Radial Basis Function Expansion

• Radial Basis Function Expansion is a method to represent functions using weighted, radially symmetric kernels centered at scattered points, enabling mesh-free interpolation and operator learning.
• It supports extensions like polynomial augmentation and Hermite interpolation, which improve numerical stability and accuracy in high-dimensional applications.
• Applications span solving PDEs, scattered data interpolation, surface reconstruction, and modern operator learning in machine learning frameworks.

Radial basis function (RBF) expansion refers to the representation of a function (or, more generally, an operator) as a linear combination or sum of translated and scaled copies of a fixed, radially symmetric kernel. The value of the kernel at any point depends only on its distance from a designated center. RBF expansions underpin a broad spectrum of methods in interpolation, approximation theory, scientific computing, and machine learning. They provide mesh-free, dimensionally robust frameworks for both classical function interpolation and modern operator learning, with rigorous theoretical underpinnings and high flexibility in practical implementation.

1. Mathematical Foundations and Classical RBF Expansion

At the core of RBF expansion is the approximation of functions by a sum of radially symmetric kernels centered at a set of points. For a function $f$ defined on $\mathbb{R}^d$ and a set of centers $\{x_j\}_{j=1}^N$, the standard (scalar) RBF expansion is:

$s(x) = \sum_{j=1}^N w_j \, \phi(\|x - x_j\|)$

where:

• $\phi: [0,\infty) \to \mathbb{R}$ is the RBF kernel (e.g., Gaussian $\phi(r) = \exp[-(\varepsilon r)^2]$, multiquadric, thin plate spline, etc.),
• $w_j$ are the interpolation coefficients determined by the data,
• $\|\cdot\|$ is a chosen norm (typically Euclidean).

To fit prescribed function values $f(x_i)$ at node locations $x_i$, coefficients $w_j$ are determined by solving the interpolation system $A w = f$ with $A_{ij} = \phi(\|x_i-x_j\|)$.

RBF interpolation is universally approximating under mild regularity of $\phi$ and dense enough data, and the expansion generalizes to higher-dimensional spaces and scattered data without requiring a mesh or grid.

2. Extensions: Hermite, Polynomial Augmentation, and Modified RBFs

RBF expansions can be extended by:

• Incorporating polynomial terms for improved reproduction of low-degree behavior and numerical stability (Majdisova et al., 2018). The RBF interpolant then takes the form

$$s(x) = \sum_{j=1}^N w_j \phi(\|x-x_j\|) + \sum_{k=1}^M \lambda_k p_k(x)$$

where $p_k$ are monomials in $x$.

• Hermite RBF (HRBF) expansions further generalize the approach by interpolating both the function and selected derivatives (e.g., gradients):

$$s(x) = \sum_{j=1}^N \left[ w_j \phi(\|x-x_j\|) + \mathbf{b}_j \cdot \nabla\phi(\|x-x_j\|) \right] + \text{(polynomial terms)}$$

This improves accuracy and derivative continuity, crucial for PDE solvers.

• Modified HRBF expansions (MHRBF) add monomial scaling terms to the kernels (e.g., $(x-x_j)^n \phi(\|x-x_j\|)$), which counteract ill-conditioning for small shape parameters (flatter kernels) and reduce the reliance on higher derivatives (Fashamiha et al., 21 Feb 2025).
• Specialized expansions (e.g., in quantum or matrix contexts) utilize alternative inner structures or reformulated kernels, as in quantum RBF networks or the nonlinear RBF matrix decompositions (Shao, 2019, Rebrova et al., 2021).

