Multi-scale Adaptive KAN (MAN)

• Multi-scale Adaptive KAN (MAN) is a framework that integrates adaptive KAN blocks with explicit multi-scale attention to redistribute representational capacity across different scales.
• Its canonical form in U-MAN employs a dual-branch module in U-Net-like architectures to fuse refined nonlinear features with multi-scale spatial extraction for improved segmentation.
• Empirical results demonstrate that MAN's adaptive fusion consistently yields gains in performance metrics such as IoU, while efficiently handling diverse tasks from imaging to forecasting.

Multi-scale Adaptive KAN (MAN) denotes a family of Kolmogorov-Arnold Network (KAN) designs in which adaptivity is coupled to explicit scale handling. In the narrow sense used by U-MAN, MAN is a dual-branch module inserted into the deep encoder of a U-Net-like segmentation architecture, where a KAN branch and a multi-scale attention branch are fused by learnable weights (Huang et al., 26 Sep 2025). In a broader sense, later KAN literature uses the same phrase for training-aware grid adaptation, adaptive basis allocation, and coarse-to-fine or multi-hop constructions that redistribute representational capacity across spatial, temporal, or graph scales (Rigas et al., 26 Jan 2026). Across these usages, the unifying idea is not merely the replacement of fixed activations with learnable univariate functions, but the coordinated adaptation of basis functions, neighborhoods, or fusion pathways so that local complexity at different scales can be modeled more effectively than with flat MLP or fixed-grid formulations.

1. Conceptual basis and terminology

KANs are motivated by the Kolmogorov-Arnold representation theorem and operationalize it by learning adaptive basis functions for each input dimension rather than relying on fixed nonlinearities such as ReLU or GELU. A representative layer expression given in the KANMixer work is

$$y=\sum_{i=1}^{n} g_i\left(\sum_{j=1}^{m} f_{ij}(x_j)\right),$$

where the $f_{ij}$ are learnable univariate functions and the $g_i$ are combining functions (Jiang et al., 3 Aug 2025). This construction is repeatedly associated in the cited literature with granular or local modulation of nonlinearities, local function shaping, and improved flexibility relative to conventional MLP blocks.

The designation “MAN” is not attached to a single universally standardized architecture. In U-MAN, it is the explicit name of a module added to a medical image segmentation network (Huang et al., 26 Sep 2025). In “A Dynamic Framework for Grid Adaptation in Kolmogorov-Arnold Networks,” the conclusion refers to a “Multi-scale Adaptive KAN (MAN) framework” in the sense of generalized grid adaptation via Importance Density Functions (IDFs) (Rigas et al., 26 Jan 2026). This suggests two levels of usage: MAN as a concrete architectural block, and MAN as a broader design principle for multi-scale, training-aware KAN adaptation.

A recurring technical motivation is the same across tasks: conventional backbones are described as limited by flat architectures, simple skip connections, local receptive fields, naive patchification, or fixed-grid basis choices. MAN-style constructions address these limits by introducing explicit multi-scale decomposition, hierarchical locality, multi-hop aggregation, adaptive grid allocation, or coarse-to-fine supervision (Jiang et al., 3 Aug 2025, Huang et al., 26 Sep 2025).

2. Canonical MAN formulation in U-MAN

The most explicit definition of MAN appears in “U-MAN: U-Net with Multi-scale Adaptive KAN Network for Medical Image Segmentation” (Huang et al., 26 Sep 2025). There, MAN is an enhancement block placed in the deeper stages of the encoder to address two limitations attributed to conventional U-Nets: simple skip connections that ignore the encoder-decoder semantic gap, and insufficient multi-scale feature extraction in deep layers.

Its architecture is dual-branch. Given an embedded feature tensor $$\mathbf{X}_{\text{embed}} \in \mathbb{R}^{H_{\text{embed}} \times W_{\text{embed}} \times C}$$, the KAN branch applies a KANBlock and the second branch applies a Multi-Scale Attention Block (MSAB):

$$\mathbf{F}_{\text{kan}} = \text{KANBlock}(\mathbf{X}_{\text{embed}}),$$

$$\mathbf{F}_{\text{msab}} = \text{MSAB}(\mathbf{X}_{\text{embed}}).$$

The two outputs are fused by trainable scalar weights:

$$\mathbf{F}_{\text{man}} = w_1 \otimes \mathbf{F}_{\text{msab}} + w_2 \otimes \mathbf{F}_{\text{kan}},$$

where $w_1$ and $w_2$ are learnable parameters and $\otimes$ denotes elementwise scaling with scalar broadcasting (Huang et al., 26 Sep 2025).

