{"id":"698174a1-5d29-4c2b-929b-22dc9c9945e8","arxiv_id":"2506.07549","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"MetaKANs generates each KAN activation function from a shared prompt-conditioned meta-learner, cutting trainable parameters toward MLP level while retaining comparable or better accuracy on tested benchmarks.","lead":"This paper replaces the many per-connection coefficient vectors in Kolmogorov-Arnold Networks with outputs of a small meta-network that reads a one-number prompt per connection. The resulting MetaKANs use far fewer trainable parameters and less memory, while keeping accuracy on regression, PDE, and classification benchmarks.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Scalar-prompt assumption (Eq. 9) is contradicted by the paper's own ablation: the best CIFAR-100 results use 2–4 dimensional prompts, not z in R.","rationale":"The reader identified the scalar-prompt sufficiency as the weakest assumption. My stress-test concurs and sharpens it with direct evidence from the manuscript: the method as defined in Eq. (9) uses z in R, but the experimental section explicitly uses dim(z)=2 or 4 for the best-performing deep networks, and the ablation in Table 9 shows a monotone accuracy benefit from increasing dim(z). This internal inconsistency is more load-bearing than the Table 8 parameter-count typo or the missing memory-measurement protocol, because it challenges the very mechanism that justifies the parameter-count formula and the memory-efficiency narrative. If a scalar prompt were sufficient, the model's performance should not improve when prompt dimension grows; it does, and the improvement is large (e.g., ~8 points on CIFAR-100 for MetaFastKAN). Consequently, the paper's headline '1/3 to 1/9 parameters' does not apply to the actual models that attain the reported accuracy, and the conceptual grounding of the meta-learner as learning a scalar-identified function class is not validated. The concrete test I propose (retraining with dim(z)=1) would settle whether the scalar-prompt mechanism can work at all in these settings. Since the reader already conditioned the verdict on fixing these issues, my analysis does not move the verdict; it strengthens the condition and provides a specific falsifiable check. I do not see an internal logical contradiction in the parameter-count derivation itself when dim_z=1; the problem is the disconnect between the theoretical formulation and the experimental implementation. That is a correctness concern, not a matter of outside consensus, so it is appropriate to require the authors to reconcile the formulation with the experiments. No ad hominem is intended; the critique is on the argument and its evidence.","tokens_in":24905,"tokens_out":7307,"duration_ms":85453,"concrete_test":"Retrain the 8-layer MetaKANConv and MetaFastKANConv models on CIFAR-100 using exactly the best-reported configurations (C, d_hidden, grid G) but with dim(z)=1 only, keeping all other hyperparameters fixed. If accuracy drops to the level of the dim=1 rows in Table 9 (e.g., ~30–44% for MetaKAN, ~40–44% for MetaFastKAN) and remains below the dim=2/4 results, the scalar-prompt assumption is empirically refuted. Additionally, recompute the actual parameter counts for the reported models using sum_l (n_l*n_{l+1})*dim_z + C*(d_hidden+1)*(G+k+1) with dim_z=2 and 4, and verify whether the claimed '1/3 to 1/9' reduction versus KAN still holds for the exact models that produced the reported accuracies.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central parameter-reduction claim (Table 1, abstract '1/3 to 1/9 parameters') rests on the formulation in Section 3.2.2, Eq. (9), where the task prompt z is a single scalar in R, giving prompt cost sum_l (n_l*n_{l+1}) and meta-learner cost C*(d_hidden+1)*(G+k+1). If scalar prompts were sufficient identifiers of the univariate function class, this would be a clean win. However, the paper's own experiments undermine this assumption. The best 8-layer results in Table 3 use dim(z)=2 for MetaKANConv and dim(z)=4 for MetaFastKANConv (footnote), and the ablation in Table 9 shows accuracy rising sharply with prompt dimension: MetaFastKAN at C=7 goes from 44.16% (dim=1) to 52.02% (dim=4). This directly demonstrates that a single scalar prompt is not