{"id":"9e3ee719-5e94-48be-9a1c-9ab043b9d8f9","arxiv_id":"2506.16392","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"State-Space Kolmogorov-Arnold Networks fit a linear state-space model plus sparse learnable univariate functions, and the shapes of those functions recover the known cubic and saturation nonlinearities in two benchmark systems.","lead":"This paper builds a neural network that combines a linear state-space model with small learnable functions, so nonlinear physical systems can be identified from data while showing which nonlinear terms matter. It demonstrates the idea on two standard electronic test systems, recovering a cubic spring effect and a saturation curve.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Interpretability claim rests on learned states retaining physical coordinates; state-space similarity invariance makes this unproven, and Sec. III-C admits the assumption.","rationale":"The paper's central contribution is interpretability: learned univariate KAN functions are claimed to reveal the physical nonlinearities of the Silverbox and Wiener-Hammerstein benchmarks. The reader's weakest assumption identifies the precise condition on which this claim depends: the learned state variables must retain their physical meaning after training. My review agrees. The state-space realization is not unique; any invertible linear transformation of the state produces the same input-output behavior while changing the meaning of the KAN inputs. The paper provides no proof, no identifiability analysis, and no diagnostic showing that training preserves the physical coordinate. Section III-C explicitly frames physical interpretability as an assumption rather than a result, and Section IV-B invokes an omitted complementary analysis to support the claim that other KAN inputs are negligible. These are flagged as missing support, not as evidence of fraud or carelessness. The empirical demonstrations are plausible and the accuracy numbers are honestly reported, but the interpretability conclusion is not yet established beyond the specific favorable initialization. This is exactly the kind of gap that warrants a conditional verdict: the architecture and experiments are promising, but the central claim needs either a formal argument, a similarity-invariance test, or a softened claim that interpretability is initialization-dependent and empirical rather than guaranteed. Since the reader's verdict is already CONDITIONAL and identifies the same assumption, my stress-test does not move the verdict.","tokens_in":9333,"tokens_out":4295,"duration_ms":50597,"concrete_test":"Train the Silverbox SS-KAN from a rotated state-space initialization: choose a fixed orthogonal T (e.g., a 45-degree rotation) and initialize A' = T A T^{-1}, B' = T B, C' = C T^{-1}, with KAN_f(x,u) replaced by T KAN_f(T^{-1} x, u) so the initial input-output map is identical. After training to the same test RMSE, reproduce Fig. 2 in the transformed coordinates. If the dominant term is no longer cubic in the first transformed state, or the fitted cubic coefficient changes materially, the claimed physical interpretability is initialization-dependent rather than guaranteed by the architecture and regularization.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that plotting KAN_f(x(k), xdot(k), u) reveals the physical cubic stiffness and saturation nonlinearity. This requires that the trained state x(k) remains the physical position/velocity rather than a linear mixture. Equations (4)-(5) define a state-space model whose input-output behavior is invariant under any invertible similarity transform (x -> T x, A -> T A T^{-1}, B -> T B, C -> C T^{-1}), with the KAN inputs transforming accordingly. Nothing in the cost function (7), the L1 sparsity penalty, or the 'close to identity' initialization proves or tests that training selects the physical coordinate. The paper explicitly hedges in Sec. III-C: 'under the assumption of a well-approximated nonlinearity, this structure may help preserve or even promote physical interpretability of the state variables themselves.' The Silverbox evidence is also partly supported by an omitted complementary analysis (Sec. IV-B), and the plotted function is fit by a polynomial whose cubic term is interpreted physically. If a similarity transform mixes position and velocity, the plotted function of the first state is a mixture of stiffness and damping and need not be cubic; another equally valid realization would 'reveal' a different nonlinearity. The interpretability evidence is therefore anecdotal to the chosen initialization and is not an architectural guarantee.