Uniting Nonempirical and Empirical Density Functional Approximation Strategies Using Constraint-Based Regularization

Abstract

In this work, we present a general framework that unites the two primary strategies for constructing density functional approximations (DFAs): nonempirical (NE) constraint satisfaction and empirical (E) data-driven optimization. The proposed method employs B-splines, bell-shaped spline functions with compact support, to construct each inhomogeneity correction factor (ICF). This choice offers several distinct advantages over traditional polynomial expansions by enabling explicit enforcement of linear and nonlinear constraints as well as ICF smoothness using Tikhonov and penalized B-splines (P-splines) regularization. As proof-of-concept, we use the so-called CASE (constrained and smoothed empirical) framework to construct a constraint-satisfying and data-driven global hybrid that exhibits enhanced performance across a diverse set of chemical properties. We argue that the CASE approach can be used to generate DFAs that maintain the physical rigor and transferability of NE-DFAs while leveraging high-quality quantum-mechanical data to remove the arbitrariness of ansatz selection and improve performance.

Machine learning for accuracy in density functional approximations

“…129 The performance of the strongly constrained and appropriately normed (SCAN) meta-GGA 34 clearly showed the benefits of fulfilling an increasing number of known analytical constraints for XC functionals. 130,131 Sparrow et al 132 expressed inhomogeneity factors for GGA exchange and correlation in a spline basis facilitating straightforward enforcement of equality and inequality constraints, resulting in the CASE21 functional for molecular chemistry. In addition to coefficient elimination for equality constraints, inequality constraints were implemented as penalties during exchange enhancement factor optimization of the meta-GGA MCML by Brown et al 37 for bulk, surface, and gasphase chemistry.…”

Artificial intelligence in recommender systems

“…Recently, thanks to the rapid development of deep learning methods and their surprising capability to capture nonlinear patterns, data-driven approaches have been used to estimate the precise exchange-correlation energy functional [Bogojeski et al 2020;Nagai et al 2020;Dick and Fernandez-Serra 2021;Kirkpatrick et al 2021;Bystrom and Kozinsky 2022;Trepte and Voss 2022;Sparrow et al 2022;Bystrom and Kozinsky 2023;Fernandez-Serra 2019, 2020;Ryabov et al 2020;Lei and Medford 2019], which demonstrates exceptional accuracy on main-group chemistry and represents a state-of-the-art achievement in the field. Unlike approximation techniques, data-driven approaches can learn a theoretically unbiased (exact) estimator of the XC energy functional from real data because they do not impose any approximations on the functional form.…”

“…Several works combine data-driven XC energy fitting with exact physical constraints, including fractional electron constraint [Kirkpatrick et al 2021], linear/nonlinear constraints [Sparrow et al 2022], physical asymptotic constraints [Nagai et al 2022], and others [Trepte and Voss 2022;Brown et al 2021]. For example, one essential constraint of the DFT system is that electrons are treated as a continuous charge density rather than discrete particles.…”