sufficient to represent the required weight space for the tasks where the method is actually competitive. Moreover, the reported parameter counts for those models are consistent with the higher prompt dimension: e.g., 8-layer MetaKANConv has ~9.1M parameters, while the scalar-prompt formula would give ~4.5M. Thus the headline parameter-efficiency and memory claims are overstated for the configurations that deliver the stated accuracy; the theoretical formula only applies to a special case that the experiments do not validate. This is not a question of tuning or implementation but of the core premise: the shared meta-learner with scalar identifiers is empirically insufficient, contradicting the paper's central mechanism as defined in Eq. (9).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces MetaKAN, a meta-learning framework that generates KAN edge weights with a smaller meta-learner instead of optimizing each spline coefficient directly. A scalar learnable prompt is associated with each activation, and a two-layer MLP meta-learner maps that prompt to the spline coefficient vector. The claimed parameter count drops from sum_l (n_l n_{l+1})(G+k+1) to sum_l (n_l n_{l+1}) plus a meta-learner cost, with a layer-clustering extension for deep networks. The method is applied to standard KANs, FastKAN, WavKAN, and ConvKAN, and experiments cover symbolic regression, high-dimensional function fitting, PDEs, and image classification. The paper reports comparable or better accuracy than the corresponding KAN baselines while using substantially fewer parameters, and the code is released.","tokens_in":25325,"tokens_out":5844,"duration_ms":65927,"significance":"If the parameter-efficiency claims hold, this is a practically useful contribution: it gives a simple, model-agnostic recipe for reducing KAN memory overhead and narrowing the training-cost gap with MLPs. The parameter-count arithmetic in Sections 3.2.2 and 3.3 is transparent and checkable, the experiments span several KAN variants and application domains, and the code availability is a concrete strength. However, the central parameter-reduction formula is stated only for scalar prompts, while the experiments that produce the best accuracy use prompt dimensions 2 and 4; the paper never reconciles this with the headline formulas. There is also an algebraic sign error in the memory-efficiency condition and an apparent inconsistency in the PDE parameter table. These issues affect the load-bearing claim that MetaKAN scales to MLP-level memory usage, and they need to be corrected before the contribution can be fully assessed.","major_comments":[{"comment":"The paper's central parameter-count formula assumes a scalar prompt z in R, giving |Z| = sum_l (n_l n_{l+1}) and meta-learner cost C*(d_hidden+1)*(G+k+1). However, Section 4.2.1 states that the 8-layer MetaKANConv and MetaFastKANConv used in Table 3 set the prompt dimension to 2 and 4, and Table 9 shows that the best CIFAR-100 accuracies occur at dim_z=2 or 4 (e.g., MetaFastKAN at C=7 improves from 44.16% at dim_z=1 to 52.02% at dim_z=4, with parameters increasing from 4,576,073 to 18,140,617). For dim_z=d, the prompt term should be d * sum_l (n_l n_{l+1}) and the meta-learner input layer has d*d_hidden additional parameters, so the formulas in Table 1 and Section 3.2.2 are not applicable to the reported best configurations. The headline '1/3 to 1/9 parameters' claim therefore needs to be restated with the correct dimension-dependent formula, and the main text should explain when the scalar special case is actually used.","section":"Section 3.2.2, Eq. (9); Table 1; Table 9"},{"comment":"The stated condition for memory efficiency is algebraically reversed. With S = sum_l (n_l n_{l+1}) and m = G+k+1, MetaKAN has S + C(d_hidden+1)m parameters and KAN has mS. Requiring MetaKAN < KAN gives (d_hidden+1)m < (m-1)S, i.e., d_hidden+1 < (G+k)/(G+k+1) * S, not d_hidden ≳ (G+k)/(G+k+1) * S as written. Since this inequality is the paper's explicit justification for when MetaKAN improves memory efficiency, the error should be corrected and the surrounding discussion adjusted accordingly.","section":"Section 3.4"},{"comment":"In the 100-dimensional Allen-Cahn row, both KAN and MetaKAN are listed with 47,520 parameters. This is inconsistent with the method: