Virtual sample generation in machine learning assisted materials design and discovery

Bootstrap

Bootstrap is a resampling-based technique in statistical learning to estimate standard errors, confidence intervals, and biases^[26]. As shown in Figure 1, the core of Bootstrap is to calculate the confidence interval of an estimator by repeated sampling from the original data. A new dataset formed from the original data by repeated sampling with replacement is called Bootstrap data, with the true values of the unknown information estimated by calculating the statistical parameters of these Bootstrap samples^[27,28]. In Bootstrap, there exist columns of independently and identically distributed samples X = [x₁, x₂, … x_m] with the distribution function of F_m. By sampling X^* = [x₁^*, x₂^*, …, x_m^*] from X, and the distribution function $$ \hat{F}_{m}^{2} $$ of the X^* could be obtained. If $$ \hat{F}_{m}^{2} $$ is a good estimate of F_m, then the relationship between X and F_m can be fully reflected in the relationship between X^* and $$ \hat{F}_{m}^{2} $$, and the statistical distribution of the overall samples can be estimated by repeated sampling. The Bootstrap method can be used to generate virtual samples when the data size cannot meet the modeling needs. The detailed steps of using Bootstrap to generate virtual samples from small original data are described as follows:

Figure 1. The scheme of Bootstrap.

Arrange the original dataset X = [x₁, x₂, … x_m] in ascending order to obtain ascending statistics and randomly generate integers i₁, i₂, … i_m ∈ [1, m].

According to the subscripts corresponding to the generated integers i₁, i₂, … i_m ∈ [1, m], perform the repeated sampling with replacement on the original small-sample dataset X = [x₁, x₂, … x_m] to obtain a new Bootstrap dataset X^* = [x_i1, x_i2, …, x_im].

Repeat step (2) n times to obtain a virtual dataset X^* = [x₁^*, x₂^*, …, x_m^*] that can reflect the overall characteristics, and the generated virtual sample data volume is m*n.

The advantage of Bootstrap lies in the fact that it does not require the assumptions about the distribution of samples in advance. In general, accurately determining the distribution of samples can be challenging. When analyzing the distribution of a dataset, if the assumptions made deviate significantly from the true distribution, substantial errors may occur. Furthermore, results obtained based on assumptions about the sample distribution often necessitate hypothesis testing to assess the reasonableness of those assumptions. Bootstrap is a non-parametric method by repeated sampling only with replacement, where the population estimators can be derived from sample estimators. In addition, Bootstrap has obvious advantages over other statistical methods under the conditions of complex or unknown data distribution. As long as there exist the original samples, plenty of virtual samples could be generated by simply repeated sampling with Bootstrap for modeling. However, it is also important to acknowledge the limitations of the Bootstrap method. Since Bootstrap relies heavily on repeated sampling, the accuracy and reliability of the virtual samples can be significantly compromised when the dataset is extremely small or not representative of the overall population. Moreover, Bootstrap could not analyze and extract potentially valuable information from the samples. Consequently, the generated virtual samples may not effectively bridge the information gap arising from the limited size of the original sample set. This limitation may lead to significant errors when employing the model to predict unknown datasets beyond the scope of the original data. Furthermore, the characteristic of random repeated sampling could result in substantial variability in the distribution of the generated samples. Particularly when dealing with sparse sample sizes, the probability density distribution of the virtual samples may deviate significantly from the assumed Gaussian distribution. This deviation can affect the accuracy and reliability of statistical analysis and predictions based on the generated samples. Rubin et al. proposed an enhanced version of the Bootstrap method named Bayesian Bootstrap, which combines Bootstrap with Bayesian algorithms to sample the original data^[29]. Different from the traditional Bootstrap approach, which randomly samples with replacement from the data, Bayesian Bootstrap treats the weight vector in the empirical distribution function as an unknown parameter and simulates its posterior distribution. This integration of Bayesian principles enables a more refined sampling process. Notably, the samples generated by Bayesian Bootstrap tend to exhibit a probability density distribution that closely approximates a Gaussian distribution. This improvement enhances the reliability and accuracy of statistical analyses and predictions based on the generated samples.

The VSG method based on Bootstrap has been applied to materials machine learning with small data. Zhu et al. used Bootstrap-VSG combined with an extreme learning machine (ELM) to construct a prediction model of acetic acid consumption in a purified terephthalic acid (PTA) process^[27]. The results show that compared with the original 30 samples, the ELM model constructed with 90 samples improves the prediction accuracy of the model with 30 samples by 27.48% after Bootstrap generates 60 virtual samples. Based on Bayesian Bootstrap, Han et al. proposed a Bootstrap-Bayesian dynamic modification model for the small sample size^[28]. Firstly, Bootstrap confidence interval estimation was used to obtain the interval range at the mean 95% confidence level and generate Bootstrap samples that are uniformly distributed over the interval. Secondly, Bayesian parameter estimation was used to achieve Bayesian posterior estimates of the Bootstrap-based mean as virtual samples to extend the training sample set. The advantage of the Bootstrap-Bayesian dynamic modification model lies in that the virtual samples generated based on Bootstrap confidence intervals can provide more valid sample information. The Bootstrap-Bayesian dynamic modification model has been successfully applied to the underwater target classification, but few relevant applications have been found in the materials field, indicating the great potential for applications in materials research.

Monte Carlo

The basic principle of Monte Carlo is to use random sampling techniques to approximate the simulation probability, obtaining the simulation results on the basis of random repeated sampling and statistical analysis^[30]. In terms of calculating the probability of an event occurrence, the Monte Carlo method can estimate the corresponding probability by repeated sampling and calculating the frequency of the event. Monte Carlo is based on a probability model. Different from Bootstrap, which samples from the original sample, Monte Carlo analyzes the data distribution of the samples and then conducts random repeated sampling according to the data distribution. Consequently, the virtual samples generated by Monte Carlo align more closely with the probability distribution of the original data. According to the process described by the probability model, the results are simulated as an approximate solution to the problem, which includes three main steps:

The construction of the probability distribution model: Select an appropriate probability distribution to create a probability model to convert a non-random problem into a random problem.

Sampling according to the probability distribution: The core of the Monte Carlo method is sampling, which can make the obtained random numbers conform to the constructed probability distribution model to obtain samples that satisfy a certain probability distribution. Commonly used Monte Carlo sampling methods include rejection sampling, inverse sampling, acceptance-rejection sampling, importance sampling, Markov Monte Carlo sampling, and Latin Hypercube sampling.

