{"id":"d864f13e-d41d-4a52-bcf9-419ccc193c59","arxiv_id":"2505.13315","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"KHRONOS builds surrogates from sums of products of per-coordinate B-spline expansions and reports large accuracy gains over KAN and FEM on smooth 2D Poisson and regression benchmarks, though key results lack code and contain an impossible noisy-regression number.","lead":"KHRONOS is a proposed neural architecture that approximates fields as sums of products of one-dimensional B-spline expansions, tested on regression, a 2D Poisson equation, and inverse level-set recovery. It claims orders-of-magnitude accuracy and speed gains over KAN, FEM, and standard ML models, but major reported results are not reproducible from the manuscript.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 20D noisy-regression result in Table 2 is below the sigma=0.01 additive-noise floor, so the stated protocol cannot produce a test MSE of 6.8e-7; the model-free high-dimensional claim is internally inconsistent.","rationale":"The most load-bearing concern is the internal inconsistency in the 20D noisy-regression benchmark. This number is not a matter of tuning or implementation detail; it violates a basic statistical lower bound under the protocol stated in the paper. The abstract's broad claim of a unified framework for model-free, model-based, and model-inverse tasks leans heavily on the model-free results, and Table 2 is the only evidence that KHRONOS scales well in 20 dimensions with noise. If that table is unreliable, the model-free pillar of the central claim collapses. I checked whether other concerns are more central. The '1e6x over FEM' claim is arithmetically plausible: with P1 FEM on a 2D grid and a smooth solution, L2^2 error at 512 DoF can be on the order of 1e-4-1e-5 with a standard constant, so a 1e6 ratio to 6e-11 is not impossible. The separability limitation is explicitly admitted in Section 6, but the 2D Poisson source f(x,y) = pi^2(1+4y^2) sin(pi x) sin(pi y^2) - 2pi sin(pi x) cos(pi y^2) is separable, so it does not invalidate the central PDE benchmark. The absence of code and data is an evidence-quality problem, but the noise-floor contradiction is a decisive technical flaw. The reader's weakest_assumption identified the same noise-floor issue, and I agree with the REJECT verdict; no adjustment is needed.","tokens_in":11642,"tokens_out":10194,"duration_ms":99784,"concrete_test":"Re-run the Section 3.1.2 experiment exactly as written: sample 100,000 LHS points in [0,1]^20, compute u(p), add independent Gaussian noise with sigma=0.01 to form u_noisy, split 70/30 into train/test, and train KHRONOS with the stated 1560-parameter setup for 1000 epochs. Compute test MSE on the held-out noisy targets, and separately on the clean targets u(p). If the noisy-target MSE is at or above 1e-4, Table 2 is misreported; if it is below 1e-4, the noise was not actually present in the test labels, contradicting Eq. (52).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (52) states that noisy targets are u_noisy = u(p) + epsilon with epsilon ~ N(0, sigma^2), sigma = 0.01, so the irreducible variance of any held-out target is 1e-4. For any predictor f, the expected test MSE on such targets is E[(f(p)-u(p))^2] + 1e-4, which is at least 1e-4. Table 2 reports KHRONOS test MSE = 6.8e-7, a factor of about 147 below this floor. This is impossible if the written protocol was followed. The only ways to obtain 6.8e-7 are (a) the test targets were not actually corrupted with noise, (b) the reported MSE was computed against the clean function u(p) rather than u_noisy, or (c) the entry is misreported. Each possibility invalidates the claim that KHRONOS 'maintained near-perfect accuracy (R2=0.9994)' on the 20D noisy benchmark and undermines the general model-free comparison against RF, XGBoost, and MLP in Section 3.1.2. No code or data are provided to disambiguate. The separability limitation in Section 6 is also real, but the Poisson benchmark uses a separable source, so that limitation is not the primary load-bearing issue for the headline PDE result.