Robust cluster expansion of multicomponent systems using structured sparsity

Leong, Zhidong; Tan, Teck Leong

doi:10.1103/physrevb.100.134108

Cited by 32 publications

(32 citation statements)

References 70 publications

(104 reference statements)

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“…Hence, we do not have to worry about dimensionality reduction, i.e., the discrimination between relevant and irrelevant many-body terms, a topic that is beyond the scope of this study, but has been adressed in the dedicated literature. 37,[66][67][68] Of course, these contributions are not known beforehand by the framework which has only access to a limited number of predictions from the reference model Hamiltonian (with eventual Gaussian errors added), just as if the predictions were computed from first-principle calculations.…”

Section: Setup For Proof Of Principlesmentioning

confidence: 99%

Replacing Chemical Intuition by Machine Learning: a Mixed Design of Experiments - Reinforcement Learning Approach to the Construction of Training Sets for Model Hamiltonians

Staub

Steinmann

2021

Preprint

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Model Hamiltonians based on the so-called cluster expansion (CE), which consist of a linear fit of parameters corresponding to geometric patterns, provide an efficient and rigorous means to quickly evaluate the energy of diverse arrangements of adsorbate mixtures on reactive surfaces as typically relevant for heterogeneous catalysis. However, establishing the model Hamiltonian is a tedious task, requiring the construction and optimization of many geometries. Today, most of these geometries are constructed by hand, based on chemical intuition or random choices. Hence, the quality of the training set is unlikely to be optimal and its construction is not reproducible. Herein, we propose a reformulation of the construction of the training set as a strategy-based game, aiming at an efficient exploration of the relevant patterns constituting the model Hamiltonian. Based on this reformulation, we exploit a typical active learning solution for machine-learning such a strategy game: an upper confidence tree (UCT) based framework. However, in contrast to standard games, evaluating the true score is computationally expensive, as it requires a costly geometry optimization. Hence, we augment the UCT with a pre-exploration step inspired by the variance-based Design of Experiments (DoE) methods. This novel mixed UCT+DoE framework allows to automatically construct a well adapted training set, minimizing computational cost and user-intervention. As a proof of principle, we apply our UCT+DoE approach on the CO oxidation reaction on Pd(111), for which a relevant model Hamiltonian has been established previously. The results demonstrate the effectiveness of the custom built UCT and its significant benefits on a DoE-based approach.

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Section: Setup For Proof Of Principlesmentioning

confidence: 99%

Replacing Chemical Intuition by Machine Learning: a Mixed Design of Experiments - Reinforcement Learning Approach to the Construction of Training Sets for Model Hamiltonians

Staub

Steinmann

2021

Preprint

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“…2 Although conceptually well established and practically widely used, the CE method, especially regarding what is the optimal strategy to build CE models, has continuously attracted a lot of interest in the recent decades. [6][7][8][15][16][17][18][19][20][21][22][23][24][25][26][27][28] The main challenge is to build an accurate (unbiased) and robust (with low variance) CE model based on a limited number of training data obtained from…”

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confidence: 99%

Machine Learning Force Field Aided Cluster Expansion Approach to Configurationally Disordered Materials: Critical Assessment of Training Set Selection and Size Convergence

Xie

Zhou

Liu

et al. 2022

Preprint

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Cluster expansion (CE) is a powerful theoretical tool to study the configuration-dependent properties of substitutionally disordered systems. Typically, a CE model is built by fitting a few tens or hundreds of target quantities calculated by first-principles approaches. To validate the reliability of the model, a convergence test of cross-validation (CV) score to the training set size is commonly conducted to verify the sufficiency of training data. However, such test only confirms the convergence of the predictive capability of the CE model within the training set and it is unknown whether the convergence of the CV score would lead to robust thermodynamic simulation results such as order-disorder phase transition temperature $T_{\rm c}$. In this work, using carbon defective MoC$_{1-x}$ as a model system and aided by the machine-learning force field technique, a training data pool with about 13000 configurations has been efficiently obtained and used to generate different training sets of the same size randomly. By conducting parallel Monte Carlo simulations with the CE models trained with different randomly selected training set, the uncertainty in calculated $T_{\rm c}$ can be evaluated at different training set size. It is found that the training set size that is sufficient for the CV score to converge still leads to a significant uncertainty in the predicted $T_{\rm c}$, and that the latter can be considerably reduced by enlarging the training set to that of a few thousand configurations. This work highlights the importance of considering large training set for building the optimal CE model that can achieve robust statistical modeling results, and the facility provided by the machine-learning force field approach to efficiently produce adequate training data.