3. Analytical, Numerical, and Algorithmic Properties

Stability and Conditioning

• Infinitely smooth kernels (e.g., Gaussian) exhibit "flat limit" ill-conditioning for small shape parameters. Stabilization strategies include basis transformation (Hermite- or Chebyshev-based expansions) which mathematically separate the ill-conditioned part from the remainder, leading to efficient algorithms suitable for high-precision and high-dimensional interpolation (Yurova et al., 2017, Drake et al., 2020).
• Band-limited approximations allow the use of fast multipole methods (FMM) and FFT-based solvers by expressing RBFs as convolutions with compactly supported mollifiers. This enables efficient hierarchical or block-circulant algorithms, reducing computational cost from $\mathcal{O}(N^2)$ to $\mathcal{O}(N\log N)$ or lower for large scale problems (Zhao et al., 2016, Zhou et al., 2023).

Preconditioning and Hierarchical Bases

• Toeplitz and block Toeplitz structures in regularly gridded RBF systems facilitate sharp spectral characterizations and enable the design of preconditioners (e.g., via the symbol of the infinite matrix), with the number of iterative solver steps being independent of the number of interpolation points for certain kernels (Baxter, 2010).
• Hierarchical basis (HB) and polynomial-orthogonal bases decouple RBF and polynomial parts of the solution, improving numerical stability and the efficiency of iterative solvers. HB constructions aligned with kernel seed functions and node locations are particularly effective for large, variable-order polynomial RBF interpolation (Castrillon-Candas et al., 2011).

Error and Convergence

• Rigorous error analysis is available for RBF interpolation in both pointwise and energy norms, with convergence often expressed as:

$\|f - s\| \leq C\, h^{s-\tau} \|f\|_{W_2^s}$

where $h$ is the node spacing ("fill distance") and $\tau$ is linked to the smoothness of the RBF (Heuer et al., 2012, Zhou et al., 2023).

• Modified RBFs (MHRBF) achieve lower interpolation errors across a wide range of shape parameters and system sizes, outperforming standard HRBFs in both function and derivative approximation, with optimal monomial degree and shape parameter combinations being robust across domain scales (Fashamiha et al., 21 Feb 2025).

4. Applications: From Function and Operator Approximation to Scientific and ML Tasks

Scattered Data and Surface Reconstruction

• RBF expansion excels in interpolating scattered, high-dimensional geometrical data due to minimal assumptions on node placement and direct construction without meshing (Majdisova et al., 2018). Applications include surface reconstruction, image restoration, and meshless PDE solvers.

Partial Differential Equations (PDEs)

• RBF and HRBF expansions (including MHRBF) are foundational for meshfree solutions of PDEs, including those with complex boundary or regularity requirements. They are deployed for the Helmholtz, Laplace, convection-diffusion, wave, and higher-order equations.
• Specialized techniques handle boundary conditions either by extension and weak enforcement via Lagrange multipliers or through augmented fictitious/resampling points in the collocation method, particularly for irregular domains (Heuer et al., 2012, Safdari-Vaighani et al., 2017).

Operator Learning and Neural Operators

• RBF expansion is generalized in operator learning architectures, such as the Radial Basis Operator Network (RBON). Here, operators mapping between function spaces (e.g., solution maps for PDEs) are learned using a dual "branch-trunk" structure with RBFs in both functional and spatial domains:

$$G^\dagger(u^m)(x) = \sum_{i=1}^M \sum_{k=1}^N \xi_{ik} g(\lambda_i \|u^m - \mu_{ik}^m\|) \, g(\omega_k \|x - \nu_k\|)$$

This structure enables learning of both time- and frequency-domain operators and achieves small $L^2$ errors for both in-distribution and out-of-distribution data, even with compact, single-layer architectures (Kurz et al., 6 Oct 2024).

• RBF expansions underpin both classical RBF networks and their quantum generalizations, offering speedups and parametric compression in high-dimensional classification problems (Shao, 2019).

Matrix and Graph Representations

• RBF-based decompositions generalize the SVD by using nonlinear RBF outer products in place of rank-1 matrix factors, enabling memory-efficient, expressive, and interpretable factorizations for matrices arising in graph, molecular, or image data (Rebrova et al., 2021, Sledge et al., 2019).

Reinforcement Learning and Control

• Hybrid controllers blend linear model-based policies and RBF-expansion-based universal nonlinear controllers, orchestrated by smooth distance-dependent interpolation. The RBF component ensures global approximation capacity, allowing retention of stability guarantees local to an operating point (Capel et al., 2020).

