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  "body": "\n## Summary\n\nA radial basis function is a function on $\\R^d$ whose value depends only on the distance from a centre point.\nStandard examples are the Gaussian $e^{-(\\varepsilon r)^2}$, the multiquadric $\\sqrt{1 + (\\varepsilon r)^2}$ and the [[thin plate spline|P9ZPH7EC8K]] $r^2 \\log r$, where $r$ is the distance.\nShifted copies centred at data points are used as building blocks for [[radial basis function interpolation|XZ33P3HBB9]] of scattered data in any dimension.\nThey also appear as kernels in machine learning and in meshfree methods for differential equations.\n\n## Explanations\n\n### L1 human\n\nDrop a stone in a still pond: the height of the water depends only on the distance from where the stone fell.\nA radial basis function has the same symmetry: it looks the same in every direction from its centre.\nThe Gaussian bump $e^{-\\norm{x}^2}$ is the typical example; it equals $1$ at the centre and decays as one moves away.\nPlacing such bumps at many points and adding them with suitable weights gives flexible surfaces.\nThe [[definition|3SS5K8MHPK]] makes this precise.\n",
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