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces KHRONOS, a kernel-based surrogate architecture that projects each input dimension through hierarchical B-spline kernel expansions, combines them via tensor products into modes, and superposes the modes. The authors evaluate KHRONOS on three task classes: model-free regression (8D borehole and 20D noisy Sobol-G), model-based PDE solving (2D Poisson via a Galerkin/energy-minimization loss), and model inversion (batched Gauss-Newton level-set recovery). The main claims are that KHRONOS achieves L2-square errors from 5e-4 to 6e-11 on the Poisson benchmark with 16 to 512 degrees of freedom, outperforming KAN and linear FEM by orders of magnitude, and that it reaches near-perfect accuracy on the 20D noisy regression benchmark while using fewer parameters than baselines.","tokens_in":11961,"tokens_out":7326,"duration_ms":66200,"significance":"If the results were reproducible, the architecture would be an interesting contribution to scientific machine learning: the Galerkin-style variational training with hierarchical B-spline bases is principled, and the emphasis on parameter efficiency and fast inference is timely. The 2D Poisson energy-minimization derivation is internally coherent, and the batch-inversion procedure is a natural use of the differentiable surrogate. However, the high-dimensional regression result is mathematically impossible under the stated protocol, the baseline comparisons are not reimplemented or statistically characterized, and the absence of code and data makes verification difficult. These issues undermine the paper's central claims as presented.","major_comments":[{"comment":"The reported KHRONOS test MSE of 6.8e-7 is below the irreducible noise floor sigma^2 = 1e-4. Since the test targets are stated to be the noisy outputs unoisy = u(p) + epsilon, epsilon ~ N(0, 0.01^2), the expected test MSE of any predictor is at least sigma^2 = 1e-4 (because E[(f(p) - u_noisy)^2] = E[(f(p) - u(p))^2] + sigma^2). Thus the value 6.8e-7 is impossible if the protocol was followed. Moreover, the reported R2 = 0.9994 is inconsistent with this MSE: the Sobol-G function has variance considerably larger than 1e-3, so an MSE of 6.8e-7 would imply R2 much closer to 1. The authors must either clarify that the test labels were not corrupted, re-run the experiment with the stated protocol, or correct the numbers; no code or data are provided to disambiguate. This invalidates the model-free high-dimensional claim in the abstract and conclusions.","section":"Section 3.1.2, Eqs. (51)-(52), Table 2"},{"comment":"The headline comparisons against KAN and FEM are not reimplemented or described with sufficient detail. The text states that KAN 'constructed with the same second order b-splines' achieves certain scalings, and Figure 4 includes P1/P2 FEM curves, but the manuscript does not specify how these baselines were configured, how the degrees of freedom were matched, whether the results are from the authors' own runs or cited from the literature, or what error bars or repeated trials were used. Without this information, the claimed '>100-fold gain over KAN' and '1e6-fold improvement over FEM' cannot be independently verified, and the lack of error bars makes it impossible to assess statistical significance of the scaling laws.","section":"Section 3.2, Table 3 and Figure 4"},{"comment":"The text claims that 'KHRONOS achieved an R2-score of 0.9935 with as few as 64 trainable parameters,' but Table 1 reports a KHRONOS configuration with 240 parameters and test R2 = 0.9998, and no 64-parameter result is shown. This is an unsupported claim; the reader cannot tell which configuration corresponds to the 64-parameter model, what the test error was, or why it is omitted from the table.","section":"Section 3.1.1, text and Table 1"}],"minor_comments":[{"comment":"The heading 'Forward Propogation' contains a typo; it should read 'Forward Propagation'.","section":"Section 2.2.1"},{"comment":"The sentence 'S. Saha gratefully acknowledges the start-up fund provided by the by the Kevin T. Crofton Department...' contains a duplicated 'by the'.","section":"Section 7 Acknowledgments"},{"comment":"The complexity expression O(ngaussne(2M2 +MN)) is written without a space and the symbol n is not defined; please clarify what n represents and use consistent notation (e.g., O(n_gauss n_e (2M^2 + MN))).","section":"Section 2.2.2"},{"comment":"The column headers 'Epoch Time (µs) Inference (µs)' are ambiguous because the table body appears to interleave time and error values in a way that is difficult to parse; please reformat the table so that each column has a clear header and the numerical entries are unambiguous.","section":"Table 3"},{"comment":"The manuscript does not include a data availability or code availability statement, which is important given that the central regression result appears to violate the stated protocol and no implementation is provided for verification.