# Physical interpretation of biharmonic operator

In the book Mathematics of Classical and Quantum Physics, the authors give an (enlightening) interpretation of the Laplace Operator $$\nabla^{2}$$ of a field $$f(\mathbf{x})$$,

$$\nabla^{2}f(\mathbf{x})$$ as the difference between the average value of $$f$$ over a infinitesimal element around the point $$\mathbf{x}$$ and the value of $$f$$ at the point.

$$\nabla^{2}f(\mathbf{x})=\bar{f}_{dv}-f(\mathbf{x})$$

This interpretation clarifies, for example, the heat equation as saying that the temperature at a point increases if the average around it is larger and vice versa.

What about the biharmonic operator $$\nabla^{4}\equiv\nabla^{2}\nabla^{2}$$? The obvious answer would be to interpret it like the Laplace operator but recursively (the difference between [the average of the difference between the average of $$f$$ and the value of $$f$$ at $$\mathbf{x}$$] and [the difference between the average of $$f$$ and the value of $$f$$ at $$\mathbf{x}$$]) but this would be very convoluted... is there a simpler interpretation?

I know that the biharmonic is classified along the Laplace operator as a diffusion-type term, so maybe the interpretation is similar, but I could not find anythin online.