The calculation of the estimator: After obtaining the samples that conform to the probability distribution model, it is required to calculate the statistical parameters of each sample as the estimator of the overall samples to investigate the simulation results and obtain the solution to the problem.

The VSG could be achieved with the Monte Carlo method by treating each input variable as a random variable, constructing the probability model based on the statistical characteristics^[31]. Then the virtual samples reflecting the spatial characteristics of the probability distribution could be generated by sampling. The advantage of Monte Carlo lies in the principle of only repeated sampling, leading to a simpler calculation method and procedure. For stochastic tasks, Monte Carlo methods excel in the ability to directly simulate solutions that closely approximate the real solution. However, Monte Carlo relies on random sampling, while computers can only generate pseudo-random numbers instead of true random numbers. Moreover, unlike traditional methods that can precisely estimate the error, Monte Carlo is based on a probabilistic model that can only guarantee the error meets the accuracy requirement with a certain probability. The sampling technique in Monte Carlo is completely random, indicating that the samples can be located anywhere within the range of the input data distribution. Samples are more likely to be drawn from regions of the distribution with a high probability of occurrence. However, if the number of samples is small, samples with low probability may not be drawn in sufficient numbers. As a result, the samples tend to cluster in areas with a high probability of occurrence and may not accurately represent the distribution of the original sample.

Particle swarm optimization

PSO originates from the study of predation behavior of birds to seek the optimal solution through cooperation and information sharing among individuals in the group^[32]. It is assumed that a flock of birds is randomly searching for a piece of food in an area. All the birds do not know the exact location of the food, but they know how far the current location is from the food. The flocks let other birds know their location by passing the respective information to each other during the searching process to figure out whether they have found the optimal solution for the food location. At the same time, the information of the optimal solution could also be transmitted to the whole group. Eventually, the entire flock of birds can gather around the food to find the optimal solution. In PSO, every solution to the optimization problem is a bird in the search space, which could also be called a “particle”. All the particles have the respective fitness value determined by the optimized function and a velocity to determine the searching direction and distance. PSO is initialized as a group of random particles to iterate to find the optimal solution. In the iteration, each particle is characterized by its position and velocity, which use their individual historical optimal positions (p_best) and the group historical optimal positions of all other members (g_best) to find the searching direction and update their positions and velocities until the convergence conditions are met. The virtual sample generated by PSO is to obtain the optimal virtual sample by searching in the sample space. The particle swarm algorithm aims to minimize the error between the actual value and the expected value in the generation of virtual samples^[33-34]. In order to reduce the error between the actual value and the predicted value, PSO searches for a better combination in the input data to ensure the validity of the virtual samples, where the process is transformed into a nonlinear constrained optimization problem:

Where f(x) is the objective function; x_iL and x_iU are the lower and upper limits of the search area of the input variable x; x_i is the input variable; a is the number of input attribute variables; c is the number of constraints; G_k(x) is a nonlinear constraint. The solutions to the problem obtained by PSO are the optimal virtual samples. The advantages of PSO include its minimal number of adjustable parameters, the ability to generate virtual samples that closely approximate real values, and the absence of a requirement to know the distribution of samples in advance. However, there are also limitations to consider. Due to the limited number of adjustable parameters, the selection of parameter settings can greatly affect the model performance. Furthermore, the particle swarm tends to exhibit “convergence” as it approaches the optimal solution, causing the particle speeds to decrease and potentially resulting in a local optimal solution. Currently, PSO has been successfully applied to multi-objective optimization tasks, and this PSO-based VSG technique offers valuable insights for the multi-objective optimization of materials.

Gong et al. proposed the VSG technique of MC-PSO, which efficiently combines Monte Carlo and PSO algorithms^[31]. Firstly, the ELM is combined with the original training set of small data to construct a model for prediction. Then the probability distribution of the original dataset is evaluated and sampled using the Monte Carlo method, with the sampled data used as the initial points for the PSO searching. The feature values of the virtual samples could be obtained by PSO searching. The process of PSO searching is terminated when the expected value of the fitness function or the maximum number of iterations is satisfied. The target variable of the virtual samples is calculated by the constructed ELM model. Finally, the virtual samples would be added back to the original dataset to train the new ELM model to evaluate the performance with the test set. MC-PSO has been successfully applied to the ethylene industry capacity prediction. Compared with the initial 30 samples, the root mean square error (RMSE) of the ELM model decreased from 0.0387 to 0.0220 after 60 virtual samples were generated and added to the dataset by MC-PSO.

Mega trend diffusion

The theoretical basis of the MTD technology is diffusion neural networks, which combine information diffusion with neural networks and regard data points as a data center with a fuzzy normal distribution in a certain interval^[35]. Two new sample points are spread symmetrically on both sides of these data points using a symmetric spread function. Therefore, each sample can obtain two virtual samples after the diffusion to expand the data size of the original small data and fill the information gap caused by small data. Li et al. took the integrity of the sample data distribution into consideration and proposed the method of MTD based on diffusion neural networks^[36]. Different from diffusion neural networks, MTD considers the integrity of the data and generalizes the single-point diffusion to the overall diffusion, which could asymmetrically expand the attribute domain of the sample with the possibility of the occurrence of the sample reflected by the triangular membership function value^[37]. The scheme of MTD is shown in Figure 2, and the steps to generate virtual samples with MTD technology are as follows:

Figure 2. The scheme of MTD. MTD: mega trend diffusion.

In the dataset X = {x₁, x₂, …x_n}, max and min represent the maximum and minimum values of a certain property of the sample; CL is the center point of the sample property to be calculated by the formula of (max+min)/2;N_L and N_U are the sample sizes smaller and larger than CL, respectively.

The lower bound LB and upper bound UB of the acceptable domain can be calculated by the following formulas:

Where Skew_L and Skew_U represent the left and right skewness of sample diffusion, respectively; $$ \hat{S}_{x}^{2} $$ is the sample variance; n is the total number of samples.

Randomly draw n_v virtual samples according to a uniform distribution.

Calculate the membership function value MF of the observation point x_i to express the importance of the sample and the possibility of occurrence.