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The impossible 20D regression result (Table 2 vs. Eq. (52)) is a load-bearing inconsistency that cannot be fixed by a small edit; the authors would need to re-run the experiment under a clarified protocol. The Poisson benchmark derivation appears sound, but the lack of reimplemented baselines and error bars further weakens the manuscript. I recommend rejection, though the authors might resubmit after correcting the experimental reporting and adding reproducibility artifacts."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nTwo things to know about KHRONOS. First, the architecture is essentially a low-rank separated representation — a sum of products of per-coordinate hierarchical B-spline expansions. That idea is well known in proper generalized decomposition and tensor methods, and the paper does not cite that literature; the specific KAN-like hierarchical twist is a modest engineering contribution. Second, the headline model-free result, a 20D Sobol-G regression with test MSE 6.8e-7 under stated additive noise sigma=0.01, is impossible: the irreducible MSE floor on noisy targets is sigma^2 = 1e-4. The paper's own Eq. (52) makes that explicit. So the Table 2 comparison and the \"only KHRONOS retains near-perfect accuracy\" conclusion are internally inconsistent.\n\nWhat the paper does well: the model-based derivation for the 2D Poisson energy functional is clean. When the source term is separable, the Gram-matrix integration is correct and the complexity reduction to O(n_gauss n_e (2M^2+MN)) is real. The reported Poisson errors show attractive scaling, and the batched Gauss-Newton inversion idea is sensible. If those numbers hold up, there is a useful low-parameter surrogate for smooth separable problems, which has genuine practical value for edge computing.\n\nWhere it is soft:\n\n- The 20D result is not a missing error bar; it is a contradiction with the stated protocol. Either the noise was not applied to test targets or the MSE was computed against the clean function. Either way the claim as written is wrong.\n- The KAN and FEM comparisons are taken from the cited KAN paper, not from reimplementation. No code, no data, no error bars. The >100x and ~1e6x gain claims are therefore not independently checkable. For an empirical paper, that is a serious gap.\n- The novelty is oversold. Eq. (23) is a classical separated representation; the authors should engage with the tensor/PGD literature. This does not kill the paper, but it misplaces the contribution.\n- The inversion section is a toy example with no baseline, so sub-microsecond latency is not contextualized.\n\nThe separability limitation noted in Section 6 is real, but it is not the primary problem for the headline PDE result because the chosen Poisson source is separable.\n\nThe central Poisson idea looks sound, and the authors clearly understand the math. But the load-bearing empirical evidence as presented does not support the abstract's claims. I would send this to peer review, not desk reject it, so a referee can force the authors to correct or remove the 20D result, provide code/data, and rerun baselines. As it stands, it should not be accepted.\n\nRecommendation: engage with it in peer review, but expect heavy revision.","headline":"The 20D noisy-regression headline is mathematically impossible under the paper's own noise model; the separable Poisson core is plausible but the empirical evidence as presented does not support the claimed gains.","tokens_in":12496,"tokens_out":4518,"would_cite":false,"duration_ms":43841,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N30","68T07","41A15","65D07"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper argues that KHRONOS, a sum of tensorized per-coordinate B-spline modes, solves a 2D Poisson benchmark down to $6\\times10^{-11}$ squared L2 error at 512 degrees of freedom, while also handling regression