Shape-Adapting Hub (SA-Hub) Overview

• SA-Hub is an adaptive interface module that dynamically reconciles heterogeneous domains through tunable geometry and mechanics.
• In neural architectures, SA-Hub enables dynamic routing between convolutional and transformer features, improving segmentation and feature fusion.
• In fluid and kirigami applications, it optimizes morphing and localized deformation, yielding enhanced flow efficiency and structural resilience.

A Shape-Adapting Hub (SA-Hub) denotes a class of functional coupling or interface modules whose geometry, internal mechanics, or representational form adapt to mediate between heterogeneous domains or structures. Across application domains, the SA-Hub architecture provides adaptive reconfiguration of shape, function, or information flow, enabling either structural morphing (in mechanical or fluid systems) or dynamic representation fusion (in neural networks). Three archetypal realizations are: (1) as a module that reconciles and bridges convolutional (local) and transformer-based (global) features in deep neural networks for medical visual reasoning (Thai et al., 23 Nov 2025), (2) as the shape-morphable central component in fluid-mechanical device optimization (Sheikh et al., 2022), and (3) as the locus of localized deformation in hub-and-spoke kirigami mechanics (Barckicke et al., 9 Apr 2025).

1. Motivation and Conceptual Foundation

The unifying theme in SA-Hub constructions is the need to adaptively interface or reconcile competing requirements or modalities via modifiable structure. In deep neural networks, static hybridization of feature extractors (e.g., CNNs and Transformers) is rigid and non-adaptive; an SA-Hub enables dynamic routing and feature translation layer by layer (Thai et al., 23 Nov 2025). In turbine and kirigami applications, geometric adaptability is central: hub profiles are engineered, via morphing or local strain localization, to mediate complex boundary or load transitions (Sheikh et al., 2022, Barckicke et al., 9 Apr 2025). In all cases, the SA-Hub’s function is to harmonize entities that are locally optimal but globally disparate, by mediating between them with an intermediate, shape- or function-adaptive element.

2. SA-Hub in Neural Architectures: Harmonizing CNNs and Transformers

In the Shape-Adapting Gated Experts (SAGE) framework (Thai et al., 23 Nov 2025), an SA-Hub acts as the critical bridge between convolutional (ConvNeXt) and transformer (ViT) modules:

• At each network layer, two parallel branches receive the incoming feature map $z_{i-1}$: a main path ($z_i^{(\text{main})}$) and an expert path composed of a sparse mixture-of-experts (MoE) pool.
• The SA-Hub serves as the adaptation interface, with input adapters $S_\text{in}$ mapping spatial feature maps into the expert's expected domain (e.g., patchify for ViT, project for ConvNeXt), and output adapters $S_\text{out}$ deprojecting the expert outputs into the common fusion space (Equations 9, 10).
• SA-Hub–mediated routing enables dynamic, per-sample selection of experts, utilizing hierarchical gating: a shared-expert gate $g_s$ modulates the selection probability between generalist and specialist experts based on global-pooled features (Equation 7), and Semantic Affinity Routing (SAR) introduces softmax-based, Top-K sparse routing to maximize representational specificity and computational efficiency (Equations 8–13).
• The outputs of all selected experts, upon adaption, are aggregated as a convex weighted sum and fused with the main path output through a learnable gate $\alpha_i$: $$z_i = \alpha_i z_i^{(\text{main})} + (1-\alpha_i) z_i^{(\text{expert})}$$ (Equation 6).
• The adapters encapsulated by the SA-Hub permit cross-module communication and feature aggregation even when the experts are structurally heterogeneous (e.g., differing spatial or channel dimensionality), offering shape- and modality-adaptive translation that is key for dynamic expert routing.

The empirical impact in SAGE-UNet—where SA-Hub is instantiated at every encoder layer—yields state-of-the-art Dice scores on colonoscopic lesion segmentation benchmarks (EBHI 95.57%, DigestPath 95.16%, GlaS 94.17%), outperforming static CNN-Transformer hybrids by 1–3% and demonstrating enhanced generalization by adaptively combining local detail (CNN) with global context (ViT) (Thai et al., 23 Nov 2025).

3. Geometric Shape-Adaptive Hubs in Fluid-Mechanical Optimization

SA-Hub is realized in hydrokinetic turbine design as an axisymmetric, morphable-solid interface mediating rotor and draft-tube flows (Sheikh et al., 2022). The target is to maximize pressure recovery coefficient $C_{prm}$ via hub shape optimization:

• The hub’s geometry is parameterized by Design-by-Morphing (DbM): two baseline hubs (topologically homeomorphic, with identical inlet/outlet radii and axial length) define the extremes; the SA-Hub interpolates and extrapolates between these using morphing weights $\alpha_1$ and $\alpha_2$, constrained such that $\alpha_1 + \alpha_2 = 0.5$, with $\alpha_1 \in [-0.5, 1.0]$.
• The final shape at each spatial collocation $(s_j,\varphi_k)$ is $$R^\text{morph}(s_j,\varphi_k) = [\alpha_1 R^1 + \alpha_2 R^2]/(\alpha_1 + \alpha_2)$$.
• A mixed-variable, multi-objective Bayesian optimization (MixMOBO) drives the search for optimal weights, combining a Gaussian process surrogate with a hedged acquisition function portfolio over 6 variables (5 draft-tube, 1 hub), evaluated via CFD with only 425 runs (50 initial random, 75 epochs × 5 batches).
• The optimized SA-Hub achieves $C_{prm}^{\text{opt}} = 0.9620$ compared to $C_{prm}^{\text{base}} = 0.9370$, a 2.7% power gain over the industry baseline (Sheikh et al., 2022).
• Morphological analysis reveals the optimal SA-Hub shape to be an extrapolation toward the pure cone baseline (nearly $\alpha_1 \approx 0.99$), featuring a more gradual inlet fillet and widened outlet radius, yielding a “softer” interface facilitating flow attachment and efficient pressure recovery.

