Journal of Materials Informatics

Open Access Research Article

Correspondence to: Prof. Junjie Li, State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, 127 West Youyi Road, Xi'an 710072, China. E-mail: lijunjie@nwpu.edu.cn ;Prof. Jincheng Wang, State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, 127 West Youyi Road, Xi'an 710072, China. E-mail: jchwang@nwpu.edu.cn

© The Author(s) 2022. **Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, sharing, adaptation, distribution and reproduction in any medium or format, for any purpose, even commercially, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Construction of the structure-property (SP) relationship is an important tenet during materials development. Optimizing microstructural information is a necessary and challenging task in understanding and improving this linkage. To solve the problem that the experimental microstructures with a small size usually fail to represent the entire sample structure, a data-driven scheme integrating two-point statistics, principal component analysis, and machine learning was developed to reasonably construct a representative volume element (RVE) set from the small microstructures and extract optimized structural information. Based on the elaborate quantitative metrics and method, this kind of RVE set was successfully constructed on an experimental microstructure dataset of ferrite heat-resistant steels. Moreover, to remove redundant information included in two-point statistics, the critical threshold of the tolerance factor related to the coherence length in microstructures was determined to be 0.005. An accurate SP linkage was finally established (mean absolute error < 6.28 MPa for yield strength). This scheme was further validated on two other simulated and experimental datasets, which proved that it can offer scientific nature, reliability, and universality compared to traditional strategies. This scheme has a bright application prospect in microstructure classification, property prediction, and alloy design.

Microstructural information, structure-property linkage, two-point statistics, machine learning

A great acceleration in target prediction and alloy design can be realized through materials informatics including multitudinous advanced data-driven technologies and theories related to materials science, which has received extensive attention in recent years^{[1-3]}. Establishing process-structure-property (PSP) linkages represents a recognized core task for achieving this ambitious goal. Indeed, significant efforts have been made to pursue such an accurate and universal linkage^{[3-5]}.

Traditionally, materials development is largely completed by a mix of Edisonian approaches and serendipity, which can extract experiential process-property (PP) relationships from existing experimental data, and then studies to understand and explain the dominant mechanism leading to the expectations or serendipity through investigating microstructural features^{[6]}. Using an informatics strategy that is different from the traditional experiment methods simply guided by physical metallurgy knowledges, we previously demonstrated the improvement effect of microstructural information in predicting the hardness of austenite steels by comparing PP and PSP linkages^{[7]}. Similarly, Molkeri *et al*. proposed a novel microstructure-aware framework for materials design and rigorously confirmed the importance of microstructure information in alloy design^{[8]}. One can find that the focus of attention on microstructure has gradually shifted from providing scientific explanations to practically promoting the forward and reverse process of PSP. Therefore, it is necessary to quantificationally extract microstructure information^{[9-13]}. Generally, some physical parameters based on statistical average (phase fraction, grain size, *etc*.) are used to simply characterize the microstructural features of materials, which nevertheless ignores the correlation and heterogeneity of these features in terms of spatial distribution. In addition, considering the entire discrete and highly nonlinear micrograph data as partial input of a PSP linkage, one may be at risk of dimensional disaster due to the inapplicability of some common dimension reduction algorithms^{[14]}. Therefore, a challenge to be addressed is how to extract sufficient and effective microstructure information including correlation and heterogeneity of spatial features in a quantitative and low-dimensional manner.