for the structure [100,32,32,32,1], the KAN count equals (100*32 + 32*32 + 32*32 + 32*1)*(G+k+1) = 5,280*9, and a MetaKAN with any reasonable meta-learner should have far fewer than 47,520 parameters (the neighboring 100D rows report 6,697). The identical count suggests a typo or a misconfigured meta-learner, and it directly undermines the PDE parameter-efficiency claim in Table 8. The entry needs to be corrected or, if 47,520 is genuinely the value, the configuration and formula must be explained.","section":"Table 8"}],"minor_comments":[{"comment":"The heading contains a typo: 'UNDERSTANING' should be 'UNDERSTANDING'.","section":"Section 3.2.1, heading"},{"comment":"The loss expression 'ℓ(MetaKAN(x;Z,θ)−f(x))' should be 'ℓ(MetaKAN(x;Z,θ), f(x))' to match the usual supervised-loss notation used elsewhere.","section":"Eq. (12)"},{"comment":"The line 'Compute the output of MetaKAN using Eq. (3.2.2)' references an equation number that does not exist; it should refer to Eq. (10) or the relevant forward formula.","section":"Algorithm 1, line 5"},{"comment":"The FMNIST 32,32 row contains the stray text '85.81S'; this should be cleaned up.","section":"Table 5"},{"comment":"The text first says 'across 16 Feynman equations' and then reports 'lower MSE in 11 out of 17'; the counts should be made consistent.","section":"Section 4.1.2"},{"comment":"The caption says 'pairwise absolute distances' while the text discusses 'similarity'; clarify whether darker regions indicate smaller distance or higher similarity.","section":"Figure 10 caption"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this if you care about KANs. It is a clean application of hypernetworks to generate KAN activation weights, and the parameter-count identity is real: instead of storing (G+k+1) coefficients per edge, you store one prompt per edge plus a small meta-learner. The arithmetic in Table 1 checks out, and the experiments are broad—Feynman symbolic regression, high-dimensional fitting, PDEs, and ConvKAN/FastKAN/WavKAN variants on six image datasets. On many tasks the accuracy is competitive or better with large parameter reductions. The layer-clustering trick for deep KANs is a sensible engineering choice, and the code is public.\n\nThe soft spots are serious but fixable. The central formulation in Eq. (9) uses a scalar prompt z in R, and the abstract's \"1/3 to 1/9 parameters\" is computed for that case. But the paper's own ablation (Table 9) shows accuracy climbing sharply with prompt dimension: MetaFastKAN goes from 44.16% (dim=1) to 52.02% (dim=4) at C=7, and the best 8-layer ConvKAN results use dim=2 or 4. With dim=4 the parameter count doubles relative to scalar, so the headline reduction is overstated for the configurations that actually deliver the reported accuracy. The method still gives a real reduction, just not the 1/9 claimed in the abstract. This needs an honest rewrite of the parameter-count formula to include dim(z), plus results reported at the dimensions actually used.\n\nThere is also a clear data error in Table 8: for Allen-Cahn at 100D, KAN and MetaKAN are both listed at 47,520 parameters, which cannot be right under the method's own counting. And the memory-efficiency claim (Figure 4) has no measurement protocol—batch size, whether optimizer states are counted, how peak memory was obtained. As written, the memory argument is not independently checkable.\n\nOn citations: the paper appropriately cites Ha et al., Zhao et al., and Navon et al.; the novelty is the application to KANs plus the per-activation prompt and cluster partitioning, not the hypernetwork idea itself. That is fine, but it should be framed as an application.