MTD is a VSG technique developed and proposed based on information diffusion. The value of the triangular membership function is used to represent the likelihood of occurrence of sample points. The advantage of MTD to generate virtual samples lies in considering the integrity of the samples, taking the whole samples as the object of diffusion, making full use of the distribution trend of the samples, and deeply exploring the useful information between the samples. However, the virtual samples generated by MTD are randomly drawn according to a uniform distribution, which may cause the obtained virtual samples to not conform to the distribution of the original samples. Due to the non-uniformity of realistic sampling, the values of N_L and N_U can be very different, resulting in unreasonable diffusion areas due to large left and right offsets, Skew_L and Skew_U. Zhu et al. proposed a novel multi-distribution MTD technique by adding a correction quantity factor S_p on the calculation of Skew_L and Skew_U to prevent excessive Skew_L and Skew_U due to the non-uniformity of sampling^[38]. The improved Skew_L and Skew_U formulas are shown below:

Yu et al. combined the advantages of both the MTD technique and the Monte Carlo method to develop the VSG technique of MC-MTD for generating effective, high-quality virtual samples to expand the original sample set to improve the learning ability and generalization ability of the model^[39]. First, the MTD is used to estimate the acceptable range of each dimensional attribute of the sample, calculate the relevant data, and model the probability distribution of the sample according to the triangular affiliation function. Then, the Monte Carlo method is used for sampling to generate virtual samples. MC-MTD has been successfully applied to the multilayer ceramic capacitor (MLCC) dataset and the PTA dataset. The mean absolute percentage error of ten independent tests was reduced from 5.3% to 3.7% after adding 100 virtual samples using MC-MTD to the 25 initial samples of MLCC. The mean absolute percentage error of ten independent tests was reduced from 1.25% to 0.94% after adding 250 virtual samples using MC-MTD to the 25 initial samples of PTA.

Gaussian mixture model

As shown in Figure 3, a GMM is a probabilistic model that assumes that all data points are generated from a mixture of a finite number of Gaussian distributions^[40,41]. If n observations X = {X₁, …, X_n} are generated by a mixture distribution P, where each vector X_i is p-dimensional, and the distribution P is composed of G components. Then the maximum mixture likelihood function of the distribution is shown in the formula:

Figure 3. Flowchart of the proposed GMM-VSG. Reproduced with permission from ref.^[40] Copyright 2022, Elsevier. GMM: Gaussian mixture model; VSG: virtual sample generation.

Where f_k(x_i|θ_k) represents that X_i is the k-th density function; θ_k is the corresponding parameter; π_k is the weight parameter. If f_k(x_i|θ_k) is a multivariate normal distribution, then P is the Gaussian mixture distribution, where θ_k consists of the mean value µ_k and the covariance matrix ∑_k. The density function f_k(x_i|θ_k is shown in the formula:

The Gaussian mixture distribution P can be described by the probability density function represented by the weighted average of G Gaussian density functions, and the specific description is shown in the following formula:

A GMM is a density estimation algorithm that can be used to construct a probability model for small data. After calculating the parameters with the EM algorithm, the virtual samples that meet expectations could be generated through the model of the probability density distribution. Virtual samples generated by GMMs have been used for materials process optimization and industrial optimization. While GMM-based VSG techniques have shown success in material design and discovery, there is still potential for improvement in the GMM approach. One limitation is the potential presence of outliers in the virtual samples generated by GMM that may not accurately represent actual materials. Additionally, materials data often exhibit a range of eigenvalues, making it challenging to ensure that the generated data align with the characteristics of real materials. Currently, the analysis of anomalies in GMM-generated virtual samples relies heavily on empirical domain knowledge. However, in the future, objective analysis tools can be developed to better identify and analyze anomalies in these virtual samples.

In 2022, Shen et al. used a GMM combined with XGBoost to construct a machine learning model for predicting the wear resistance of rubber materials through mechanical properties^[40]. The authors collected 24 rubber Acrolon abrasion test data as target property and corresponding six mechanical properties as descriptors from the publications and applied the algorithms of XGBoost, LASSO, support vector regression (SVR), and RF to construct the model. The results show that after generating 295 virtual samples with a GMM, the model constructed by XGBoost has the best prediction accuracy, with the R² of the test set reaching 0.95, which has been improved by 41% compared with the prediction accuracy before the original 24 samples. Our team also successfully designed the yttria-stabilized zirconia (YSZ) thermal barrier coating materials with high bonding strength using GMMs combined with machine learning^[42]. First, eight experimental bonding strengths of YSZ thermal barrier coatings under four different atmospheric plasma spraying (APS) parameters were collected as the target property, and the corresponding APS parameters were taken as descriptors. Then, after expanding the data size from 8 to 400 with the GMM-VSG, the models were constructed and compared with various algorithms, including the ordinary least square (OLS), linear regression (LR), RF regression (RFR), decision tree regression (DTR), partial least squares (PLS), multiple LR (MLR), artificial neural network (ANN), and SVR with different kernel functions. The R and RMSE of leaving-one-out cross-validation (LOOCV) before and after data expansion by GMM are shown in Figure 4A and B. The results indicate that the SVR with polynomial kernel has the highest accuracy after a GMM generates virtual samples, and the R of LOOCV is as high as 0.989, while the R of LOOCV of the optimal ANN constructed by the original eight datasets could only reach 0.758. After sensitivity analysis and virtual sample analysis, we broke through the limit of the maximum bonding strength of 46.6 MPa in the original eight data and designed a sample with a predicted bonding strength of 53.137 MPa. After experimental validation, the experimental value of bonding strength has reached up to 55.5 MPa, and the absolute error with the model predicted value is only 2.363 MPa.

Figure 4. Experimental bonding strength vs. predicted bonding strength of LOOCV with corresponding R and RMSE based on (A) original data and (B) after data expansion by a GMM. Reproduced with permission from ref^[42]. Copyright 2022, American Chemical Society. GMM: Gaussian mixture model. LOOCV: leaving-one-out cross-validation.

Random forest

RF primarily learns the distribution and features of samples by constructing multiple decision trees to generate a new set of virtual samples^[43,44]. In RF, multiple decision trees are combined for prediction and classification, with each decision tree constructed based on different random samples and features. Virtual samples are generated through the Bagging process in RF, where different random samples and feature sets are used to train models in each decision tree, resulting in the generation of diverse virtual samples^[45]. The basic process of RF includes: (1) Using the RF model to train the original data; (2) Generating a specified number of virtual samples, where the target variable of each virtual sample is predicted by the RF model, and the feature is obtained by random extraction from the original data. The RF model is used to predict the target variable of virtual samples as it can effectively deal with high-dimensional features and nonlinear relationships. Generating features by random sampling of the original data can increase the diversity of virtual samples and improve the generalization ability of the model. However, the RF-based VSG may lead to a concentration of generated samples and a lack of diversity. More randomness can be introduced when generating virtual samples, such as introducing noise, random perturbation, or other data enhancement techniques to increase the diversity of the generated samples.