and batch inversion.","keywords":["surrogate modeling","kernel expansion","B-splines","Poisson equation","Kolmogorov-Arnold networks","separable tensor modes","Gauss-Newton inversion","curse of dimensionality"],"falsifier":"Re-run the 20D Sobol-G benchmark with $\\sigma=0.01$ noise explicitly added to both training and test targets; if the test MSE is reproducibly below $10^{-4}$, the reported protocol is inconsistent. Separately, solve the 2D Poisson problem with a non-separable source such as $f(x,y)=\\exp(xy)$ and check whether the pre-asymptotic $\\mathrm{DoF}^{-6}$ scaling still appears or falls back to $\\mathrm{DoF}^{-4}$.","tokens_in":11425,"feed_emoji":"🧮","tokens_out":5681,"duration_ms":53672,"temperature":0.7,"pith_summary":"KHRONOS is a neural surrogate that represents a target field as a sum of separable modes, where each mode is a tensor product of per-coordinate expansions built from hierarchical quadratic B-spline kernels. The paper claims this single architecture handles three tasks: supervised regression, PDE solving, and inverse problems. On a 2D Poisson benchmark, the claimed squared L2 error falls from $5\\times10^{-4}$ at 16 degrees of freedom to $6\\times10^{-11}$ at 512, a more than 100-fold gain over KAN and about a million-fold gain over linear FEM at comparable degrees of freedom, with sub-millisecond full-field inference. The same ansatz matches or beats Random Forest, XGBoost, and MLP on 8D and 20D regression, and supports batched Gauss-Newton level-set inversion at sub-microsecond per-sample latency. A sympathetic reader would care because the result suggests that tensorized kernel expansions can sidestep the parameter explosion of dense networks while retaining differentiability and speed.","feed_headline":"KHRONOS claims Poisson errors down to 6e-11 at 512 DoF","feed_subtitle":"A sum of per-coordinate B-spline modes beats KAN and linear FEM while keeping inference sub-millisecond.","key_machinery":"The load-bearing object is a mode: each input coordinate $x_p$ passes through $L$ stacked kernel-expansion layers using quadratic B-spline kernels with compact support, producing a scalar feature $f_p^{(L)}(x_p)$; a mode multiplies one such feature per coordinate, $M_j(x)=\\prod_p f_{p,j}^{(L)}(x_p)$, and the surrogate is $\\hat u(x)=\\sum_j M_j(x)$. Because each mode is a product of one-dimensional functions, integrals of $\\hat u$ and its derivatives factor into products of one-dimensional inner products, which become entries of Gram matrices $G, G', H, H'$ and cross matrices $A, B$. This separability is what carries the argument: it turns the 2D variational energy into a trace formula and reduces the cost from $O(n^2)$ to roughly $O(n_{\\mathrm{gauss}}n_e(2M^2+MN))$ with $n_{\\mathrm{gauss}}, n_e, M, N \\ll n$.","core_discovery":"The central discovery is that hierarchical composition of per-dimension B-spline expansions, combined across dimensions by tensor products and superposed over a small number of modes, yields a differentiable surrogate with a favorable accuracy-versus-complexity tradeoff. The paper argues that because each mode is separable, the variational energy for a 2D Poisson problem collapses to Gram-matrix traces, $\\tfrac12(\\operatorname{tr}(G'^T H)+\\operatorname{tr}(H'^T G))-\\operatorname{tr}(A^T B)$, reducing the integral cost from $O(n^2)$ to $O(n_{\\mathrm{gauss}}n_e(2M^2+MN))$. This lets KHRONOS reach squared L2 errors of $5\\times10^{-4}$ at 16 degrees of freedom and $6\\times10^{-11}$ at 512, with empirical scaling laws of approximately $\\mathrm{DoF}^{-6}$ pre-asymptotically and $\\mathrm{DoF}^{-4}$ asymptotically. The paper further claims the same architecture achieves $R^2=0.9994$ on a noisy 20D Sobol-G benchmark with 1560 parameters and enables batched Gauss-Newton inversion at sub-microsecond per-sample latency.","pith_inferences":["Editorial inference: the steep pre-asymptotic $\\mathrm{DoF}^{-6}$ rate is probably tied to the benchmark's exact separability, $\\sin(\\pi x)\\sin(\\pi y^2)$; on non-separable fields the generic $\\mathrm{DoF}^{-4}$ rate is the likely ceiling.","Editorial inference: if the noisy-regression result survives a corrected noise protocol, the architecture behaves like a low-rank tensor decomposition of high-dimensional functions, so its real utility may lie in problems with moderate