Approximation of Anisotropic Pair Potentials Using Multivariate Interpolation

The interaction between two particles with shape or interaction
anisotropy can be modeled by using a pairwise potential energy function
that depends on their relative position and orientation; however,
this function is often challenging to mathematically formulate. Data-driven
approaches for approximating anisotropic pair potentials have gained
significant interest due to their flexibility and generality but often
require large sets of training data, potentially limiting their feasibility
when training data are computationally demanding to collect. Here,
we investigate the use of multivariate polynomial interpolation to
approximate anisotropic pair potentials from a limited set of prescribed
particle configurations. We consider both standard Chebyshev polynomial
interpolation and mixed-basis polynomial interpolation that uses trigonometric
polynomials for coordinates along which the pair potential is known
to be periodic. We exploit mathematical reasoning and physical knowledge
to refine the interpolation domain and to design our interpolants.
We test our approach on two-dimensional and three-dimensional model
anisotropic nanoparticles, finding that satisfactory approximations
can be constructed in all cases.