Generative adversarial networks

The VSG technology based on GANs is a deep learning technique that generates virtual samples similar to the original data through adversarial training^[46]. A GAN comprises two neural networks: a generator and a discriminator. The generator generates virtual samples from random noise vectors, while the discriminator distinguishes between real and virtual data by detecting differences between them^[46,47]. During the training process, the generator network continuously generates more realistic virtual samples, and the discriminator network continuously identifies differences between real and virtual data. This competitive process gradually leads to the generation of more realistic virtual samples by the generator network and improved accuracy of the discriminator network. Once both networks are optimally trained, the generator network can produce high-quality and diverse virtual samples, while the discriminator network can accurately differentiate real and virtual data. GAN technology has been widely applied in various fields, such as image, video, and audio generation^[48,49]. In cases of insufficient data, GAN-based VSG technology can expand the dataset and enhance the performance and generalization ability of the model. However, the training process of GANs may not be stable enough, and the balance between the generator and discriminator is difficult to achieve, resulting in the generator generating low-quality samples or the discriminator not being able to effectively distinguish between real and virtual samples. In order to improve the stability of training and the quality of generated virtual samples, different generator and discriminator architectures can be applied, such as deep convolutional generative adversarial networks (DCGAN), Wasserstein generative adversarial networks (WGAN), and conditional generative adversarial networks (CGAN), etc.

Zhu et al. proposed a CGAN-based VSG method (CGAN-VSG) to identify sparse regions of the data and generate the output of new virtual samples^[50]. First, the local anomaly factor algorithm is used to identify discrete points in the dataset, which are also the potentially sparse regions. Then, the k-means++ is used to collect the discrete points to obtain the clustering centers^[51]. Intermediate interpolation is performed between the centroids to generate target values for the virtual samples. The eigenvalues of the virtual samples are then generated using a CGAN. The datasets of two-dimensional criterion functions and three-dimensional (3D) criterion functions are used to verify the validity of the CGAN-VSG method. In addition, CGAN-VSG was used to predict the melt index (MI) for practical industrial applications in the production of high-density polyethylene (HDPE). After generating 120 virtual samples using five methods, including Bootstrap, MTD, etc., the MI prediction model for HDPE was developed by combining back-propagation networks. The comparison shows that the RMSE of the model trained by CGAN-VSG is the lowest at 0.4995, while the RMSE of the model without the VSG technique is as high as 0.5630.

Particle swarm optimization

The PSO algorithm is a population-based optimization algorithm to be used not only for generating virtual samples but also for materials searching^[52-54]. The basic idea of PSO is to transform the optimization problem into a searching problem in a multidimensional space, where each materials sample is treated as a particle and the properties or structures to be searched are regarded as the objective function. The algorithm updates the position and velocity of each particle continuously to search for the optimal solution. The choice of the objective function should take into account the practical application requirements of the materials.

In the PSO algorithm, the particle swarm is composed of multiple particles, where each particle represents a material sample. Each particle has an initial randomly generated position vector and velocity vector, where the position represents the structural parameters of the materials and the velocity represents the change of the position. Based on the current position and velocity, the particle position is updated with the value of the objective function calculated using machine learning models. The fitness of each particle is then obtained based on the objective function, where higher fitness represents a performance indicator closer to the desired materials properties. The PSO algorithm is then used to update the position and velocity of each particle, taking into account the historical best solution of each particle and the entire particle swarm. The algorithm continuously calculates fitness to update the position and velocity, ultimately outputting the properties of the materials corresponding to the optimal solution in the particle swarm, which is the desired virtual material.

In materials searching, the PSO algorithm can be used to search for the optimal materials structure and virtual materials with target properties by continuously adjusting the particle position and velocity. Additionally, a multi-objective PSO algorithm can be used to transform multiple objective function optimization problems into a multidimensional space, thus searching for the structure of the optimal materials more comprehensively. With the PSO algorithm, we can quickly and efficiently search for the target structure and property of the virtual samples, providing more reliable theoretical support for materials design. Zheng et al. used the PSO algorithm to optimize the reflectivity band of absorptive materials and compared the bandwidth under different reflectivity conditions between the PSO algorithm and the genetic algorithm (GA)^[55]. The study mainly aimed to optimize the bandwidth of materials with reflectivity less than -15 dB in the range of 2-18 GHz. It was found that the PSO algorithm had good frequency bandwidth under the conditions of reflectivity less than -10 dB and -20 dB, and corresponding dielectric constants, isolation layer thicknesses, and impedance values could be obtained. After three consecutive optimization searches, the bandwidths of two-layer absorptive materials with reflectivity less than -15 dB were compared. It was found that the PSO algorithm had a bandwidth of about 2 GHz larger than the GA and had better stability. Under the condition of reflectivity less than -20 dB, the bandwidths of two optimization algorithms were compared. The results showed that the bandwidth of the PSO algorithm was 3 GHz larger than that of the GA, and wider bandwidths could be obtained. Finally, under the constraint condition that the reflectivity of three-layer absorptive materials was less than -15 dB, the bandwidths of two optimization algorithms were compared to reveal that the PSO algorithm had a wider bandwidth and the sum of reflectivity was 2 dB less than that of GA. Overall, using the PSO algorithm to optimize the reflectivity band of absorptive materials has a good optimization effect.

Efficient global optimization

The EGO algorithm is a Bayesian optimization method that uses a Kriging surrogate model to predict the unknown objective function values and select the best parameter combination to optimize the objective function^[56,57]. The Kriging surrogate model can be used to map the relationship between material properties and parameters^[58]. By fitting the given data to the Kriging surrogate model, the EGO algorithm can predict the material properties of unknown virtual samples. In each iteration, the EGO algorithm would select the optimal parameter combination and generates new sample points near the combination to update the surrogate model. This strategy can help the algorithm converge quickly to the global optimal solution. The EGO algorithm has wide application potential in materials design and material searching fields^[59]. By modeling and optimizing the objective function, the EGO algorithm can quickly search for materials with excellent performance with fewer computational resources.