Some admirable efforts have been made to overcome this challenge. Sangid *et al*. used crystal plasticity simulations to identify the stress concentration around pores of various sizes and quantify the pore with the smallest size that results in a debit in the fatigue performance of IN718 alloy^{[15]}. Zinovieva *et al*. proposed a multi-physics methodology combining physically based cellular automata to simulate the grain structure evolution^{[16]}. They successfully uncovered the effects of scanning pattern on the microstructure and elastic properties of 316L austenitic stainless steel prepared by powder bed-based additive manufacturing. It is noted that, although the two works mentioned above used advanced simulation methods to investigate the PSP relationship, the high computational cost and high-dimensional data analysis process were not completely avoided. Popova *et al*. developed a data-driven workflow and applied it to a set of synthetic AM microstructures obtained using the Potts-kinetic Monte Carlo (kMC) approach^{[17]}. They finally correlated process parameters in the kMC approach with the predicted microstructures. The chord length distributions method used in their workflow addresses the quantification of the grain size and shape distributions and their anisotropy in a microstructure. However, other important microstructural features, such as the volume fraction of the phase of interest, fail to be extracted by this method^{[18-21]}. Fortunately, Fortunately, a rigorous quantitative framework based on the ^{[22-25]}. As the basis of the ^{[26, 27]}. However, an experimentally obtained microstructure with a small size always includes limited structure information and cannot be used as a representative volume element (RVE); it thus can hardly be associated with the macroscopic mechanical properties. One may emphasize a compromised scheme of combining chemical compositions and experimental conditions to fill the gaps^{[7, 9]}. Unfortunately, this indirect strategy cannot essentially eliminate the statistical error caused by the small size of microstructures. Another surrogate solution is to approximate the statistics of an RVE by averaging that of multiple subdomains of the entire sample based on the assumption of statistical homogeneity. It is conceivable that more details of structure will be captured by these subdomains [a single domain refers to a statistical volume element (SVE) in this study] with a higher resolution. Niezgoda *et al*. proposed a novel concept called the RVE set consisting of a certain number of SVEs with the minimum size^{[28]}. Through accessing the convergence of a quantitative metric ^{[29-32]}. Therefore, it is necessary to explore new and universal methods for optimizing microstructure information to construct an RVE set and thus build a more reliable PSP linkage.

The effective information of two-point statistics is compressed in the central area after centralized transformation, leading to the statistics of an RVE containing a great deal of redundancy in the area with a large length of ^{[7, 33, 34]}. Determining the boundary of these two areas is beneficial for analyzing and understanding structure features, especially for the features related to length scales such as average grain size. Through an example of Al-alloy matrix composites, Tewari *et al*. found that numerous length parameters that characterize spatial heterogeneity and clustering of SiC particles can be extracted from two-point statistics^{[35]}. Niezgoda *et al*. further defined a concept named coherence length,

where ^{[22]}. ^{[14, 30, 34, 36, 37]}. Thus, determining the threshold of

Principal component analysis (PCA)^{[38]}, a popular dimensionality reduction algorithm, can effectively address the above challenge of high-dimensional data and has been widely applied to many fields, such as grain coarsening^{[39]}, microstructure evolution during creep^{[31]}, nonmetallic inclusions in steels^{[37]}, *etc*. Interestingly, one can project statistical features of microstructures into a PCA space and compare them using some common distance metrics such as Euler distance^{[40-43]}, which provides a potential solution for optimizing microstructural information by constructing an RVE set and removing redundancy. More importantly, low-dimensional features of microstructures obtained by PCA can be input into machine learning (ML) models to establish high-fidelity PSP linkages^{[13, 44-48]}.

In the present study, we developed a new scheme to build a more reliable structure-property (SP) linkage by optimizing microstructural information, which can be extended to a higher-ordered PSP linkage in the future. Taking an example of ferrite heat-resistant steels, we performed a series of experiments and built a small dataset. This kind of steel has become one of the main materials for the heavy and thick components of advanced ultra-supercritical (A-USC) power plants due to its high thermal diffusivity and low cost^{[49]}. Significant efforts have been made to understand the SP linkage and improve mechanical properties at a high temperature (650 ℃) for the steels^{[50-53]}. Using PCA and two-point statistics, we propose a new method and metric to construct an RVE set from the small SVEs of the steels. We also explored the effect of different redundancy-truncation levels of two-point statistics on the established ML model and determined the acceptable threshold of the tolerance factor *et al*.^{[54]} and simulated data by phase field method (PFM), respectively.

Five alloys were prepared using the raw metals with purity higher than 99.99% by smelting, followed by casting into ingots of

Table 1

Nominal chemical compositions and yield strength (

Label | Mn | Ni | Al | Ti | Mo | W | (MPa) |

Alloy 1 | 1.06 | 2.64 | 0.80 | 0.24 | 0.08 | 0.04 | 254 |

Alloy 2 | 0.96 | 3.20 | 1.12 | 0.08 | 0.24 | 0.08 | 299 |

Alloy 3 | 1.54 | 3.04 | 1.20 | 0.08 | 0 | 0.32 | 276 |

Alloy 4 | 0.60 | 3.12 | 1.12 | 0.08 | 0.32 | 0 | 336 |

Alloy 5 | 1.44 | 2.56 | 1.20 | 0 | 0.04 | 0.52 | 325 |

All microstructures were binarized by an image processing technique named Otsu's threshold processing^{[55]}. This technique is a nonparametric and unsupervised method of automatic threshold selection for picture segmentation. It selects a threshold automatically from a gray level histogram, and the threshold is equal to the one specific pixel value ^{[56]}. Moreover, the discrete microstructures were transformed to a uniform dimension using the transform.rescale(

Figure 1. Experimental microstructures and corresponding binary results for Alloy 1. The magnification of the microscope and the number of microstructures at each magnification is listed at the top, while the grid number is given at the bottom.