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9655,"tokens_out":4179,"duration_ms":46257,"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":[{"comment":"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.","section":"Sec. IV-B, Eq. (5)"},{"comment":"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.","section":"Sec. IV-B, Silverbox results"},{"comment":"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.","section":"Eqs. (5) and (7), Fig. 2"},{"comment":"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.","section":"Sec. V-A, Eq. (10)"},{"comment":"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.","section":"Tables I and II"}],"minor_comments":[{"comment":"There is a typo in 'Deep Enconder'; it should read 'Deep Encoder'.","section":"Table I paragraph"},{"comment":"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...'.","section":"Abstract"},{"comment":"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.","section":"Eq. (5)"},{"comment":"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.","section":"Sec. IV-A and V-A"},{"comment":"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.","section":"Fig. 2 caption"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The manuscript is publishable in principle, but the central interpretability claim requires either a concrete state-coordinate test or an explicit downgrading of the claim to an illustrative, initialization-dependent result. The omitted Silverbox complementary analysis must be included, and the Wiener-Hammerstein section should clarify the amount of a priori structural information used. I would be satisfied with a focused revision that adds one experiment addressing the similarity-transform question and includes the missing analysis; given the venue is a letters journal, this is proportionate, but without it the main novelty of the paper is not demonstrated beyond anecdotal evidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing to know: SS-KAN is a genuine, modest architectural novelty—KAN univariate functions inside a linear state-space backbone, with L1 regularization to sparsify the learned nonlinearities. The two benchmark demonstrations are convincing as case studies: the cubic shape in Silverbox matches the Duffing equation, and the saturation curve matches the Wiener-Hammerstein diode-resistor circuit. The paper also deserves credit for being honest about the accuracy trade-off; it does not pretend to beat PNLSS or deep encoders on RMSE, and it clearly positions itself as a grey-box interpretability tool.\n\nThe paper is well-written and does not oversell. The visualization of the learned univariate functions is the core evidence, and on its face it is compelling, especially the Silverbox cubic term with the near-zero position-state update. The acknowledgement of the assumption in Section III-C is also appreciated—the authors know interpretability is not automatic.\n\nThe main soft spot is exactly that assumption: interpretability depends on the learned states keeping their physical meaning, and nothing in the cost function or initialization guarantees that. State-space models are similarity-invariant, so a mixed realization could just as easily produce plots that look different. The paper says the structure 'may help preserve' physical interpretability, which is a hedge, but it does not test whether the trained A, B, C, D actually correspond to position and velocity. The Silverbox evidence is further weakened by the omitted complementary analysis in Section IV-B—the claim that only position drives the nonlinearity rests on an analysis the reader cannot check. That omission should be fixed or removed.\n\nThe RMSE comparisons come from heterogeneous literature sources, which is typical for benchmark papers, but the differences in training protocols make the numbers only roughly comparable. Minor: no code or exact initialization details are given, so reproducibility is limited.\n\nThis paper is for researchers in system identification who want a transparent alternative to black-box models. It is not a breakthrough, but it is a solid and honest demonstration. A serious referee should engage with it; the main revision would be to either add a state-coordinate test (e.g., comparing learned states to the known physical states) or to limit the interpretability claim to 'demonstrated on these benchmarks rather than guaranteed by the architecture.' I would take it in with expectation of moderate revision.","headline":"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.","tokens_in":10127,"tokens_out":2181,"would_cite":true,"duration_ms":23119,"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":"From raw data, SS-KAN recovers a Duffing cubic and a diode saturation.","keywords":["nonlinear system identification","state-space models","Kolmogorov-Arnold networks","interpretability","sparsity regularization","Duffing oscillator","Wiener-Hammerstein benchmark","grey-box modelling"],"falsifier":"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.","tokens_in":9189,"feed_emoji":"⚙️","tokens_out":10774,"duration_ms":93135,"temperature":0.7,"pith_summary":"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.","feed_headline":"From raw data, SS-KAN recovers a Duffing cubic and a diode saturation.","feed_subtitle":"It costs some accuracy next to black-box models; the payoff is visible physics.