After the target properties and features of material design are determined, the EGO algorithm initializes a search space for materials composition and structure based on these features. Next, sampling points would be selected in the search space of virtual samples with their corresponding objective function values calculated by the surrogate model. Then, based on the objective function values of the sampling points, a surrogate model is fit, and the next sampling point is selected. Common selection strategies include confidence intervals and the expected improvement (EI) strategy. Through iterations of selecting sample points and fitting the Kriging surrogate model, the algorithm would keep operating until it reaches a preset stopping criterion, such as the maximum number of sampling points or convergence of the objective function value. Finally, by analyzing the relationship between the objective function values of the sampling points and the material parameters, the EGO algorithm finds the optimal material parameter combination and carries out material preparation and property testing validation.

The EGO algorithm uses the Kriging surrogate model to fit and predict the objective function, enabling it to quickly search for materials with excellent performance with fewer computational resources, and has the advantages of high efficiency and high accuracy. It can also improve search efficiency by introducing multiple surrogate models and has good scalability. However, if the sample dimension in the search space is too high, the search efficiency of the EGO algorithm may be limited, and the optimization results of the EGO algorithm may be affected by the initial sampling points.

In addition to being used for target materials searching, the EGO algorithm can also be combined with an active learning framework. In the active learning process, the core step is to sample important samples from the unlabeled sample pool, and the EGO algorithm can accomplish this task. Zhao et al. combined the EGO algorithm as a representative sampling strategy with an active learning strategy to develop an effective machine learning model for predicting the hardness of 6061-aluminum alloy elements^[60]. First, they used a full-process high-throughput alloy preparation and characterization system to prepare 32 different composition ratios of 6061-aluminum alloys and characterized their hardness. The initial 309 descriptors were constructed by the composition of elements and the knowledge of the alloy field. After feature selection, the remaining five important features were used for modeling. After comparing multiple algorithms, the SVR algorithm with a radial basis kernel function was used to construct the aluminum alloy hardness prediction model. Artificial experience sampling and Bayesian optimization sampling were used to select samples from candidate materials for labeling and subsequent experiments. The Bayesian sampling strategy used four methods, the EGO algorithm, the knowledge gradient (KG) algorithm, the maximum hardness point method, and the maximum error point method, with four data points for each method and a total of 16 experimental alloy compositions were designed for the next round of experiments. Before reaching the convergence condition, the experimental data was returned to the initial dataset for further feature selection and model construction. After three iterations, the results indicated that the adaptive sampling strategy based on Bayesian optimization can more effectively guide experiments than artificial experience sampling, with a 63.03% decrease in an average absolute error (MAE) and a 53.85% decrease in RMSE. The RMSE was 4.49 HV, which was close to the experimental error of 4.05 HV for the test samples. Xue et al. collected 22 Ni-Ti-based shape memory alloys and used atomic parameters as descriptors and thermal hysteresis as the target variable to establish a hysteresis prediction model using the SVR algorithm^[61]. The EGO algorithm was used to search for four samples with low hysteresis from a search space of 800,000 samples, which were then synthesized and characterized through experiments. After experimental validation, the four samples were put back into the training set for iterative modeling, searching, and experimentation. After nine iterations, a total of 36 samples were searched, of which 14 had lower hysteresis than any of the 22 samples in the original dataset. The Ti_50.0Ni_46.7Cu_0.8Fe_2.3Pd_0.2 sample had the lowest hysteresis, which was only 1.84K.

Proactive searching progress

The proactive searching progress (PSP) is a materials search method developed based on the Sequential Model-Based Optimization (SMBO) method^[62,63]. Its core idea is to treat the materials composition to be optimized as parameters and use a lower-cost proxy model to learn the relationship between the existing parameters and properties, and quickly search for the approximate trend of the optimized composition in the chemical space. The PSP method uses a pre-constructed high-precision machine learning model to predict the precise performance of the optimized composition, and after repeated iterations, it can gradually approach the optimized composition o_* with the desired performance^[64].

Assuming that a materials composition space is constructed as {γ_i | γ_i = (γ_i1, γ_i2, …γ_α)^T, i, d ∈ R}, where i and d are positive integers, T represents transpose, and γ_i is any materials composition represented by a vector. γ_i1,γ_i2, …γ_id are the specific components of the materials composition γ_i. Based on this materials composition space {γ_i | i ∈ R}, the pre-constructed high-precision machine learning model can be used to predict the performance values {o_i | i ∈ R} of each composition in this space, forming a set of compositions and performance {(γ_i, o_i) | i ∈ R}. In this process, the pre-constructed high-precision machine learning model actually plays the role of a function mapping between the composition and the performance, that is, o = f(γ), which can be used to describe the distribution of materials performance samples in the global materials composition space.

Combining the idea of SMBO in machine learning model parameter optimization, a proxy model can be constructed to search for optimized compositions with desired properties. In order to shift the materials composition exploration approach from traversal to search, the function mapping relationship between the composition and the performance o' = g(γ) constructed by the proxy model requires continuity and differentiability so that its derivative g'(γ) can be used to determine the direction of the predicted performance o', in order to meet the necessary conditions for search. Since the GPR algorithm has the characteristics of a simple algorithm, parameter-free modeling, and high accuracy, it can be used to construct the proxy model.

The process flowchart for the PSP method is shown in Figure 5. In the first step, several materials components {γ_i | i ≥ 2} are randomly generated from the existing performance distribution in the materials composition space. The number of materials components should be greater than or equal to 2, and their performance is predicted using a high-precision machine learning model to form the dataset {(γ_i EE_i) | i ∈ R} for fitting the surrogate model. In the second step, a GPR model is built based on the existing dataset {(γ_i, EE_i) | i ∈ R}. In the third step, the materials composition that minimizes EE_GP is found in the GPR model and predicted using a high-precision machine learning model. If the prediction error is less than a pre-defined threshold, the sample is outputted. Otherwise, it is added to the existing dataset {(γ_i, EE_i) | i ∈ R}, and a new GPR model is constructed for the next round of search.

Figure 5. General principle of the PSP method applied for searching HOIPs with targeted band gaps. Reproduced with permission from ref^[64]. Copyright 2022, American Chemical Society. HOIPs: Hybrid organic-inorganic perovskites; PSP: proactive searching progress; WVR: weighted voting regressor.