To eliminate the impact of data magnitude differences on the performance of the model, before establishing the SP linkage by a ML model, all of the microstructural features (low-dimensional representativeness of the microstructures)

The circular sample, as displayed in Figure 2A, may rotate during OM characterization, leading to inconsistent statistics of the same microstructure in different reference frames. To filter out the dependence of the statistics on the observer reference frame, we employed rotationally invariant two-point statistics (RI2SS) to capture the important structural details^{[23]}. For a certain local state

Figure 2. Example of a microstructure and corresponding autocorrelations: (A) schematic diagram of sample preparation; (B) binarized micrograph; (C) autocorrelations of the black phase; and (D) enlarged drawing of the central domain in (C) labeled by the dotted black circle.

where *etc*.) of the microstructure are contained in the central area that is enlarged in Figure 2D. In addition, the invariant value in the blue area is approximately equal to the square of the peak and represents redundant information.

PCA was used to reduce the dimensionality of autocorrelations. One can obtain principal component scores (PCs), i.e., low-dimensional features of a microstructure, through projecting its autocorrelations into a new space supported by several orthogonal basis vectors. The vectors are ordered and selected according to their explained variance that reflects the main variation of the samples. Mathematically, the original autocorrelations can be reconstructed by

where

We used a classical ML regression model, Ridge regression^{[57]}, to build the SP linkage. By imposing the penalty

where ^{[58]}. All of the models keep the default hyperparameters.

The performance of these models was quantified by root mean square error (

We propose a data-driven scheme for building SP linkage including five modularity: dataset preparation, data preprocessing, microstructural information extraction, microstructural information optimization, and SP linkage construction [Figure 3]. All parameters and corresponding explanations are listed in Table 2. The details of applying this scheme to the Ferrite steels are as follows:

Figure 3. Scheme of optimizing microstructural information and constructing SP linkage. Note that the arrows with a colored filling point to the modeling path of SP linkage, and the arrow with dotted lines represents that the ML modeling is also used to determine the optimized

Table 2

Parameters used in this study and corresponding explanation list

Parameters | Explanation |

Random vector thrown into a microstructure | |

Local state of interest (austenite phase and ferrite phase in our case) | |

One-point statistics of local state | |

Cell node indexed the spatial domain of a microstructure | |

The total number of valid trials associated with discrete vector | |

Microstructure function representing the volume fraction of local state | |

Two-point cross-correlation statistics of local state | |

Two-point autocorrelation statistics of local state | |

Coherence length of a realistic structure | |

Tolerance factor related to redundant information in two-point statistics. | |

The number of a microstructure sample or two-point statistics in a set | |

The number of all two-point statistics in a set | |

Pixel value in a digital microstructure | |

Inputted feature array of the ML model | |

Outputted property array of the ML model | |

Predicted property of the | |

The average property of the samples | |

Mean of inputted features | |

Variance of inputted features | |

Yield strength at 650 ℃ | |

The coefficients of | |

The complexity parameter that controls the amount of shrinkage in Ridge model | |

The | |

The | |

The ensemble average of the two-point statistics | |

The dimensionality of the two-point statistics | |

The experimental dataset in this study | |

The set of true size of the SVEs, i.e., | |

The optimal size of the SVEs in an RVE set | |

The set of the number of SVEs with different sizes, i.e., | |

The optimal number of the SVEs in an RVE set | |

The number of randomly selected samples in the subset, here | |

The number of primary grains in the PFM model | |

Anisotropy coefficient of the solid-liquid interfacial energy in PFM model | |

Nucleation supercooling in PFM model | |

Pair correlation function of a microstructure | |

Root-mean-square error between the two-point statistics of each SVE and the target ensemble-averaged statistics |

1) Creating experimental dataset

2) Preprocessing microstructure and property data by Otsu's threshold processing and normalization operation mentioned above.

3) Extracting quantitative information of all microstructures by RI2SS.

