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The Atomic Cluster Expansion (ACE) (Drautz, Phys. Rev. B 99, 2019) has been widely applied in machine learning of high energy physics, quantum mechanics and atomistic modeling to construct many-body interaction models respecting physical symmetries. Computational efficiency is achieved by allowing non-physical self-interaction terms in the model. In this thesis, we propose and analyze an efficient method to evaluate and parameterize an orthogonal, or, non-self-interacting cluster expansion model, which also leads to an efficient algorithm for constructing a high order symmetric tensor product basis from conventional polynomials. We further present numerical experiments demonstrating improved conditioning and more robust approximation properties than the original expansion in regression tasks, both in simplified toy problems and in applications in the machine learning of interatomic potentials.

The temperature-dependent behavior of defect densities within a crystalline structure is intricately linked to the phenomenon of vibrational entropy. Traditional methods for evaluating vibrational entropy are computationally intensive, limiting their practical utility. We show that total entropy can be decomposed into atomic site contributions and rigorously estimate the locality of site entropy. This analysis suggests that vibrational entropy can be effectively predicted using a surrogate model for site entropy. We employ machine learning to develop such a surrogate model, specifically the Atomic Cluster Expansion model. We supplement our rigorous analysis with an empirical convergence study. In addition we demonstrate the performance of our method for predicting vibrational formation entropy and attempt frequency of the transition rates, on point defects such as vacancies and interstitials.

In recent decades, Ab initio quantum chemistry methods have played a crucial role in elucidating the electronic structures of molecules and materials. Despite their widespread application, the constant emergence of new methods lacking rigorous mathematical foundations raises concerns about their reliability.This thesis seeks to bridge this gap by addressing the electronic structure problem through a mathematical lens, leveraging existing numerical analysis results to address the electronic structure problem. Focusing on wavefunction methods, specifically Variational Monte Carlo, our investigation delves into the distinctive features of the Schrodinger equation across different chemical systems, encompassing variations between molecules and materials. We conduct a numerical analysis to derive the convergence of the Ewald Summation of the simulated potential. Subsequently, we employ the Atomic Cluster Expansion to parameterize the trial wavefunction, demonstrating its adaptability to different chemical systems. Building upon existing numerical analyses of the Atomic Cluster Expansion, we argue that our methodology not only provides superior parameterization flexibility compared to existing methods but also ensures a systematicand mathematically sound process for deriving the parameterizations. Finally, we implement simulations to assess the effectiveness of our proposed wavefunction architectures in the 1D uniform electron gas, a system allowing us to probe electronic correlation effects. Through this comprehensive approach, our research aspires to contribute to the development of more robust and mathematically justified methods in the field of Ab initio quantum chemistry.

We are interested in exploring the laws of interaction between atoms and molecules. In this thesis, we construct a model to predict the many-body interaction potential between atoms, given the averaged total potentials in an atomic cluster. This is an inverse problem because we are fitting against averaged observations of energy between many-body interactions. In a simplified setting with identically and independently distributed data, and defining an averaged basis that has the appropriate orthogonal property, the inverse problem is well-posed and can be solved with the least squares approximations in a numerically stable way. We perform numerical analysis to estimate the parameters and study the convergence of the approximation. Two-body and three-body interactions potential are studied as a motivation to generalize the framework for higher-order many-body interaction potentials with tensor product bases in the future.

In this thesis, we present a proof of concept implementation of linear and nonlinlear models based on the Atomic Cluster Expansion (ACE) introduced in [16]. We introduce machine-learned interatomic potentials and derive the ACE as an atomic descriptor. This produces a model linear in its coefficient that serves to approximate the energies and forces of an atomic configuration. We train its coefficients for Silicon, Copper, and Molybdenum, and analyze the fit accuracy for energiesand forces benchmark training sets [37]. Furthermore, we extend the ACE model to approximate energies and forces through a nonlinear combination of linear ACE models. We describe how to implement this model, and in particular, how to efficiently compute the derivatives, and present example results for the same data sets. We summarize the Julia implementation of these nonlinear models and provide an overview of the direction the code base will take in the future.