Lu et al. of our team used integrated machine learning techniques and the PSP method to predict the band gaps of hybrid organic-inorganic perovskites (HOIPs)^[64]. The author collected 1,201 samples from publications with 129 atomic descriptors, including atomic radius, electronegativity, tolerance factor, tao factor, and octahedral factor. Then, various ML algorithms were used to construct models. The top 4 models (including CatBoost, XGBoost, LightGBM, and Gradient Boosting Machine) were selected and integrated using weighted voting regression to achieve better performance. The weighted voting regressor (WVR) model achieved R² and RMSE of 0.95 and 0.079 in LOOCV and 0.91 and 0.106 in the test set. Based on the ions collected from the formulas in the dataset, the author constructed a vast material space, including over 8.20*10¹⁸ possible combinations, to explore new HOIP structures with suitable band gaps. As shown in Figure 5, the PSP method effectively searched for material combinations with expected band gap values from the generalized chemical space. As a result of the PSP method, 20,242, 733,848, 764,883, and 746,190 lead-free candidates were designed for HOIPs with band gaps of 1.20, 1.34, 1.70, and 1.75 eV, respectively. To validate the PSP searching results and the prediction ability of the WVR model, the author synthesized new HOIP compositions MASn_xGe_1-xI₃ (x = 0.85, 0.74, 0.66) and conducted experimental validation, with an average error between the experiment and prediction of only 0.07 eV. Although the PSP method enables rapid exploration of a vast space of material components to identify target materials with desirable properties, the SMBO search method employed in PSP can be substituted with alternative search methods. By utilizing different search techniques, not only can diverse search requirements be fulfilled, but also the search efficiency can be further enhanced. Some commonly employed search methods include gradient descent and Newton-like methods.

Genetic algorithm

GA is a computational method that simulates the process of biological evolution, following the principle of natural selection described in Darwin’s theory of evolution^[65,66]. By simulating basic genetic operations, GA can seek optimal or near-optimal solutions to problems. Material inverse design based on GA uses computer simulation and optimization to design virtual materials, with a focus on the materials encoding and evolutionary mutation^[67,68]. In GA, material structures are encoded as a string, such as a binary string or a string of characters. For each structural parameter of the materials, such as lattice constants or atomic positions, a binary digit or character is used for encoding. Materials encoding is more focused on digitizing and compiling the composition and structural information of materials, which is different from material descriptors. For example, organic structures can be encoded using a simplified molecular input line entry system (SMILES)^[69]. Encoding molecules in the SMILES format can effectively save storage space and has been widely used in chemical databases and data-driven molecular design. The evolution process of GA mainly includes selection, mutation, and crossover to generate virtual samples. Through the evaluation and screening of newly generated virtual samples using fitness functions, the virtual samples with high fitness could be left behind and put into the next generation for evolution. The materials inverse design strategy based on GA includes the following steps:

Design of initial population: The initial population is the starting point in the materials inverse design process. It can be generated by random generation or by mutation and recombination based on existing materials.

Definition of fitness function: The fitness function is the criterion for evaluating the quality of each individual. Depending on the characteristics of the materials, goals, and constraints of the inverse design, a fitness function can be constructed. The fitness function is usually the error between the expectation and prediction via the machine learning model.

Execution of genetic operations: Selection, crossover, and mutation are the three basic operations of GAs. These operations can produce new individuals, and selection is performed based on the fitness function in each iteration to evolve more suitable individuals.

Setting the end condition: The inverse design process can be iterated multiple times until the set end condition is reached, such as reaching the maximum number of iterations or the fitness reaching a certain threshold. Materials structures that meet the fitness function are decompiled and outputted.

The GA has been successful in the applications of materials inverse design. For example, the highly efficient mechanical performance of minimal surface structures has drawn widespread attention from scholars due to their lightweight and high strength. However, the existing research has mainly focused on the mechanical properties of different minimal surface configurations. Therefore, the inverse design of surface-like metamaterial configurations that meet the requirements based on mechanical properties is still a research gap. Wang et al. proposed a GA-based algorithm that combines efficient machine learning methods and globally optimal solutions to achieve the inverse design of mechanical metamaterials based on load curves^[70]. To generate a dataset for constructing the machine learning model, a numerical model with different mechanical metamaterial configurations was constructed through the finite element method (FEM) to predict the load curve of the shell-based mechanical metamaterials (SMMs). A total of 7,000 SMM configurations were generated, with 5,500 used for the training set and 1,500 used for the test set. ANNs were used to map the relationship between the load curve and the geometric configuration to achieve forward prediction of the load curve through geometric configuration. The GA was used to search for the SMM configuration that most closely matched the target load curve. The flowchart of the GA-based algorithm for the inverse design of SMMs is shown in Figure 6. The individuals and the chromosomes in a GA are SMM configurations and geometric parameters, respectively. A total of 10,000 random samples were generated as the initial population. 10,000 and 1,000 samples were generated through crossover and mutation, respectively. Finally, 21,000 samples were selected, with 10,000 samples selected as the next population based on the load curve calculated by the previously trained ANN model. The results showed that regardless of considering strain hardening or softening, SMM configurations that closely match the target load curve can always be obtained through GA inverse design. To explore the relationship between the load curve and the deformation mode, the authors designed several SMM configurations based on different types of load curves and then simulated the deformation mode under 4*4*4 unit compression using the FEM. The results showed that single cells with a “hardening” load curve tend to undergo overall deformation during the macroscopic overall structural deformation process, while those with a “softening” load curve tend to undergo layer-by-layer deformation during the macroscopic overall deformation mode. This work has filled the gap in the inverse design of SMMs and can be used to design SMM materials with specific load curves. It also contributes to the structural design concept that combines machine learning and traditional optimization methods.

Figure 6. The flowchart of the genetic algorithm for the inverse design of SMM. Reproduced with permission from ref^[70]. Copyright 2022, Elsevier. SMM: Shell-based mechanical metamaterial.

Maurizi et al. combined machine learning methods with GAs to construct a composite model for the inverse design of non-uniformly assembled lattice materials, with flexural strength as the target variable^[71]. Deep neural networks (DNNs) were used to construct the model to predict the yield strength to optimize the desired lattice architecture. Then, the GA was used to combine simple and diverse basic building blocks into various architectural itineraries to design spaces with DNNs to explore the design space to find the optimal architecture. For the GA-based inverse design, the fitness function is set to consider the average flexural strength and flexural isotropy in different load directions. Given the random nature of the GA algorithm, 100 iterations of optimization rounds were conducted until 12 different candidate designs were obtained. The best performing six design candidates were selected after ranking the 12 candidates based on their fitness values. Three-dimensional printing was used to obtain samples of the designed lattice architectures for mechanical testing, and the test results were consistent with the simulation results, with flexural properties 30%-90% and 10%-30% higher than those of conventional reinforced lattice-like lattices and bionic lattices, respectively. The composite model proposed in this work for the inverse design of material structures can autonomously select basic components to construct more complex primitives to arrange into periodic structures, fully exploiting the periodic and local properties. Based on this composite model, truss grids with better buckling resistance than conventional reinforced lattices or bionic sponge constructions are obtained.

