MultiPatch-RBFKAN Frameworks

• MultiPatch-RBFKAN is a framework that leverages partitioned local RBF transforms to achieve robust, efficient, and flexible mappings across various domains.
• It partitions input domains into patches using anisotropic RBF kernels, enabling precise modeling in mesh, point cloud, and feature-based applications.
• The approach supports applications from PDE mesh coupling to attention-free vision token mixing, ensuring computational efficiency and high accuracy.

MultiPatch-RBFKAN is a family of frameworks that leverage local Radial Basis Function (RBF) transforms over domain or feature map partitions (“patches”) to achieve highly expressive, efficient, and flexible mappings. These methods are unified by the partitioning of the input domain—either physical (in mesh or point cloud contexts), computational (for non-matching mesh coupling), or feature-space (for vision architectures)—and the use of RBF kernels as nonlinear basis functions or interpolants on each patch. MultiPatch-RBFKAN finds applications in mesh interfacing for complex PDEs (Moretto et al., 2024), spatial data interpolation with anisotropic geometry (Cavoretto et al., 2018), and as a token-mixing operator in attention-free vision backbones (Yang et al., 29 Jan 2026).

1. Mathematical Foundations

MultiPatch-RBFKAN couples local RBF-based approximants or transforms with a domain partition, typically employing:

• A covering of the full domain (Ω ⊂ ℝ^d) by M overlapping or non-overlapping patches {Ωᵢ}. Patches may be geometrically structured (ellipsoidal (Cavoretto et al., 2018), mesh facets (Moretto et al., 2024), or image grid regions (Yang et al., 29 Jan 2026)).
• On each patch Ωᵢ, an RBF basis is constructed. For interpolation problems, this yields a patchwise approximant: $$Rᵢ(x) = \sum_{j=1}^{Nᵢ} c_{ij} φ(‖Aᵢ(x - x_{ij})‖_2)$$, where $Aᵢ$ is a local (potentially anisotropic) shape matrix.
• The global functional or field is represented as a sum or aggregation over patchwise results, with partition-of-unity (PU) weights $Wᵢ(x)$ to enforce consistency: $s(x) = \sum_{i=1}^M Wᵢ(x) Rᵢ(x)$.
• In finite element mortar coupling, the RBF interpolant is used for mesh interface projection, with the RBF operator replacing the classical geometric $L^2$ projector in weak interface constraints (Moretto et al., 2024).
• For token mixing (in ViT-like architectures), partitioning is over feature maps (e.g., into $p \times p$ patches), and RBF networks serve as the atomic nonlinearity for patchwise transformations; higher-level propagation (axis-separable mixing, low-rank global operators) is then layered atop (Yang et al., 29 Jan 2026).

Key kernels include Gaussian, inverse multiquadric, and compactly supported Wendland functions, with hyperparameters (e.g., shape, width) tuned per patch or globally.

2. Partitioning Strategies and RBF Construction

Partitioning is crucial for tractability and performance:

• Mesh and Point Cloud Domains:

Overlapping ellipsoidal patches $\Omega_i$ are used to capture anisotropic geometry or data sparsity. Each patch has a center $c_i$, axes $\delta_{ik}$ (possibly optimized by LOOCV), and an associated RBF shape matrix $A_i$ (Cavoretto et al., 2018).

• Mortar Coupling for PDEs:

Patch boundaries are the mesh interface facets $\Gamma_k$. On each, a set of interpolation nodes $\{\xi_m\}$ determines the RBF basis. Shape parameter $\varepsilon$ is typically tied to the facet diameter (Moretto et al., 2024).

• Vision (KAN-based Mixing):

Non-overlapping grid patches of fixed shape (commonly $p \times p$) are used, yielding tractable per-patch nonlinear mapping without quadratic scaling with input size. Each channel and patch has learnable centers $\mu_{cj}$, widths $\sigma_{cj}$, and weights $w_{cj}$ for the RBF expansion (Yang et al., 29 Jan 2026).

On each patch, the local system for the RBF coefficients can be written as: $$\sum_{n=1}^M φ(‖\xi_i - \xi_n‖, \varepsilon) \gamma_n = f_i$$ and advanced scaling schemes (e.g., partition-of-unity normalization) are advised for accurate constant reproduction and stability.

3. Algorithmic Architecture and Implementation

Mesh Coupling/Mortar Formulation (Moretto et al., 2024):

• Two subdomains $\Omega_1$, $\Omega_2$ share interface $\Gamma$; weak continuity is enforced via a mortar constraint $b(u, \mu) = \langle \mu, u_2 - u_1 \rangle_\Gamma$.
• The classical $L^2$ projector $\Pi$ is replaced by an RBF projection constructed at discretization nodes. The discrete transfer operator projects master-side FE basis functions onto the slave side via: $$[N_{\Gamma_2}^\Pi]_{g,l} = \frac{\sum_{m=1}^M W_{m,l} φ(‖\zeta_g - \xi_m‖, \varepsilon)}{R_{g,g}}$$
• The global FE system couples domain and multiplier dofs; static condensation is possible when dual multipliers are diagonal.