4) Optimizing microstructural information to represent the structural features in the whole sample for each alloy. This procedure includes two sub-paths labeled by the colored arrows in the orange box in Figure 3:

a) Constructing RVE set (confirming the size and number of the included SVEs). We randomly selected different numbers

where

We also propose a novel method called "recursive addition" to confirm the optimal number

b) Removing redundant information of the autocorrelations. We truncated the autocorrelations in the constructed RVE set by controlling the different maximum lengths of

5) Establishing SP linkage. By inputting the low-dimensional features of the RVEs for the five alloys, we trained a Ridge regression model and assessed its accuracy in predicting yield strength. It is noted that this process was also used to validate the reliability of the methods proposed in Procedure (4).

This scheme shown in Figure 3 was also performed on the dendrite solidification data from PFM for Al-Cu alloys and experimental data of Ni-Fe-based superalloys. The reliability and generalization ability of the scheme were also considered in this study.

Following the workflow shown in Figure 3, we traversed all possible combinations with the variation of

Figure 4. The distribution of the averaged autocorrelations of a certain number SVEs in the PCA space. The PCs are grouped into five clusters that are distinguished by the size

To quantify the interclass and intraclass convergence observed above, we used Equation (5) and (6) to calculate the normalized average distances,

Figure 5. Evaluation of convergence and reliability of ensemble averaged autocorrelations: (A) schematic diagram of evaluation metrics,

As for the volume of the RVE set, we propose a novel method named recursive addition based on Euler distance in the PCA space. Figure 6B explains the rationality of the method by using a defined distance,

Figure 6. (A, C-F) Distance

We then employed Ridge regression to extract SP linkage. The inputs of the model are the low-dimensional features (

Figure 7. Distribution of the alloys #1-#5 in the PCA space and prediction accuracy (

To verify the generalization ability of the proposed method in constructing the RVE set, we performed this method on a dataset of dendrite solidification of Al-Cu alloys simulated by PFM^{[59-63]}. The parameters of PFM are listed in Table S1. The dataset includes 48 microstructures that are produced by controlling solidification parameters including the number of primary grains (

Two-point statistical autocorrelations contain valuable information concentrated in the central area and vast redundant information in the peripheral area. Following the procedures shown in Figure 3, we truncated the autocorrelations of the microstructures in the RVE set, as displayed in Figure 8A. The maximum modulus of the vector

Figure 8. Variation of autocorrelations and PC variance with different lengths of

While removing redundancy in statistical autocorrelations, the distribution of the SVEs in the RVE set in the PCA space was also altered, as shown in Figure S6. When

Figure 9. Variation of intraclass variance, interclass variance and

Tolerance factor

By calculating pair correlation function (PCF) of the average autocorrelations in the RVE set, we modified the left-hand side of Equation (1) as ^{[23]}. For convenience, the item is simply expressed as

Figure 10. Variation of

Using the autocorrelations with different

We further employed the procedure in Figure 3 on a Ni-Fe-based superalloy dataset to explore the impact of redundancy removement on the accuracy of SP linkage and the threshold ^{[54]}. It is noted that the microstructures shown in Figure S7 are extremely different from our experimental ones shown in Figure 1 in terms of morphology. Extraction of low-dimensional features and construction of SP linkage on this superalloy dataset are visualized in Figures S8 and S9. Eventually, the accuracy of the models along with the change of

Quantitative comparison between two microstructures has always been a fascinating issue. To complete this task, Niezgoda *et al*. developed a metric ^{[28]}. When the errors for all SVEs are small and close to each other, the amount of information included in the two-point statistics of these SVEs will be saturated and independent of the size and number of the SVEs. Niezgoda *et al*. used ^{[22]}. Figures S10E provides the variation of

Our proposed quantitative metrics based on PCA successfully overcome the defects above. From Equation (5) - (7), we can find that the metrics are distance measurement between the low-dimensional features ^{[29, 34, 43]}. For the constructed RVE set in the present study,

In summary, the differences between our metrics and

Optimization of microstructural information in this study includes two aspects: construction of RVE set and removement of redundancy. There are two premises for an RVE set: (1) the size of members is large enough to ensure statistical homogeneity in the spatial distribution of structural features that can be mapped to macroscopic mechanical properties; and (2) the dispersion of structural features in the RVE set should match the entire material sample^{[28]}. The microstructures contained in the RVE set are independent of their size and location in the samples [Figure 4 and 5], which meets the first condition. The RVE set absorbs enough structural features that its average autocorrelations converge in PCA space [Figure 6], indicating the second condition has been met. In addition, the method was successfully applied to the datasets of experimental ferrite steels, dendrite solidification of Al-Cu alloys simulated by PFM and experimental Ni-Fe-based superalloys collected in the literature, and SP linkage with high precision was established by Ridge regression. These impressive results demonstrate the advantages of scientific nature, reliability, and universality.