Nigam et al. proposed JANUS, a GA based on the SELFIES representation of molecules^[72]. JANUS consists of iterations triggered by random structures or provided molecules. In each iteration, two fixed-size molecular populations are maintained. Members of each population compete with each other to enter the next generation. In a mutation and crossover design, JANUS relies on the STONED algorithm for the efficient generation of molecules. In terms of selection pressure, JANUS trains a DNN model for each generation to accurately predict the fitness function and uses a classifier to classify high-fitness samples from low-fitness samples. The trained DNN model evaluates the generated offspring samples and adds the highest fitness samples to the population. Inspired by parallel tempering, JANUS maintains two populations that use different genetic manipulations. One population performs a local search of the chemical space, using molecular similarity as a selection pressure. The other performs a global exploration using DNNs as selection pressure. JANUS has been successfully applied to maximize the performance of penalty logarithms for octanol-water partition coefficient scoring, protein inhibitor design, and docking scores of protein targets in molecular docking. The successful application of JANUS in the above molecular design cases also demonstrates the great potential in materials inverse design.

Bayesian optimization

Bayesian optimization is an optimization method based on the Bayes theorem. By combining prior knowledge and new observations and continuously updating the posterior distribution, it finds the global optimal or approximate optimal solution^[73]. Compared to traditional optimization algorithms such as gradient descent, the Bayesian optimization algorithm has better performance in finding global optimal or approximate optimal solutions, especially in high-dimensional, noisy, or non-differentiable situations^[74]. Therefore, it has been widely used in many application fields, such as materials inverse design and hyperparameter optimization.

In the field of materials inverse design, Bayesian optimization follows the following Bayesian theorem^[75]:

Y, U, and S represent material properties, target material properties, and material structure encoding, respectively. S can also be encoded as a SMILE file. p(Y ∈ U | S) represents the probability of target properties given a known structure, which can be obtained by training a machine learning model with structure-property relationship data. The prior distribution p(S) is the probability of constructing a structure, which is used to reduce the occurrence of invalid or unstable chemical structures by setting zero or lower probability mass for them. According to the above Bayesian law, as long as p(Y ∈ U | S) and p(S) are obtained, the probability of structure appearance under the target property p(S | Y ∈ U) can be obtained, thereby achieving the inverse design of the materials.

The core of Bayesian optimization can be divided into a generator and an evaluator. The generator refers to the posterior distribution p(S) generated by the algorithm, where p(S) is the most critical factor that affects the structural features of the generated samples, and its calculation formula is as follows:

Where s_i represents the i_th block that forms the materials. The probability of the i_th letter appearing depends on the previous s_i-1,…s₁. As mentioned above, the probability of unstable chemical structures can be effectively reduced by observing the records of subsequent characters given a given substring. The generator could be constructed with the extended n-gram model consisting of a table to record the probability of observing a subsequent character given a substring and a function to modify a given SMILES string based on the stored n-gram probability table. The evaluator refers to the likelihood function that evaluates the fitting degree of S with the attributes, which is the process of obtaining p(Y ∈ U | S) by constructing machine learning models, defined as follows:

The term φ(S) refers to the material descriptor. The property of the i_th sample Y_i is a function of the descriptor φ(S). The σ_i(φ(S)) and μ_i(φ(S)) are the mean value and standard deviation of property Y_i, respectively, predicted by the forward machine learning model. With known p(S) and p(Y ∈ U | S), the posterior model p(S | Y ∈ U) can be obtained, thus enabling the design of material structures with ideal performance.

As shown in Figure 7, Wu et al. developed a Bayesian-based inverse molecular design algorithm called iQSPR-X, which has been integrated into the materials informatics platform XenonPy, and successfully used to design polymers with specific band gaps and dielectric constants^[75]. First, the authors collected 854 polymer structures composed of nine types of atoms from the Polymer Genome (PG) database and used the hybrid Heyd-Scuseria-Ernzerhof (HSE06) exchange-correlation functional to calculate their band gaps and the density functional perturbation theory (DFPT) to calculate the sum of electronic and ionic dielectric constants. Two strategies were considered for training the generator: one was to use the 854 samples collected from PG as the training set, and the other was to add samples collected from PubChem to the PG samples as the training set. Since polymers containing F atoms generally have high band gaps and dielectric constants, 2,485 samples were collected from PubChem, with at least six F atoms in one molecule. The results showed that the generator trained on PubChem data would contain more structures with F fragments during the molecule modification process. In the training of the estimator, a well-trained neural network was first used to evaluate ten descriptors. The results showed that the atomic pair fingerprint with MACCS keys exhibited high performance in predicting band gaps and dielectric constants and was, therefore, selected for Bayesian molecular design. Gradient boosting, RF, and Bayesian LR were used to establish prediction models for band gaps and dielectric constants by combining the optimal descriptors selected by the estimator. The results showed that the gradient boosting model had the best overall performance, with MAEs of 0.320 and 0.142 for band gaps and dielectric constants, respectively, in 5-fold cross-validation, and was, therefore, selected as the optimal algorithm for inverse design. The goal of this work was to design polymers that simultaneously satisfy high band gaps and high dielectric constants, and three different strategies were used for inverse design. Strategy (1) trained the generator using the samples in PG and randomly selected 100 samples as initial samples for inverse design. Strategy (2) trained the generator using the samples in PG and selected 100 initial samples with low band gaps and dielectric constants (ε < 4 or E_g < 4.5 eV) for inverse design. Strategy (3) trained the generator using the samples from PG and PubChem and randomly selected 100 samples from the 854 samples as initial samples for inverse design. The results showed that different strategies led to the design of different polymers. Strategy (2) had difficulty converging to candidate molecules similar to those of strategy (1) and showed difficulties in capturing trends in complex ring structures. Strategy (3) showed a clear trend of attaching F fragments to the chemical structure.

Figure 7. Computational workflow in iQSPR-X. Reproduced with permission from ref^[75]. Copyright 2020, John Wiley and Sons.