\n\nBottom line: the core mechanism holds up and the experiments are extensive, but the paper overclaims in the abstract and has at least one inconsistent table entry. A serious referee should engage with it—the revision path is clear and the method is likely to be useful for the KAN community. I would send it to review, with the expectation of a major revision.","headline":"A real hypernetwork-for-KANs idea with broad experiments, but the scalar-prompt formulation in Eq. (9) is not what the best results use, making the headline parameter reduction overstated.","tokens_in":25823,"tokens_out":4546,"would_cite":true,"duration_ms":48064,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","41A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"KANs' trainable parameters can be cut to roughly MLP level by replacing direct weight optimization with a small meta-learner that generates every spline coefficient from a per-connection scalar prompt, while the paper reports matching or…","keywords":["Kolmogorov-Arnold networks","meta learning","hypernetworks","weight generation","memory efficiency","B-spline activations","learnable prompts","parameter efficiency"],"falsifier":"Train a MetaKAN and a direct KAN on a target where two edges in the same layer must represent sharply different univariate functions, such as one highly oscillatory edge and one nearly constant edge, using the same optimization budget. If the shared meta-learner with scalar prompts cannot match the direct KAN's accuracy at any hidden width, the scalar-prompt assumption fails.","tokens_in":24720,"feed_emoji":"🧠","tokens_out":9823,"duration_ms":96437,"temperature":0.7,"pith_summary":"Kolmogorov-Arnold Networks (KANs) replace fixed activations with learnable spline functions, so each connection carries $G+k+1$ coefficients and a KAN has far more trainable parameters than an MLP of the same shape. This paper proposes MetaKANs, which do not optimize those coefficients directly. Instead, a two-layer MLP meta-learner generates the coefficients for every activation from a single learnable scalar prompt per connection. The parameter count falls from $\\sum_{l=0}^{L-1}(n_l n_{l+1})(G+k+1)$ to $\\sum_{l=0}^{L-1}(n_l n_{l+1}) + C(d_{\\mathrm{hidden}}+1)(G+k+1)$, where $C$ is the number of layer clusters, which is approximately the parameter count of an MLP, and experiments on symbolic regression, PDE solving, and image classification report comparable or superior accuracy. The same recipe is applied to FastKAN, WavKAN, and ConvKAN, so the memory saving transfers across KAN variants.","feed_headline":"Meta-learner generates KAN weights, cutting parameters to MLP level","feed_subtitle":"Kolmogorov-Arnold networks keep accuracy while trainable parameters drop to roughly one per connection.","key_machinery":"The load-bearing object is the weight-generation map $M_\\theta(z)=w$, a two-layer MLP with hidden dimension $d_{\\mathrm{hidden}}$ that takes the scalar prompt of one activation and outputs its $G+k+1$ spline coefficients. The prompt $z$ is the learned identifier of the univariate function, the meta-learner is the shared rule for translating identifiers into coefficients, and the activation becomes $\\phi(t; z,\\theta)=M_\\theta(z)^\\top B(t)$. For deep KANs the machinery adds a clustering step that groups layers by their channel sizes and assigns each cluster a separate $M_\\theta^{(c)}$, trading a small constant parameter overhead for the ability to learn layer-dependent rules.","core_discovery":"The paper's central claim is that the univariate functions inside a KAN are not independent tasks: they all belong to a shared functional class $\\mathcal{F} = \\{f \\mid f(t)=w^\\top B(t), w\\in\\mathbb{R}^{G+k+1}\\}$, so the rule that maps a function's identity to its spline coefficients can be learned once and reused. MetaKANs implement that rule as $M_\\theta: \\mathbb{R}\\to\\mathbb{R}^{G+k+1}$, with a learnable scalar prompt $z$ for each activation standing in for the function's identity. Training proceeds end-to-end on the original loss, updating both prompts and meta-learner. For deep KANs, layers with similar channel dimensions are clustered by width and each cluster receives its own meta-learner, keeping cost under control. The claimed outcome is that the generated weights match or beat directly optimized KAN weights while the trainable parameter count drops to approximately that of an MLP.","pith_inferences":["If scalar prompts truly identify the needed univariate functions, then prompt geometry is a compact summary of what the network learned; a natural test is whether interpolation or clustering of prompts transfers between tasks or architectures.","The meta-learner imposes a strong shared-structure prior, which may explain the accuracy gains and improved compactness; a direct check is whether MetaKANs beat KANs on out-of-distribution inputs, not just on the training distribution.","The