---
title: Multi-scale Adaptive KAN (MAN)
url: https://www.emergentmind.com/topics/multi-scale-adaptive-kan-man
type: topic
---

# Multi-scale Adaptive KAN (MAN)

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 [2509.22444]. 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 [2601.18672]. 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 [2508.01575]. 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 [2509.22444]. 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) [2601.18672]. 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 [2508.01575; 2509.22444].

## 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” [2509.22444]. 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 [2509.22444].

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 $1\times1$, $3\times3$, and $5\times5$ [2509.22444]. 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 [2509.22444].

## 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 [2601.18672], knot placement in spline-based KANs is recast as a density-estimation problem driven by Importance Density Functions. For a batch of samples $\{x_d^{(s)}\}_{s=1}^{N_b}$ in input dimension $d$, each sample receives a weight $w^{(s)}$, producing an empirical IDF and weighted empirical CDF, after which knot positions are allocated by CDF inversion:
$$
t_m = \hat{F}_d^{-1}(q_m), \qquad q_m = \frac{m}{G+1}.
$$
The paper’s curvature-based strategy computes weights from the sum of absolute second partial derivatives along a coordinate, with a small $\epsilon$ 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 [2601.18672].

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
$$
\sigma = \exp(\tilde{\sigma})
$$
for positive trainable smoothness and a bounded reparameterization for centroids [2601.07760]. 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 [2601.07760].

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 $7.2 \pm 2.3$ rules while handling locally complex or discontinuous functions that are difficult for global KANs [2505.14273]. 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 [2509.22444] | Dual-branch MAN with KANBlock + MSAB, fused by learnable $w_1,w_2$ | IoU gains on BUSI, GLAS, CVC |
| KANMixer [2508.01575] | Multi-scale mixing backbone for LTSF | State-of-the-art in 16/28 experiments |
| PoseKAN [2511.08809] | Learnable functions on graph edges + multi-hop aggregation | Competitive 3D pose estimation |
| KAN-FPN-Stem [2512.22228] | KAN-based smoothing at final FPN fusion stage | Up to +2.0 AP over ViTPose-S |
| KANMultiSign [2605.09572] | 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 [2508.01575].

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,
$$
\mathbf{H}^{(\ell+1)}=\text{KAN}^{(\ell)}\left(\left((1-s)\hat{\mathbf{A}}+s\hat{\mathbf{A}}^2\right)\mathbf{H}^{(\ell)}+\mathbf{X}\right),
$$
mixes one-hop and two-hop neighborhoods through a tunable scaling parameter $s$, thereby introducing a graph-scale analogue of MAN [2511.08809].

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 $3\times3$ 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 [2512.22228].

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 [2605.09572].

## 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 [2509.22444]. 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 [2509.22444].

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 [2508.01575]. 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 [2511.08809].

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 [2605.09572].

## 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 [2509.22444]. In KAN-FPN-Stem, the intervention is concentrated at the final fusion-smoothing interface of an FPN rather than spread across the network [2512.22228]. In KANMultiSign, the primary gain is attributed to multi-scale supervision, with KAN contributing compactness rather than being the main driver of accuracy [2605.09572].

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 [2508.01575; 2601.18672; 2511.08809; 2512.22228; 2605.09572]. 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 [2508.01575; 2601.18672; 2505.14273; 2512.22228]. 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.

Source: https://www.emergentmind.com/topics/multi-scale-adaptive-kan-man