. VI. CONCLUSION We introduced State-Space Kolmogorov-Arnold Networks (SS-KAN), a new approach for interpretable nonlinea… view at source ↗

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Reviewed August 15, 2026 · model on record in the stance chip above.