layer-clustering rule suggests that the number of meta-learners should track the diversity of layer widths rather than depth itself, so adding layers within a width cluster could add almost no parameters.","A concrete extension is to train the meta-learner once on a family of tasks and then freeze it, using only the prompts as a task-specific interface; the paper leaves this transfer setting implicit."],"forward_implications":["Peak memory during training falls to near-MLP levels, with the largest savings reported at large grid sizes such as $G=80$ and above.","The meta-learner recipe is model-agnostic: applying it to FastKAN, WavKAN, and ConvKAN cuts their parameter counts by roughly $1/\\dim(w)$ while keeping competitive accuracy.","Deep KANs become more scalable, because the meta-learner cost $C(d_{\\mathrm{hidden}}+1)(G+k+1)$ stays small and fixed while the prompt count grows only with the number of connections.","For very small KANs the fixed meta-learner cost can exceed the spline parameters it replaces, so MetaKANs can have slightly more parameters than KANs in that regime even when accuracy stays competitive."],"supporting_citations":[{"why":"Supplies the representation theorem that motivates decomposing a multivariate target into univariate functions.","marker":"[Kolmogorov, 1957]"},{"why":"Defines the original KAN architecture, its B-spline parameterization, and the baseline that MetaKANs compress.","marker":"[Liu et al., 2024]"},{"why":"Establishes the hypernetwork mechanism of using a small auxiliary network to generate a larger network's weights.","marker":"[Ha et al., 2017]"},{"why":"Provides the FastKAN RBF-based variant whose weights MetaFastKAN generates.","marker":"[Li, 2024]"},{"why":"Provides the WavKAN wavelet variant whose weights MetaWavKAN generates.","marker":"[Bozorgasl & Chen, 2024]"},{"why":"Introduces the convolutional KAN architecture that MetaKANConv compresses.","marker":"[Bodner et al., 2024]"},{"why":"Extends KA convolutions with the KANConv/FastKANConv/KAGNConv/KALNConv modules used in the image experiments.","marker":"[Drokin, 2024]"},{"why":"Supplies the Feynman symbolic-regression benchmark used to compare fitting accuracy.","marker":"[Udrescu & Tegmark, 2020]"}],"fun_headline_variants":["MetaKAN trims KAN parameters to MLP levels","Meta-learning compresses KANs, matching MLP parameter counts","Generate KAN weights via meta-learner to slash memory","KANs become memory-efficient with meta-generated weights","MetaKAN: KAN parameter count drops to MLP scale"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"A single scalar prompt per activation carries enough information to identify the correct univariate function, so one shared meta-learner can generate near-optimal coefficients for every edge.","fun_headline_variants_meta":{"raw":{"variants":["MetaKAN trims KAN parameters to MLP levels","Meta-learning compresses KANs, matching MLP parameter counts","Generate KAN weights via meta-learner to slash memory","KANs become memory-efficient with meta-generated weights","MetaKAN: KAN parameter count drops to MLP scale"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00022,"raw_usage":{"total_tokens":1459,"prompt_tokens":969,"completion_tokens":490,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":585,"completion_tokens_details":{"reasoning_tokens":404}},"tokens_in":585,"tokens_out":490,"duration_ms":5819,"temperature":1.0,"reasoning_tokens":404,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:31:14.945468+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train a MetaKAN and a direct KAN on a target where two edges in the same layer must represent sharply different univariate functions, such as one highly oscillatory edge and one nearly constant edge, using the same optimization budget. If the shared meta-learner with scalar prompts cannot match the direct KAN's accuracy at any hidden width, the scalar-prompt assumption fails.","supporting_citations":[{"cited_title":"Unlike conventional convolution kernels, Kolmogorov- Arnold (KA) kernels consist of a set of univariate non- linear learnable activation functions","cited_arxiv_id":null,"evidence_quote":"Extends KA convolutions with the KANConv/FastKANConv/KAGNConv/KALNConv modules used in the image experiments."}],"review_version":1}