The two branches have distinct roles. The KAN branch is described as yielding refined nonlinear features through a KANBlock built from LayerNorm, a KAN layer, and DropPath. The MSAB branch provides explicit multi-scale spatial extraction through convolutions and attention, including Multi-Scale Depthwise Convolution with kernels such as $f_{ij}$0, $f_{ij}$1, and $f_{ij}$2 (Huang et al., 26 Sep 2025). The adaptive fusion step allows the network to rebalance semantic/nonlinear and spatial/multi-scale signals during training.

Within U-MAN, MAN operates jointly with Progressive Attention-Guided Feature Fusion (PAGF), which replaces the simple skip connection. The resulting design uses conventional shallow encoder-decoder processing at early stages while replacing deeper encoder blocks with MAN modules, thereby shifting multi-scale adaptive processing toward the semantic stages of the network (Huang et al., 26 Sep 2025).

3. Adaptation mechanisms beyond the canonical module

The broader MAN literature extends the notion of multi-scale adaptation from branch fusion to basis allocation itself. In the grid-adaptation framework of (Rigas et al., 26 Jan 2026), knot placement in spline-based KANs is recast as a density-estimation problem driven by Importance Density Functions. For a batch of samples $f_{ij}$3 in input dimension $f_{ij}$4, each sample receives a weight $f_{ij}$5, producing an empirical IDF and weighted empirical CDF, after which knot positions are allocated by CDF inversion:

$f_{ij}$6

The paper’s curvature-based strategy computes weights from the sum of absolute second partial derivatives along a coordinate, with a small $f_{ij}$7 added for positivity. Reported average relative error reductions are 25.3% on synthetic functions, 9.4% on the Feynman dataset, and 23.3% on Helmholtz PDE benchmarks, with Wilcoxon signed-rank tests supporting statistical significance for the synthetic and Feynman settings (Rigas et al., 26 Jan 2026).

A related but distinct route is the replacement of spline bases with adaptive radial basis functions. Free-RBF-KAN learns both centroid locations and smoothness parameters jointly with network weights, using

$f_{ij}$8

for positive trainable smoothness and a bounded reparameterization for centroids (Chiu et al., 12 Jan 2026). The paper reports that Free-RBF-KAN achieves accuracy comparable to the original B-spline-based KAN while delivering faster training and inference, and that adaptive mesh and adaptive shape are both active ingredients in multiscale function approximation, physics-informed machine learning, and PDE operator learning (Chiu et al., 12 Jan 2026).

Another form of adaptive scale handling is explicit domain partitioning. X-KAN does not build a single global MAN block; instead, it evolves multiple local KANs within XCSF, assigning a dedicated KAN to each rule-defined input region. Inference is performed by the highest-fitness matching rule, and the system reportedly uses a compact set of rules, averaging $f_{ij}$9 rules while handling locally complex or discontinuous functions that are difficult for global KANs (Shiraishi et al., 20 May 2025). A plausible implication is that MAN-style adaptivity can emerge either from intra-layer scale mechanisms or from adaptive partitioning of the input space.

4. Architectural variants across application domains

The main application areas represented in the cited literature are long-term time series forecasting, medical image segmentation, human pose estimation, ViT-based pose estimation, and sign-language-driven pose generation.

Work	Multi-scale/adaptive mechanism	Reported outcome
U-MAN (Huang et al., 26 Sep 2025)	Dual-branch MAN with KANBlock + MSAB, fused by learnable $g_i$0	IoU gains on BUSI, GLAS, CVC
KANMixer (Jiang et al., 3 Aug 2025)	Multi-scale mixing backbone for LTSF	State-of-the-art in 16/28 experiments
PoseKAN (Shahjahan et al., 11 Nov 2025)	Learnable functions on graph edges + multi-hop aggregation	Competitive 3D pose estimation
KAN-FPN-Stem (Tang, 23 Dec 2025)	KAN-based smoothing at final FPN fusion stage	Up to +2.0 AP over ViTPose-S
KANMultiSign (Du et al., 10 May 2026)	Coarse-to-fine supervision with KAN-FFN in Transformer	Lower DTW-based joint error with fewer parameters

KANMixer frames multi-scale adaptation as the modeling core for long-term time series forecasting rather than as an auxiliary module. Its architecture integrates a multi-scale mixing backbone designed to exploit KAN’s adaptive basis functions, and the paper reports state-of-the-art performance in 16 of 28 experiments across seven benchmark datasets (Jiang et al., 3 Aug 2025).

PoseKAN adapts KANs to graphs by placing learnable univariate functions on graph edges and combining them with multi-hop propagation. The propagation rule given in the summary,

$g_i$1

mixes one-hop and two-hop neighborhoods through a tunable scaling parameter $g_i$2, thereby introducing a graph-scale analogue of MAN (Shahjahan et al., 11 Nov 2025).

