An ${\ensuremath{\ell}}_{0}{\ensuremath{\ell}}_{2}$-norm regularized regression model for construction of robust cluster expansions in multicomponent systems

An $ℓ_{0} ℓ_{2}$ -norm regularized regression model for construction of robust cluster expansions in multicomponent systems

Peichen Zhong, Tina Chen, Luis Barroso-Luque, Fengyu Xie, and Gerbrand Ceder

Phys. Rev. B 106, 024203 – Published 25 July 2022

Abstract

We introduce $ℓ_{0} ℓ_{2}$ -norm regularization and hierarchy constraints into linear regression for the construction of cluster expansions to describe configurational disorder in materials. The approach is implemented through mixed integer quadratic programming (MIQP). The $ℓ_{2}$ -norm regularization is used to suppress intrinsic data noise, while the $ℓ_{0}$ -norm is used to penalize the number of nonzero elements in the solution. The hierarchy relation between clusters imposes relevant physics and is naturally included by the MIQP paradigm. As such, sparseness and cluster hierarchy can be well optimized to obtain a robust, converged set of effective cluster interactions with improved physical meaning. We demonstrate the effectiveness of $ℓ_{0} ℓ_{2}$ -norm regularization in two high-component disordered rocksalt cathode material systems, where we compare the cross-validation, convergence speed, and the reproduction of phase diagrams, voltage profiles, and Li-occupancy energies with those of the conventional $ℓ_{1}$ -norm regularized cluster expansion models.

Received 28 April 2022
Revised 27 June 2022
Accepted 13 July 2022

DOI:https://doi.org/10.1103/PhysRevB.106.024203

Physics Subject Headings (PhySH)

Thermodynamics

Disordered systems Face-centered cubic Oxides

Cluster expansion First-principles calculations Lattice models in condensed matter Machine learning

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Peichen Zhong ^1,2, Tina Chen ^1,2, Luis Barroso-Luque ^1,2, Fengyu Xie ^1,2, and Gerbrand Ceder ^1,2,*

¹Department of Materials Science and Engineering, University of California, Berkeley, California 94720, United States
²Materials Sciences Division, Lawrence Berkeley National Laboratory, California 94720, United States

^*gceder@berkeley.edu

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Vol. 106, Iss. 2 — 1 July 2022

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Figure 1
(a) The general flowchart of constructing a CE model, including initialization of input structures, DFT calculations, fitting and convergence check, and cluster expansion Monte Carlo (CEMC) for sampling. (b) An illustration of $ℓ_{2}, ℓ_{1}$ , and $ℓ_{0}$ -norm regularization in a two-parameter space $J$ = ( $J_{1}, J_{2}$ ). The blue circles represent the contours of the data term $| | E_{DFT, S} - Π_{S} J {| |}_{2}^{2}$ in cost function. The red regions represent the constraints of parameters (e.g., $J_{1}^{2} + J_{2}^{2} \leq s$ for $ℓ_{2}$ -norm, $| J_{1} | + | J_{2} | \leq s$ for $ℓ_{1}$ -norm.) The dark red point is the intersection of data term and regularization of parameters, which jointly determines the estimation of $J$ .
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Figure 2
Illustration of hierarchy relations ( $α \subset β \subset γ$ ) between pair, triplet, and quadruplet orbit. The different colors on the cluster sites represent the decorating species for a given site-basis function. The equation in red shows a pseudoactive hierarchy constraint that may appear in $ℓ_{1}$ -norm and its derivative methods.
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Figure 3
(a) An illustration of the rocksalt lattice structure. The cation sites are labeled in red and can be occupied by ${Li}^{+}$ and transition metals (TM, including ${Mn}^{2 +}, V^{3 +}$ , and ${Ti}^{4 +}$ in our example) in DRX. The anion sites are labeled in gray and can be occupied by $O^{2 -}$ and $F^{-}$ . The lower panel gives some examples of $n$ -body ( $n = 2, 3, and 4$ ) clusters used in the CE model, including intra and inter-sublattice interactions. (b) The procedure to obtain an $ℓ_{0} ℓ_{2}$ -norm regularized solution, including finding the $μ_{2}$ by minimizing the CV error in ridge regression, sparseness engineering with $ℓ_{0}$ -norm using MIQP, and terminating if the solution is converged with good sparseness, as well as good-reproduction-relevant physical properties.
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Figure 4
(a) Cross-validation error (meV/atom) of the $ℓ_{1}$ -CE and the $ℓ_{0} ℓ_{2}$ -CE. The sparseness is the number of nonzero ECIs in the fit ( ${| | J | |}_{0}$ ). The curves are generated by varying hyperparameters $μ_{0}, μ_{1}$ , and $μ_{2}$ in regularization. (b) ECIs convergence test vs training set size. $J$ is the ECIs fitted with full training data, and $J_{sub}$ is the ECIs fitted with a subset of corresponding size.
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Figure 5
(a) Phase diagram generated with DFT, $ℓ_{0} ℓ_{2}$ -CE, and $ℓ_{1}$ -CE. The DFT ground states are labeled in blue text. The incorrectly predicted ground states are labeled with red circles and text. (b) The simplified spinel voltage profile (blue line) generated by $ℓ_{1}$ -CE and $ℓ_{0} ℓ_{2}$ -CE for spinel orderings in ${Li}_{x} {Mn}_{2} O_{4}$ is compared with the DFT ground-truths (orange line). (c) Energy difference of Li occupation in octahedral and tetrahedral sites in layered ${MnO}_{2}$ (top) and spinel ${MnO}_{2}$ framework (bottom).
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Physical Review B

covering condensed matter and materials physics

An $ℓ_{0} ℓ_{2}$ -norm regularized regression model for construction of robust cluster expansions in multicomponent systems

Peichen Zhong, Tina Chen, Luis Barroso-Luque, Fengyu Xie, and Gerbrand Ceder

Phys. Rev. B 106, 024203 – Published 25 July 2022

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Physical Review B

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An ℓ0ℓ2-norm regularized regression model for construction of robust cluster expansions in multicomponent systems

Peichen Zhong, Tina Chen, Luis Barroso-Luque, Fengyu Xie, and Gerbrand Ceder

Phys. Rev. B 106, 024203 – Published 25 July 2022

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An $ℓ_{0} ℓ_{2}$ -norm regularized regression model for construction of robust cluster expansions in multicomponent systems