5. Specialized Variants and Recent Innovations

Fundamental Solution RBFs and Operator Dependence

• RBFs can be tailored to match the kernel of an integral equation by construction from the Green's function or fundamental solution associated with the differential operator. This "operator-dependent" design exploits the boundary integral connection, yielding the so-called fundamental solution RBFs (FS-RBFs) [0207016 title only], with implications for improved error bounds and operator-adapted accuracy.

Band-Limited and Fast Summation Schemes

• RBF expansions can be expressed via truncated Fourier integrals after mollification, making them compatible with FFT and multipole algorithms. Kernel-independent, fast summation is feasible for translation-invariant, rapidly decaying RBFs after appropriate bandlimiting (Zhao et al., 2016).

Quantum and Graph Formulations

• Reformulations for adjacency-matrix-based graphs enable RBF networks to operate natively in the data's relational space, with gradient-based parameter updates and prototype adjustments derived directly from matrix entries (Sledge et al., 2019, Shao, 2019).

Operator Networks and Out-of-Distribution Generalization

• RBONs extend RBF expansion orthogonally to function spaces, leveraging a fully RBF-based architecture for operator learning, with demonstrated robustness on out-of-distribution data and across function class boundaries (Kurz et al., 6 Oct 2024).

6. Limitations, Challenges, and Future Directions

• Ill-conditioning and Scalability: Infinite smoothness and small shape parameters yield high accuracy but challenging numerical stability. Stabilization via basis transformation, monomial scaling, and hierarchical methods partially mitigate these issues.
• Node Placement and Adaptivity: The distribution and local density of centers strongly influence error and matrix structure. Recent work includes localized, wavelet-based, and cluster-adapted RBF schemes, but optimization of center placement (e.g., via K-means or adaptive refinement) remains an active area (Castrillon-Candas et al., 2011, Kurz et al., 6 Oct 2024).
• High-Dimensionality: Scalable algorithms utilizing FFT, FMM, and hierarchical bases extend feasibility for very high-dimensional or large-scale problems. However, parameter selection (shape parameter, monomial degree, kernel anisotropy) and control of low-rank perturbations in irregular domains are vital for continued progress (Zhou et al., 2023, Yurova et al., 2017).
• Generalization and Universal Approximation: RBF expansion's universal approximation property is firmly established for a broad class of kernels and domains, with operator learning extensions (in RBON) bringing similar guarantees to mappings between infinite-dimensional spaces.
• Research Directions: Further development is anticipated in:
  • Robust and adaptive center selection for operator networks (Kurz et al., 6 Oct 2024).
  • Integration with deep learning frameworks and hybrid/ensemble architectures.
  • Application to increasingly complex high-dimensional PDEs, control systems, and scientific datasets.
  • New stabilized or localized kernels, and theoretical extension to broader operator classes.

7. Summary Table: Core RBF Expansion Principles

Expansion Type	Structure	Key Applications
Scalar RBF	$\sum_j w_j \phi(\|x-x_j\|)$	Scattered data interpolation, PDEs
HRBF / MHRBF	Adds derivatives, possibly monomial scaling	PDEs requiring smooth derivatives
Matrix/Graph RBF	RBF between node pairs, or nonlinear outer-products	Matrix, graph, or network approximation
Operator RBF (RBON)	$$\sum_{i,k} \xi_{ik}g(\|u^m-\mu_{ik}^m\|)g(\|x-\nu_k\|)$$	Operator learning, scientific ML
Quantum/Hybrid RBF	RBF feature maps, parameterized via unitary ops	Fast, high-dimensional classification

The radial basis function expansion framework embodies a highly adaptable mathematical formalism bridging classical approximation theory and contemporary learning of functions and operators. Its successful deployment in theoretical, numerical, and applied contexts continues to advance the frontiers of computational mathematics and data-driven modeling.