The developed method is an improved version of the average approximation method to construct an RVE set, which is also not readily applicable to the samples with microstructure gradients, for instance, some additively manufactured samples with coarse columnar grains where the "average grain size" characteristic is meaningless^{[64]}, the samples with high inhomogeneity in size or distribution of the thermodynamic phases where the average treatment loses the local variation nature of the structure^{[65]}, and so on. A rough solution to obtain the statistical information of the overall sample from the SVEs with local gradients is reserving all the original two-point statistics of the SVEs. Nevertheless, dimensional disasters are beyond the scope of conventional dimensionality reduction algorithms, such as PCA. If one insists on extracting two-point statistics of the sample by ensemble averaging the statistics from multiple SVEs, the long-range correlations may be missed. In other words, the SVEs size used to construct an RVE set must exceed the coherence length of the microstructure when the long-range order plays a significant role in the physics of the system^{[28]}. Therefore, the construction of RVE for the samples with microstructure gradients or inhomogeneity remains a challenging task, which is one of the active areas that we will investigate in the future.

Another interesting topic in this study is the tolerance factor ^{[66]}. However, the interval length ^{[67]}. The microstructures in Figure 3 by Wang *et al*. were used to analyze the problem above^{[67]}. Figures S11 provides these microstructures (with four groups of different combinations of

In the practical application of the proposed scheme, flexible feature selection is allowed, such as the addition of other necessary factors in addition to the low-dimensional PC features of microstructures. The factors here can be directly measured from the material samples or filtered by feature engineering. The factors that may be important to the yield strength (grain size, precipitated phase, dislocation, *etc*.) were indeed ignored in our study, resulting in a seemingly "capped" predictability of the final model even with the optimized parameter selection, i.e.,

A major strength of the proposed scheme comes from its ability to extract reliable low-dimensional features by optimizing structural information in an RVE set. Using the features, one can place the microstructures into correct classes by flexibly combining supervised or semi-supervised ML algorithms to study the relationship between structural features and resulting properties for a special material system such as Ag-Al-Cu ternary eutectic alloys or superalloys with multiple strengthening patterns^{[40, 43, 68]}. Therefore, the scheme can accelerate and improve the procedure of microstructure classification. In addition, our previous study proved that introducing two-point statistical information on microstructures can enhance PSP linkages, which may be further improved by the scheme in this study^{[7]}. We believe that it is competent to predict mechanical properties for most material systems, especially in the case of long-term service in a harsh experimental environment^{[31-33]}. Unfortunately, to our best knowledge, there is no effort to apply two-point statistical information to realize the goal of alloy design. Molkeri *et al*. proved that explicit incorporation of microstructure knowledge in the materials design framework can significantly enhance the materials optimization process^{[8]}, and we previously developed an iterative strategy to search ultra-strength martensitic stainless steels in a global-oriented manner^{[48]}. These two studies provide confidence that our scheme [Figure 3], combined with the previously proposed iteration strategy, can be applied to rapidly discover new alloys in experiments. In conclusion, there is a broad application prospect of our scheme in microstructure classification, property prediction, and reverse engineering for designing new materials.

We propose a novel scheme to construct SP linkage by optimizing microstructure information, which is achieved via employing RI2SS, PCA, and Ridge regression. A small experimental dataset for ferrite steels was created. We designed reliable and robust distance metrics,

Conception and design of the study: Hu X, Li J, Wang J

Data analysis, visualization, and interpretation: Hu X, Zhao J, Wang J

Method design and modeling: Hu X, Wang J

Materials preparation: Hu X, Chen y, Zhao J, Wangm Y, Wu Q

Writing: Hu X, Li J, Wang Z, Wang J

Review and editing: Li J, Wang Z, Wang J

Resources, supervision, and project administration: Li J, Wang J

Supplementary materials are available from the Journal of Materials Informatics or from the authors.

This work was supported by the National Natural Science Foundation of China (Grant No. 51871183 and 51874245). The authors also thank the High-Performance Computing Center of Northwestern Polytechnical University, China, for the computer time and facilities.

The authors declared that there are no conﬂicts of interest.

Not applicable.

Not applicable.

© The Author(s) 2022.

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Hu X,
Zhao J,
Chen Y,
Wang Y,
Li J,
Wu Q,
Wang Z,
Wang J. Structure-property modeling scheme based on optimized microstructural information by two-point statistics and principal component analysis.
* J Mater Inf* 2022;2:5. http://dx.doi.org/10.20517/jmi.2022.05

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