intrinsic dimensionality.","Editorial inference: a direct stress test is to replace the Poisson source with a non-separable function and to add noise to held-out test labels in the Sobol-G benchmark; both are cheap experiments that would separate the architecture's generic properties from benchmark-specific ones."],"forward_implications":["The same architecture can serve as a forward PDE solver, a supervised regression model, and an inverse solver, so a single differentiable representation could replace separate pipelines in control and monitoring applications.","If the Poisson error scaling holds, high accuracy becomes available at hundreds rather than thousands or millions of parameters, making full-field prediction feasible on constrained hardware.","Batched Gauss-Newton inversion at sub-microsecond per-sample latency would bring level-set recovery into real-time, online regimes for problems such as shape recovery and inverse design.","The claimed model-free results suggest that tensorized kernel modes can reach high $R^2$ with an order of magnitude fewer parameters than tree ensembles and MLPs on moderate-dimensional benchmarks."],"supporting_citations":[{"why":"Supplies the quadratic B-spline kernels and the partition-of-unity knot construction used in every per-coordinate expansion.","marker":"[19]"},{"why":"Provides the KAN baseline whose parameter-controlled L2 error scaling KHRONOS claims to beat by more than 100-fold.","marker":"[23]"},{"why":"Establishes existence and uniqueness of the energy minimizer in H0^1, justifying the variational training loss.","marker":"[24]"},{"why":"Supplies the Gauss-Newton iteration used for batched level-set inversion.","marker":"[20]"},{"why":"Provides the random-forest baseline in the 8D and 20D model-free comparisons.","marker":"[21]"},{"why":"Provides the XGBoost baseline in the model-free comparisons.","marker":"[22]"},{"why":"Defines the collocation-based physics-informed loss that KHRONOS contrasts with its Galerkin weak-form and separable integration approach.","marker":"[8]"},{"why":"Motivates the hierarchical composition of feature maps that underlies the stacked per-coordinate expansions.","marker":"[17]"}],"fun_headline_variants":["KHRONOS: 6e-11 Poisson error, 100x better than KAN","KHRONOS beats KAN by 100x, hits 6e-11 error","KHRONOS: sub-millisecond inference, 1e6x FEM improvement","KHRONOS: 20D Sobol-G solved with R^2=0.9994"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the 20D noisy-regression protocol is internally consistent: with additive Gaussian noise of $\\sigma=0.01$ in the targets, the best possible test MSE is $10^{-4}$, yet the paper reports $6.8\\times10^{-7}$, so either the held-out labels were not corrupted or the reported number is misstated.","fun_headline_variants_meta":{"raw":{"variants":["KHRONOS: 6e-11 Poisson error, 100x better than KAN","KHRONOS beats KAN by 100x, hits 6e-11 error","KHRONOS: sub-millisecond inference, 1e6x FEM improvement","KHRONOS: 20D Sobol-G solved with R^2=0.9994"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000648,"raw_usage":{"total_tokens":3038,"prompt_tokens":1074,"completion_tokens":1964,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":690,"completion_tokens_details":{"reasoning_tokens":1863}},"tokens_in":690,"tokens_out":1964,"duration_ms":12886,"temperature":1.0,"reasoning_tokens":1863,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:17:10.759812+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the 20D Sobol-G benchmark with $\\sigma=0.01$ noise explicitly added to both training and test targets; if the test MSE is reproducibly below $10^{-4}$, the reported protocol is inconsistent. Separately, solve the 2D Poisson problem with a non-separable source such as $f(x,y)=\\exp(xy)$ and check whether the pre-asymptotic $\\mathrm{DoF}^{-6}$ scaling still appears or falls back to $\\mathrm{DoF}^{-4}$.","supporting_citations":[{"cited_title":"Springer, 2nd edition, 1997","cited_arxiv_id":null,"evidence_quote":"Supplies the quadratic B-spline kernels and the partition-of-unity knot construction used in every per-coordinate expansion."},{"cited_title":"World Scientific, 2003","cited_arxiv_id":null,"evidence_quote":"Establishes existence and uniqueness of the energy minimizer in H0^1, justifying the variational training loss."}],"review_version":1}