Approximation of forces and torques from anisotropic pairwise interactions using multivariate polynomials

TL;DR

The paper addresses the challenge of representing forces and torques from anisotropic pairwise interactions with limited data by extending a multivariate-polynomial surrogate framework to include derivatives with respect to transformed coordinates. Forces and torques are connected to the energy via the Jacobian of the coordinate transformation, enabling several strategies—energy interpolation (E), force/torque interpolation (FT), partial-derivative interpolation (D), and force/torque regression (FT-R)—to approximate interactions. Across model 2D and 3D nanoparticles, energy interpolation provides the best accuracy and consistent equilibrium behavior, while FT-R offers a viable alternative that respects energy consistency; FT and D can yield nonconservative forces, underscoring the importance of energy-based or energy-consistent fitting. The work demonstrates data-efficient, physics-informed avenues for simulating anisotropic assemblies and provides guidance on when to use each strategy, along with notes on computational implementation and potential enhancements for scalability.

Abstract

The dynamics of anisotropic particles are dictated by forces and torques that can be challenging to mathematically represent in computer simulations. Several data-driven approaches have been developed to approximate these interactions, but they often rely on having large amounts of training data that may be practically difficult to generate. Here, we extend a framework we recently developed for approximating anisotropic pair potentials to the approximation of pairwise forces and torques. The framework uses multivariate polynomials and physics-motivated coordinate transformations to produce accurate approximations using limited amounts of data. We first derive expressions relating the force and torque to partial derivatives of the potential energy with respect to the transformed coordinates used to represent the particle configuration. We then explore several options for approximating the forces and torques, and we critically assess their accuracy using model two- and three-dimensional shape-anisotropic nanoparticles as test cases. We find that interpolation of the pairwise potential energy produces the best result when it is known, but force and torque matching (regression) is a viable strategy when only the force and torque is available.

Figures (4)

• Figure 1: Approximated $x$-component of the (a--d) force $\hat{F}_x$ and (e--h) torque $\hat{\tau}_x$ vs. true values $F_x$ and $\tau_x$ for the cube. The approximations were constructed by (a,e) interpolation of energy (E), (b,f) interpolation of forces and torques (FT), (c,g) interpolation of partial derivatives (D), and (d,h) regression of forces and torques (FT-R). To facilitate comparison, all panels share the same axis limits, which are set by the minimum and maximum of all true and approximated values. The root mean squared error (RMSE) for each approximation is also stated.
• Figure 2: RMSE for all force and torque components for (a) two-dimensional rod, (b) square, (c) triangle, (d) three-dimensional rod, (e) cube, and (f) tetrahedron approximated using the different strategies (see Fig. \ref{['fig:cube_parity_main']}). The unit of RMSE is $\varepsilon/\sigma$ for the force and $\varepsilon$ for the torque. Corresponding parity plots for all components of the force and torque for all nanoparticles are shown in Figs. S2--S7. Nanoparticle images were visualized using VMD 1.9.4 humphrey:jmolgrp:1996.
• Figure 3: The magnitude of the curl of the approximated forces $\mathbf{\hat{F}}$ using interpolation of forces and torques (FT) or interpolation of partial derivatives (D) for the (a) two-dimensional and (b) three-dimensional nanoparticles. The curl of the force is zero for both interpolation of energy (E) and regression of forces and torques (FT-R).
• Figure 4: Probability density function $f$ to find one three-dimensional rod a distance $r$ from the center of mass of a fixed rod. The dashed black line is the equilibrium distribution for the true interactions. The other lines are the distribution after simulating an additional $500\,\tau$ using the four different approximation strategies (see Fig. \ref{['fig:cube_parity_main']}). The normalization is such that $\int {\rm d}{r}\,4\pi r^2 f(r) = 1$.