KAN-FPN-Stem applies KAN adaptivity not to token mixing or graph propagation but to multi-scale feature fusion. It retains the classic upsample-and-add stream of an FPN while replacing the terminal linear $g_i$3 smoothing convolution with a KAN-based convolutional layer. The reported ablation identifies the post-fusion smoothing step, rather than plug-and-play attention modules such as CBAM, as the critical bottleneck, and the best variant improves ViTPose-S from 72.5 AP to 74.5 AP on COCO (Tang, 23 Dec 2025).

KANMultiSign combines coarse-to-fine multi-scale supervision with KAN-based feed-forward replacements inside a Transformer. Its key result is interpretively important: multi-scale supervision is identified as the main source of accuracy gains, whereas KAN mainly reduces parameter count while maintaining competitive performance under that supervision (Du et al., 10 May 2026).

5. Empirical profile and ablation evidence

The clearest direct evidence for MAN itself comes from U-MAN. On BUSI, GLAS, and CVC, IoU increases from 64.99 to 66.12, from 87.74 to 89.07, and from 82.69 to 86.14 when moving from U-KAN to U-MAN (Huang et al., 26 Sep 2025). In the ablation study on CVC-ClinicDB, removing MAN yields 83.82 IoU, removing PAGF yields 84.91, and the full model reaches 86.14. Within the MAN module, “KAN only” gives 85.67 IoU, a single-scale convolution gives 85.49, and the full MSAB-based multi-scale version gives 86.14; a KAN depth of 3 is reported as optimal, with deeper configurations degrading because of overfitting (Huang et al., 26 Sep 2025).

The other domains reinforce the same pattern: performance gains are rarely attributable to KAN substitutions alone. KANMixer reports that the adaptive flexibility of learnable basis functions changes the influence of network structural prior on forecasting performance and identifies critical design factors for accuracy (Jiang et al., 3 Aug 2025). PoseKAN reports improvements alongside specific architectural additions—multi-hop aggregation, residual PoseKAN blocks, and global response normalization—rather than from edge-wise KAN functions in isolation. Its reported results include 46.7 mm MPJPE on Human3.6M Protocol #1, 38.3 mm PA-MPJPE on Protocol #2, 86.0% PCK on MPI-INF-3DHP, and 5.72M parameters versus more than 9M for GraphMLP (Shahjahan et al., 11 Nov 2025).

KANMultiSign provides a particularly explicit caution against over-attribution. The model improves over a strong notation-to-pose baseline across several sign languages, but the paper states that KAN-based variants “substantially reduce parameter count while maintaining competitive performance when coupled with multi-scale supervision, rather than serving as the main driver of accuracy gains.” The reported parameter count drops from 3.8M to 1.7M in the KAN-based version, with comparable DTW metrics in the multi-scale setting (Du et al., 10 May 2026).

6. Interpretive issues, misconceptions, and methodological significance

A common misconception is that MAN simply means “use KAN instead of an MLP.” The cited literature does not support that reduction. In U-MAN, the decisive element is the adaptive fusion of a KAN branch with an explicit multi-scale attention branch (Huang et al., 26 Sep 2025). In KAN-FPN-Stem, the intervention is concentrated at the final fusion-smoothing interface of an FPN rather than spread across the network (Tang, 23 Dec 2025). In KANMultiSign, the primary gain is attributed to multi-scale supervision, with KAN contributing compactness rather than being the main driver of accuracy (Du et al., 10 May 2026).

A second misconception is that MAN is inherently tied to one data modality. The surveyed works instead show scale adaptation appearing in temporal mixing for forecasting, grid reallocation for scientific machine learning, graph-hop propagation for 3D pose estimation, pyramid fusion for vision transformers, and coarse-to-fine supervision for sequence-conditioned animation (Jiang et al., 3 Aug 2025, Rigas et al., 26 Jan 2026, Shahjahan et al., 11 Nov 2025, Tang, 23 Dec 2025, Du et al., 10 May 2026). This suggests that MAN is best understood as a design pattern in which KAN-style adaptive univariate operators are coupled to an explicit mechanism for redistributing resolution or contextual range.

Methodologically, the literature converges on a narrow set of design lessons. Scale selection matters in forecasting; curvature- or training-aware criteria can improve grid placement; local partitioning can address discontinuities that global continuous KANs cannot uniformly approximate; and KAN-enhanced components are most effective when inserted at structurally meaningful bottlenecks such as deep encoder stages, final multi-scale fusion points, or supervised coarse-to-fine transitions (Jiang et al., 3 Aug 2025, Rigas et al., 26 Jan 2026, Shiraishi et al., 20 May 2025, Tang, 23 Dec 2025). The accumulated evidence therefore treats MAN less as a monolithic architecture than as a principled combination of adaptive basis learning with explicit multi-scale organization.