Approximation of anisotropic pair potentials using multivariate interpolation

TL;DR

This paper explores using multivariate polynomial interpolation, including Chebyshev and mixed-basis methods, to efficiently approximate anisotropic pair potentials from limited data, aiding modeling of complex particle interactions.

Contribution

It introduces a novel application of polynomial interpolation techniques to approximate anisotropic pair potentials with limited data, incorporating physical insights for improved accuracy.

Findings

Successful approximation of anisotropic potentials in 2D and 3D models.

Effective use of mixed-basis interpolation for periodic coordinates.

Refined interpolation domains improve approximation quality.

Abstract

The interaction between two particles with shape or interaction anisotropy can be modeled 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 is 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 as well as 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 satisfactory approximations can be constructed in all cases.