This team also used transfer learning and Bayesian molecular design algorithms to develop a highly accurate prediction model for polymer thermal conductivity and screened thousands of polymer materials with high thermal conductivity^[76]. After screening and experimental validation, three candidate samples successfully passed the experimental evaluation. The authors collected a large number of polymer samples from the PolyInfo and QM9 databases, including melting point T_m, glass transition temperature T_g, density ρ, and heat capacity C_p and C_v data, for pre-training model construction. After model comparison, the pre-training model constructed with heat capacity C_v performed the highest prediction accuracy. Next, the parameters of the pre-training model were optimized using 28 samples with thermal conductivity data to be transferred to the prediction of thermal conductivity. The results showed that the transferred model had an MAE of 0.0204W (m·k)^-1, which was 40% lower than the direct training model using 28 data points with RF. Combining with the Bayesian algorithm to design a large number of polymers repeating unit structures, the screening was based on whether the repeating unit contained a liquid crystal structure, experimental feasibility evaluation, glass transition temperature, and the predicted thermal conductivity from the transferred model. Finally, 24 molecular structures were selected, three of which were successfully validated through synthesis and characterization experiments. The experimental results showed that the thermal conductivity of the polymers screened with the assistance of transfer learning and Bayesian molecular design was higher than that of polymer materials published in previous papers. This research also confirms the successful application of transfer learning and Bayesian molecular design in the design and discovery of polymer materials.

Pattern recognition inverse projection

Pattern recognition is a technique used to analyze and process data or signals to identify or classify different patterns or features. With the applications of computer technology, pattern recognition can extend features to a mathematical model of the distribution range of various samples in multidimensional space. Common pattern recognition methods include principal component analysis (PCA) and linear discriminant analysis (LDA).

PCA is a widely used linear dimensionality reduction technique that maps high-dimensional datasets into low-dimensional space while preserving the maximum variance information of the original data^[77]. The basic idea of PCA is to project the original dataset onto a new coordinate system through a linear transformation so that the projected data has the maximum variance to extract the most representative principal components. LDA is a classic pattern recognition method used to divide datasets into different categories and perform classification^[78]. The basic idea is to reduce the dimensionality of the data while maximizing the differences between different categories and minimizing the differences within the same category, thereby achieving effective data classification.

Pattern recognition techniques represented by PCA could compute two principal components linearly combined by descriptors, projecting the material samples onto a plane composed of different principal components for visualization to draw an optimization area according to the projection of samples and obtain the boundary equation of the optimization area. Sampling is performed in the optimization area, and the descriptor values could be calculated according to the PCA equation and deduced to obtain the chemical formula of the materials.

Pattern recognition inverse projection tends to be used in conjunction with machine learning to achieve performance breakthroughs in alloy materials. Yang et al. proposed a machine learning-based alloy design system that integrates database construction, model construction, composition optimization, and experimental validation to guide the rational design of high-entropy alloys with high hardness^[79]. Using pattern recognition inverse projection technology, they successfully designed samples with hardness beyond the original dataset and verified them with experiments. First, they collected 370 samples of high-entropy alloy compositions and Vickers hardness information from the publications. After screening using the Pearson correlation coefficient, RF feature importance ranking, forward feature selection, and best subset method, a high-entropy alloy hardness prediction model was constructed using five important descriptors combined with SVR. The R values of the independent test and LOOCV both reached 0.94. Based on the model, the pattern recognition inverse projection method was adopted to explore the optimized composition of high-hardness high-entropy alloys, as shown in Figure 8A. The inverse projection method can obtain the features of the design samples in the original space, thus obtaining the corresponding compositions of the two designed samples, which are Co₁₈Cr₇Fe₃₅Ni₅V₃₅ and Al₂₀Cr₅Cu₁₅Fe₁₅Ni₅Ti₁₀V₃₀. Using the constructed SVR model to predict the hardness of the design samples, the predicted results were 1,002 HV and 1,028 HV, respectively. The predicted hardness of both optimized samples exceeded the hardness of the highest alloy in the original dataset. These optimized samples have the potential to be high-hardness high-entropy alloy compositions. They synthesized these high-entropy alloy samples using vacuum arc melting to measure the experimental hardness. The results showed that the Co₁₈Cr₇Fe₃₅Ni₅V₃₅ alloy has ultra-high Vickers hardness, reaching 1,148 HV, which is 24.8% higher than the highest hardness alloy in the original dataset.

Figure 8. Materials pattern recognition of different samples by (A) Fisher and (B) PCA. Reproduced with permission from ref^[79]. Copyright 2021, Elsevier. Reproduced with permission from ref^[80]. Copyright 2021, Chinese Materials Research Society. PCA: principal component analysis.

Gold and its alloys, especially ternary gold alloys, have been widely used in the field of electrical contact materials. Due to the complexity of the composition and proportion of ternary gold alloys, designing ternary gold alloy electrical contact materials with low electrical resistance is still a challenge. Wang et al. proposed an inverse design method for low electrical resistance ternary gold alloys, which combines pattern recognition inverse projection with XGBoost to design new materials with lower resistivity than existing ternary gold alloys^[80]. First, the authors collected 51 ternary gold alloy samples at room temperature and pressure from the Materials Platform for Data Science (MPDS) database, including 62 atomic parameters and two-component features. After screening variables using Pearson correlation coefficients and maximum relevance minimum redundancy, five important variables were used for subsequent modeling and inverse design. Using the PCA method to establish an optimization area for low resistivity in Figure 8B, three points were selected as virtual samples in the best pattern recognition projection. Then, the pattern recognition inverse projection method was used to calculate the feature variables of the three virtual samples. Finally, by calculating the Euclidean distance, candidate samples closest to the virtual sample points were obtained, which were AuZr_1.95Cu_0.52, AuZr_1.12Cu₄, and AuSc_1.86Cu_2.75. XGBoost, SVR, multiple LR, and ridge regression were used to construct a prediction model for the electrical resistivity of gold alloys. The results showed that the XGBoost model had the highest R value and the lowest RMSE value, which were 0.850 and 0.331, respectively. The XGBoost predicted resistivity values for the three candidate materials were 6.718, 6.707, and 6.701, respectively, all of which were higher than the maximum value of 6.68 in the original dataset. Therefore, pattern recognition and its inverse projection algorithm can be used for the inverse design of low electrical resistance ternary gold alloy materials.