Patchwise RBF-PU Approximation (Cavoretto et al., 2018):

• For a given data set, patch centers are chosen (coarse grid, k-means, or skeletonization for clusters/“track” data).
• For each patch, parameters $\varepsilon_i$, $\delta_i$ are optimized for LOOCV error, subject to conditioning (optionally with penalty $\mu \log\kappa(A_i)$).
• PU weights $W_i(x)$ are anisotropic Shepard-type, normalized over all patches.
• Global evaluation is $s(x) = \sum_i W_i(x) R_i(x)$.

Feature Map Token Mixing (Yang et al., 29 Jan 2026):

• Input feature map $X \in \mathbb{R}^{H \times W \times C}$ is split into $N/F$ patches of size $p \times p$.
• For each patch and channel, $x_{i,c} \in \mathbb{R}^F$ undergoes an RBF expansion followed by axis-wise separable depthwise convolution and channel-gated mixing, and finally a low-rank ($r \ll N$) global mixing: $f_{\text{global}}(X)_c = Q_c (P_c y_c)$
• Pseudocode illustrations demonstrate drop-in replacement of standard Attention($QKV$) calls in modern ViT blocks.

4. Computational Complexity and Scalability

MultiPatch-RBFKAN achieves significant efficiency gains by restricting high-order operations to low-dimensional subdomains or patches:

Application Domain	Patch Size (F)	Dominant Per-Block Cost	Scaling vs. Standard Methods	Reference
Mesh mortar coupling	$M \ll N_G$ nodes/face	$O(M^3)$ (offline), $O(N_G M)$	5–10$\times$ faster than EB/ATTN	(Moretto et al., 2024)
RBF-PU interpolation	Local node count $N_i$	$O(N_i^3)$ (per patch, per opt)	$M$ local solves + eval $O(\sum N_i)$	(Cavoretto et al., 2018)
Vision token mixing	$F = p^2$ (small)	$O(NCM + NCk + NCr)$	$O(NC)$ vs. $O(N^2C)$ for attention	(Yang et al., 29 Jan 2026)

This linear (or near-linear) scaling is achieved without detrimental effects on numerical accuracy in PDE/mesh applications, and with competitive accuracy in large-scale vision tasks. In mesh contexts, the RBF method eliminates nonlinear closest-point search (and Newton iterations) required by conventional exact burden (EB) mortar integration.

5. Representative Numerical Results

• Interpolation/Approximation (Cavoretto et al., 2018):

On 2D “track” datasets, RMSE drops from $10^{-3}$–$10^{-4}$ (standard PU) to $10^{-7}$ (MultiPatch-RBFKAN) after per-patch LOOCV optimization of RBF shape and ellipsoidal semiaxes. For Korea topography data, substantial RMSE reduction is reported versus classical PU.

• Mortar Coupling (Moretto et al., 2024):

In 1D/2D/3D tests, MultiPatch-RBFKAN matches exact EB in $L^2$ and broken $H^1$ error, including for curved nonmatching interfaces, with absolute errors differing by only 1–2%. In complex fluid–structure interaction in porous domains, pressure and displacement coupling converges linearly with errors <1%.

• Vision KAN (Yang et al., 29 Jan 2026):

On ImageNet-1K, ViK with MultiPatch-RBFKAN mixer achieves competitive accuracy to transformer baselines, maintaining linear complexity.

A plausible implication is that MultiPatch-RBFKAN frameworks can serve as practical drop-in alternatives to either exact mortar projection or quadratic-cost attention operations, with minimal loss in accuracy and substantial reductions in computational budget.

6. Practical Implementation Guidelines

• Select patch/partition structure to match geometric anisotropy, data density, or computational efficiency requirements.
• Set RBF kernel parameters (e.g., $\varepsilon$) proportional to patch scale or feature diameter. Maintain $M \leq 10$ per patch/facet for stability in mesh contexts.
• Employ rescaled, PU-normalized RBF interpolants to ensure exact reproduction of constant fields (essential for conservation properties).
• For mesh coupling or RBF-PU, exploit sparsity (many slave-side quadrature points often have no overlap per patch/facet).
• In vision contexts, group into the smallest tractable patch size to balance expressivity and runtime.
• Modularize implementation by precomputing and re-using per-patch RBF system solves and inverses where possible.

7. Theoretical and Methodological Rationale

• The Kolmogorov–Arnold representation theorem undergirds the patchwise RBF-KAN approach: any multivariate continuous map can be represented as a finite sum of univariate nonlinearities (“sum of superpositions”). Patchwise construction enables this mapping efficiently.
• The partitioned structure addresses both tractability (mitigating “curse of dimensionality” in high-d spaces or large N) and adaptivity (local anisotropic kernels, tailored domain geometry).
• For mesh/PDE problems, weak interface continuity via mortar coupling permits robust solution transfer across nonmatching grids, and RBF-based transfer preserves accuracy and conservation properties.
• In neural architectures, the stacking of localized nonlinear mixing (patch RBF-KAN), efficient local propagation (axis-wise separable mix), and low-rank global context reconstructs both local patterns and long-range dependencies with reduced computational demands compared to full attention.

• [A novel Mortar Method Integration using Radial Basis Functions, (Moretto et al., 2024)]
• [Anisotropic weights for RBF-PU interpolation with subdomains of variable shapes, (Cavoretto et al., 2018)]
• [Vision KAN: Towards an Attention-Free Backbone for Vision with Kolmogorov-Arnold Networks, (Yang et al., 29 Jan 2026)]