RadialBasisFunctions.jl High-performance radial basis function interpolation and differential operators for Julia.

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Also known as: meshfree methods, meshless methods, RBF-FD (Radial Basis Function Finite Differences), scattered data interpolation, kernel methods, polyharmonic splines, collocation methods

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Why RadialBasisFunctions.jl?

• Truly Meshless - Work directly with scattered points, no mesh generation required
• GPU Ready - Seamless CPU/GPU execution via KernelAbstractions.jl
• Flexible Operators - Built-in Laplacian, gradient, partial derivatives, or define your own custom operators
• Boundary Conditions - Hermite interpolation for Dirichlet, Neumann, and Robin BCs in PDE applications
• Local Collocation - k-nearest neighbor stencils for scalable large-domain problems
• Differentiable - AD-compatible via ChainRulesCore and Mooncake extensions

Scattered data points (left) reconstructed as a smooth surface using RBF interpolation (right)

Quick Start

using RadialBasisFunctions, StaticArrays

# Create scattered 2D points
points = [SVector{2}(rand(2)) for _ in 1:500]
f(x) = sin(4x[1]) * cos(3x[2])
values = f.(points)

# Interpolation - evaluate anywhere in the domain
interp = Interpolator(points, values)
interp(SVector(0.5, 0.5))

# Differential operators - apply to scattered data
lap = laplacian(points)       # Laplacian operator
grad = gradient(points)       # Gradient operator
∂x = partial(points, 1, 1)    # ∂/∂x₁

lap_values = lap(values)      # Apply Laplacian to data
grad_values = grad(values)    # Returns Nx2 matrix of gradients

Supported Radial Basis Functions

Type	Formula	Best For
Polyharmonic Spline (PHS)	$r^n$ where $n = 1, 3, 5, 7$	General purpose, no shape parameter tuning needed
Inverse Multiquadric (IMQ)	$1 / \sqrt{(\varepsilon r)^2 + 1}$	Smooth interpolation with tunable accuracy
Gaussian	$e^{-(\varepsilon r)^2}$	Infinitely smooth functions

All basis functions support polynomial augmentation (enabled by default) for enhanced accuracy.

# Basis function examples
basis = PHS(3)                  # Cubic PHS (default: quadratic polynomial augmentation)
basis = PHS(5; poly_deg=3)      # Quintic PHS with cubic polynomials
basis = IMQ(1.0)                # IMQ with shape parameter ε=1.0
basis = Gaussian(0.5)           # Gaussian with shape parameter ε=0.5

Installation

using Pkg
Pkg.add("RadialBasisFunctions")

Requires Julia 1.10 or later.

Advanced Features

Hermite Interpolation for PDEs

Handle boundary conditions properly with Hermite interpolation:

# Build operator with boundary awareness
lap = laplacian(points, eval_points, PHS(3), k,
                boundary_nodes, boundary_conditions, normals)

GPU Acceleration

Operators automatically leverage GPU when data is on device:

using CUDA
points_gpu = cu(points)
lap_gpu = laplacian(points_gpu)  # Weights computed on GPU

Custom Operators

Define any differential operator using automatic differentiation:

# Example: custom operator
my_op = custom(points, basis -> (x, xc) -> your_derivative_function(basis, x, xc))

Documentation

• Getting Started - Tutorials and examples
• Theory - Mathematical background on RBF methods
• API Reference - Full function documentation

Citation

If you use RadialBasisFunctions.jl in your research, please cite:

@software{RadialBasisFunctions_jl,
  author = {Beggs, Kyle},
  title = {RadialBasisFunctions.jl: Meshless RBF interpolation and differential operators for Julia},
  url = {https://github.com/JuliaMeshless/RadialBasisFunctions.jl},
  doi = {10.5281/zenodo.7941390}
}

Contributing

Contributions are welcome! Please feel free to:

• Report bugs or request features on GitHub Issues
• Start a discussion on GitHub Discussions
• Submit pull requests

Part of the JuliaMeshless organization.