These results demonstrate that a parametrically defined, shape-adaptive hub can be systematically optimized to effect global performance improvements in complex coupled flows.

4. SA-Hub in Kirigami and Elastic Mechanics: Localized Deformation Mediation

In hub-and-spoke kirigami, the SA-Hub refers to a thin circular plate connected to multiple radially oriented, tapered spokes. Under radial compression, the system buckles out-of-plane, and the hub’s deformation becomes highly localized (Barckicke et al., 9 Apr 2025):

• The hub’s boundary experiences a bending moment $M$ imparted from the deformed spokes, which is a function of taper angle $\alpha$ and root width $w_h$.
• The balance between bending and stretching localizes significant Gaussian curvature and strain to a narrow annular boundary layer of thickness $\delta \sim (B/M)^{1/2}$, where $B$ is the plate’s bending modulus.
• The peak strain in the hub scales as $\varepsilon_{\max} \sim (1/2) (M R_h/B)$.
• Adjusting the spoke taper provides active control: decreasing $w_h$ (sharper taper) lowers the moment $M$, broadening $\delta$ and reducing $\varepsilon_{\max}$; increasing $w_h$ concentrates strain in a narrower rim. These relationships collapse to the dimensionless group $\mu = M/(B/R_h)$, yielding design charts for optimizing material selection and feature geometry with respect to yield strain and radius.
• The SA-Hub thus serves as a tunable, shape-localizing adaptation zone between distributed loading (elastic spokes) and a locally stressed interface, critically governed by geometric and constitutive parameters.

This mechanical SA-Hub paradigm enables both the programmed distribution of material strain and the mitigation of mechanical failure risks in reconfigurable structures.

5. Mathematical Formulations and Implementation Details

A selection of domain-specific mathematical operators for the SA-Hub includes (Thai et al., 23 Nov 2025, Sheikh et al., 2022, Barckicke et al., 9 Apr 2025):

Notation / Expression	Domain	Description
$$z_i = \alpha_i z_i^{(\text{main})} + (1-\alpha_i) z_i^{(\text{expert})}$$	Neural SAGE	Weighted fusion of main and expert paths
$\tilde z_{i-1}^{(k)} = S_\text{in}(z_{i-1}, e_k)$; $$\hat z_i^{(k)} = S_\text{out}(y_k, z_i^{(\text{main})})$$	Neural SAGE	Input/output adaptation in the SA-Hub
$R^\text{morph}(s_j,\varphi_k)$	Turbine optimization	Morphing interpolation of the hub surface
$\delta \sim (B/M)^{1/2}$	Kirigami mechanics	Boundary layer thickness in the hub
$\varepsilon_{\max} \sim (1/2)\theta_c^2$	Kirigami mechanics	Peak boundary-layer strain
$\mu = M/(B/R_h)$	Kirigami mechanics	Key dimensionless localization parameter

• In neural implementations, the SA-Hub’s modularity relies on lightweight spatial and channel adaptation (projection, normalization, patchify/unpatchify) to minimize overhead.
• In mechanical applications, design space morphing (DbM) and rapid-prototyping constraints guide the practical realization of the optimized hub geometry.

6. Empirical Results and Application-Specific Impact

SA-Hub deployments across domains produce quantifiable advances:

• In SAGE-UNet, integration of SA-Hub enables top-performing lesion segmentation with domain shift robustness and marked improvements in smooth-region false positive reduction and morphologically complex boundary delineation, as visualized via Grad-CAM overlays (Thai et al., 23 Nov 2025).
• Fluid-mechanical SA-Hub morphing delivers a 2.7% power gain over the industry-standard hub, achieved with minimal high-fidelity CFD calls via MixMOBO-guided DbM search (Sheikh et al., 2022).
• In kirigami, SA-Hub theory yields closed-form design criteria for tuning strain localization and avoiding yield via taper manipulation and material selection (Barckicke et al., 9 Apr 2025).

7. Cross-Domain Synthesis and Design Principles

A defining principle of the SA-Hub paradigm across neural, fluid, and mechanical systems is the explicit, parameterized, and tunable mediation between disparate modules, regions, or representations:

• The SA-Hub in neural architectures “translates” and fuses spatially organized convolutional features with globally contextualized transformer tokens—adapting dynamically to input statistics and routing decisions.
• Shape-adaptive hubs in physical structures serve as morphable, load-matching interfaces, synthesizing global form with locally optimal attachment or deformation behavior.
• The mathematical structure—modulation via gates and adapters (deep learning), or via morphing weights and boundary-layer control (mechanics)—enables both efficient optimization and interpretable design.
• In all cases, the design and placement of the SA-Hub are central to achieving high performance, robust adaptation, and domain generality, as evidenced by performance metrics, closed-form design laws, and empirical results across application areas.

A plausible implication is that SA-Hub architectural principles—dynamic adaptation, modular mediation, and tunable translation—will continue to inform next-generation designs in both intelligent computation and adaptive physical systems.