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“…Columns of Π S are the values for the included correlation functions evaluated for each of the training structures, and are referred to as correlation vectors. The several regression models that have been proposed in literature [4,[14][15][16] can be separated broadly into those dealing with over-determined linear systems and those dealing with underdetermined linear systems.…”

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confidence: 99%

“…The motivation for using an over-determined system comes from several studies showing decreasing cross validation errors with increasing number of training structures. [16,17] However this approach can be prone to over-fitting and lower prediction accuracy [18] which has usually been addressed with 2 regularization (ridge regression), [14] or by using some form feature selection, such as step-wise fitting, [17] evolutionary algorithms, [19,20] or some other form of p regularization. [16,21,22] In contrast, for the case of under-determined systems more correlation functions are included in the model than structures used when fitting.…”

mentioning

confidence: 99%

“…[16,17] However this approach can be prone to over-fitting and lower prediction accuracy [18] which has usually been addressed with 2 regularization (ridge regression), [14] or by using some form feature selection, such as step-wise fitting, [17] evolutionary algorithms, [19,20] or some other form of p regularization. [16,21,22] In contrast, for the case of under-determined systems more correlation functions are included in the model than structures used when fitting. In this case the main issue becomes degeneracy of solutions, which is also addressed by performing feature selection via 1 or 0 regularization in order to seek out sparse solutions only.…”

mentioning

confidence: 99%

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Sparse expansions of multicomponent oxide configuration energy using coherency & redundancy

Barroso-Luque,

Yang,

Ceder

2021

Preprint

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Compressed sensing has become a widely accepted paradigm to construct high dimensional cluster expansion models used for statistical mechanical studies of atomic configuration in complex multicomponent crystalline materials. However, strict sampling requirements necessary to obtain minimal coherence measurements for compressed sensing to guarantee accurate estimation of model parameters are difficult and in some cases impossible to satisfy due to the inability of physical systems to access certain configurations. Nevertheless, the dependence of energy on atomic configuration can still be adequately learned without these strict requirements by using compressed sensing by way of coherent measurements using redundant function sets known as frames. We develop a particular frame constructed from the union of all occupancy-based cluster expansion basis sets. We illustrate how using this highly redundant frame yields sparse expansions of the configuration energy of complex oxide materials that are competitive and often surpass the prediction accuracy and sparsity of models obtained from standard cluster expansions. I. INTRODUCTIONThe Cluster Expansion Method (CEM), [1-3] has become a standard tool in computational thermodynamics of multi-component crystalline systems as an effective and efficient means to compute functions of atomic configuration. The CEM is used to represent a coarse-grained lattice model of materials properties as a function of the possible atomic configurations, i.e. the occupations of crystallographic sites. The most common use of the CEM involves the expansion of the formation energy of a particular multicomponent material system. A cluster expansion is fitted using formation energies for a set of representative structures calculated from first principles electronic structure methods, most often calculated with density functional theory (DFT). Well-converged cluster expansions fitted to DFT energies have been shown to give very accurate predictions of formation energy to within a few meV/atom. [3-5] A converged cluster expansion can subsequently be used in Monte Carlo simulations to compute thermodynamic properties, such as configuration entropy, [6, 7] phase diagrams, [8] ionic percolation,[9, 10] and short range order. [11,12] * †

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Robust cluster expansion of multicomponent systems using structured sparsity

Cited by 32 publications

References 70 publications

Replacing Chemical Intuition by Machine Learning: a Mixed Design of Experiments - Reinforcement Learning Approach to the Construction of Training Sets for Model Hamiltonians

Replacing Chemical Intuition by Machine Learning: a Mixed Design of Experiments - Reinforcement Learning Approach to the Construction of Training Sets for Model Hamiltonians

Machine Learning Force Field Aided Cluster Expansion Approach to Configurationally Disordered Materials: Critical Assessment of Training Set Selection and Size Convergence

Sparse expansions of multicomponent oxide configuration energy using coherency & redundancy

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