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the KAN architecture, B-spline parameterization, and dynamic grid update used inside the state-space model.","marker":"[8]"},{"why":"Provides the efficientkan implementation actually trained in the experiments.","marker":"[13]"},{"why":"Defines the deep encoder baseline whose high accuracy and multi-day training set the black-box comparison.","marker":"[2]"},{"why":"Demonstrates that decoupled polynomial branches embedded in state-space models can yield physically meaningful intermediates, the interpretability precedent SS-KAN builds on.","marker":"[5]"},{"why":"Shows a broad class of multivariate functions can be approximated in decoupled univariate-branch form, motivating the KAN structure.","marker":"[15]"},{"why":"Supplies the Silverbox benchmark with its known Duffing-type physics, the first test case.","marker":"[16]"},{"why":"Is the source from which the Silverbox training and test data are downloaded.","marker":"[17]"},{"why":"Supplies the polynomial nonlinear state-space (PNLSS) baseline with the lowest Silverbox RMSE.","marker":"[18]"},{"why":"Defines the Wiener-Hammerstein benchmark circuit and dataset used as the second test case.","marker":"[19]"},{"why":"Supplies the best linear approximation baseline for the Wiener-Hammerstein comparison.","marker":"[20]"}],"fun_headline_variants":["SS-KAN turns black-box data into readable physics","SS-KAN exposes Duffing and diode nonlinearities","SS-KAN recovers Duffing cubic and diode saturation","State-space KAN puts nonlinearities in plain sight"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["SS-KAN turns black-box data into readable physics","SS-KAN exposes Duffing and diode nonlinearities","SS-KAN recovers Duffing cubic and diode saturation","State-space KAN puts nonlinearities in plain sight"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001086,"raw_usage":{"total_tokens":4506,"prompt_tokens":877,"completion_tokens":3629,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":493,"completion_tokens_details":{"reasoning_tokens":3565}},"tokens_in":493,"tokens_out":3629,"duration_ms":25940,"temperature":1.0,"reasoning_tokens":3565,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:27:07.299359+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"KAN: Kolmogorov–arnold networks,","cited_arxiv_id":null,"evidence_quote":"Supplies the KAN architecture, B-spline parameterization, and dynamic grid update used inside the state-space model."},{"cited_title":"efficientkan,","cited_arxiv_id":null,"evidence_quote":"Provides the efficientkan implementation actually trained in the experiments."},{"cited_title":"Nonlinear state-space identification using deep encoder networks,","cited_arxiv_id":null,"evidence_quote":"Defines the deep encoder baseline whose high accuracy and multi-day training set the black-box comparison."},{"cited_title":"Retrieving highly structured models starting from black-box nonlinear state-space models using polynomial decoupling,","cited_arxiv_id":null,"evidence_quote":"Demonstrates that decoupled polynomial branches embedded in state-space models can yield physically meaningful intermediates, the interpretability precedent SS-KAN builds on."},{"cited_title":"Decoupling multivariate functions using a non-parametric Filtered CPD approach,","cited_arxiv_id":null,"evidence_quote":"Shows a broad class of multivariate functions can be approximated in decoupled univariate-branch form, motivating the KAN structure."},{"cited_title":"Identification of the Silverbox Benchmark Using Nonlinear State-Space Models,","cited_arxiv_id":null,"evidence_quote":"Supplies the Silverbox benchmark with its known Duffing-type physics, the first test case."},{"cited_title":"Nonlinear Benchmarks,","cited_arxiv_id":null,"evidence_quote":"Is the source from which the Silverbox training and test data are downloaded."},{"cited_title":"Identification of nonlinear systems using Polynomial Nonlinear State Space models,","cited_arxiv_id":null,"evidence_quote":"Supplies the polynomial nonlinear state-space (PNLSS) baseline with the lowest Silverbox RMSE."},{"cited_title":"Wiener-hammerstein bench- mark,","cited_arxiv_id":null,"evidence_quote":"Defines the Wiener-Hammerstein benchmark circuit and dataset used as the second test case."},{"cited_title":"Modelling of Wiener- Hammerstein Systems via the Best Linear Approximation,","cited_arxiv_id":null,"evidence_quote":"Supplies the best linear approximation baseline for the Wiener-Hammerstein comparison."}],"review_version":2}