Data-Driven Models for Anisotropic Pairwise Interactions using Limited Sampling

Abstract

The interaction of two particles with shape anisotropy depends not only on their separation but also on their relative orientation. Such interactions are known to lead to interesting collective behavior such as self-assembly in both synthetic and natural materials; hence, there is great interest in developing modeling strategies to better understand and engineer these materials. Particle-based simulations provide a powerful route for this purpose, but they require accurate interaction models. Such complex interactions do not have a universal functional form, leading to great interest in approximating them from more detailed models using data-driven approaches. However, many machine learning methods require millions of training samples from higher-resolution models to reach acceptable accuracy. This computational cost becomes prohibitive when the underlying reference model is expensive to simulate, motivating the development of data-efficient strategies for learning anisotropic pair interactions.

In this dissertation, I develop a systematic data-driven framework based on multivariate polynomial surrogate models for accurately modeling anisotropic pairwise interactions using limited training samples. I first apply this framework to approximating anisotropic pairwise potentials, in which the training points are systematically prescribed by the polynomial basis. By exploiting physical symmetry and a set of physics-informed coordinate transformations, I define an optimal sampling domain that systematically concentrates more training samples in the most thermodynamically important regions of the configuration space. I investigate the effectiveness of this approach for several shape-anisotropic nanoparticles and show that accurate approximations can be obtained with substantially fewer points than black-box machine learning approaches. I then derive analytic expressions for the forces and torques, which govern the dynamical behavior of anisotropic particles, as functions of the transformed coordinates, and use them to approximate the forces and torques of nanoparticles. The results show that multivariate polynomials can accurately approximate forces and torques and reproduce structural observables, such as distribution functions. Finally, to investigate generality, I extend this framework to molecular systems, where a multipole expansion describes the slowly decaying electrostatic interactions, and achieve accurate approximations with limited training samples. Such a data-efficient approximation approach offers a practical route to simulating self-assembly of complex synthetic and biomolecular systems, and ultimately, engineering functional materials, and informs our understanding of biomolecular assembly processes.