Symbolic regression in materials science via dimension-synchronous-computation

doi:10.1016/j.jmst.2021.12.052

Abstract

Abstract:

There is growing interest in applying machine learning techniques in the field of materials science. However, the interpretation and knowledge extracted from machine learning models is a major concern, particularly as formulating an explicit model that provides insight into physics is the goal of learning. In the present study, we propose a framework that utilizes the filtering ability of feature engineering, in conjunction with symbolic regression to extract explicit, quantitative expressions for the band gap energy from materials data. We propose enhancements to genetic programming with dimensional consistency and artificial constraints to improve the search efficiency of symbolic regression. We show how two descriptors attributed to volumetric and electronic factors, from 32 possible candidates, explicitly express the band gap energy of NaCl-type compounds. Our approach provides a basis to capture underlying physical relationships between materials descriptors and target properties.

Key words: Symbolic regression, Band gap, Dimensional calculation

Changxin Wang, Yan Zhang, Cheng Wen, Mingli Yang, Turab Lookman, Yanjing Su, Tong-Yi Zhang. Symbolic regression in materials science via dimension-synchronous-computation[J]. J. Mater. Sci. Technol., 2022, 122: 77-83.

Figures/Tables 7

Fig. 1. Knowledge extraction by dimensional-synchronous-computation symbolic regression (SR-DSC).

Table 1. Potential physical constants for band gap.

Name	Abbreviation	Value	Unit
Electronic charge	E	$1.602\times {{10}^{19}} $	A⋅s
Bohr radius	$ {{a}_{0}} $	5.290×10^-11	m
Madelung constant	${{\alpha }_{\text{NaCl}}}$	1.740	1

Table 2. Parameters of SR-DSC.

Name	Value
Terminal condition	The best expression is stable for the last generation 20.
Population	1000
Operator	${+,-,\times,/,\exp,\ln,{{x}^{\frac{1}{3}}},{{x}^{0.5}},{{x}^{-1}},{{x}^{2}},{{x}^{3}},\ \sum,\ \prod,{{-}^{}},{{/}^{}},\text{exc}{{\text{h}}^{*}} }$
Terminal	15 descriptors and 3 physical constants
Score	R² + Unit filter
Upper limit of depth	2
Target unit	eV

Fig. 2. Pearson correlation map of the initial 32 descriptors. Blue and red colors denote positive and negative correlations, respectively. The fraction of the colored circular sector in each pie chart corresponds to the absolute value of the associated Pearson correlations coefficient. For each descriptor with A, B sites, we use the mean value of the A and B correlation values.

Table 3. 15 representative descriptors. ÅÅ(100 pm), $\text{eV} $(1.602 ×10-19J).

Name	Abbreviation	Unit
Cell volume	${{V}_{\text{c}}}$	ÅÅ³
Electron density	${{\rho }_{\text{e}}}$	ÅÅ^-3
Electronegativity (Martynov & Batsanov)	${{\chi }_{\text{mA}}},{{\chi }_{\text{mB}}}$	ÅÅ^-1
Atomic volume (Villars, Daams)	${{V}_{\text{aA}}},{{V}_{\text{aB}}}$	10⁷pm3
Lattice constants a	${{a}_{\text{A}}},{{a}_{\text{B}}}$	ÅÅ
Lattice constants c	${{c}_{\text{A}}},{{c}_{\text{B}}}$	ÅÅ
Covalent radii	${{r}_{\text{cA}}},{{r}_{\text{cB}}}$	ÅÅ
Ionic radii (Shannon)	${{r}_{\text{isA}}},{{r}_{\text{isB}}}$	ÅÅ
Core electron distance (Schubert)	${{d}_{\text{ceA}}},{{d}_{\text{ceB}}}$	ÅÅ
Latent heat of fusion	${{H}_{\text{fA}}},{{H}_{\text{fB}}}$	$\text{eV}$
Energy cohesive (Brewer)	${{E}_{\text{cohA}}},{{E}_{\text{cohB}}}$	$\text{eV}$
Total energy	${{E}_{\text{totA}}},{{E}_{\text{totB}}}$	$\text{eV}$.
Effective nuclear charge	${{N}_{\text{csA}}},{{N}_{\text{csB}}}$	1
Valence electron number (Slater)	$\text{VE}{{\text{C}}_{\text{A}}},\text{VE}{{\text{C}}_{\text{B}}}$	1
Electron number	${{N}_{\text{eA}}},\ {{N}_{\text{eB}}}$	1

Fig. 3. Symbolic regression flow. (A) SR-GP flow. (B) Dimension calculation system. (B1) Standard unit conversion. (B2) Transformation of Unit to the array of dimension. (B3) Dimensional operation. (B4) Transformation of the array of dimension to the unit. (C) Tree mode comparison of SR-GP. (C1) SR-BGP. (C2) Both are the tree model of a general weighted average formula. The symbol ‘$S$’ implies that these operators contain no processing but act as placeholders. For better comparison, the height of the expression, ${{h}_{\text{BGP}}}$ for SR-BGP is defined as the maximum number of connections in the double-layers, corresponding to the usual definition for SR-GP.

Fig. 4. Symbolic regression results. (A) Comparison of the iterative optimization curve for three methods, ${{R}^{2}}$ and L are scores of determination coefficient and the number of terms in the best expressions, respectively. The iteration process is repeated until the ${{R}^{2}}$ of the best expression does not change for 20 generations. (B) ${{R}^{2}}$ comparison with common machine learning models and SR-DSC. (C) Stability results with different data sizes. (D, E) Comparison of the calculated band gap using Eq. (1) and experimental band gap for (D) NaCl-type and (E) cubic ZnS-type compounds. The ${{R}^{2}}$ (Determination Coefficient) and MAE (Mean Absolute Error) for NaCl-type compounds are 0.93 and 0.65 eV, and 0.79 and 0.35 eV for cubic ZnS-type compounds, respectively. Tendency of ${{E}_{\text{g}1}}$ and ${{E}_{\text{g}2}}$ vs experimental band gap for (F) NaCl-type and (G) cubic ZnS-type compounds. The grey dotted lines are reflected trends.

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