# Principles of Machine Learning and Its Application to Thermal Barrier Coatings

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Current Status of Machine Learning and Its Application in Materials Design and Development

#### 2.1. Big Data in Materials Science

#### 2.2. Machine Learning Framework for Materials Design and Development

#### 2.2.1. Classes of ML Problems

#### 2.2.2. Feature Engineering and Dimension Reduction

#### 2.2.3. ML Algorithms

#### k-Nearest Neighbor (kNN) Method

#### Decision Tree

#### Neural Networks

#### Support Vector Machines and Support Vector Regression

#### 2.2.4. Ensemble Learning Algorithms

#### Gradient Boosting Tree Algorithm

^{2}are used to evaluate the prediction accuracy of the trained models.

#### Random Forest Algorithm

_{2}C will adopt Heusler structures, based on the composition alone. Very high performance was achieved using this model which successfully predicted 12 novel gallides, namely as Heusler compounds. The RF algorithm was used to train a model using experimentally obtained compounds to predict the stability of half-Heusler compounds [40]. This model demonstrated good performance by retrieving 71,178 compositions and yielding 30 results for further exploration. The random forest algorithm was used to identify a low-thermal-conductivity half-Heusler semiconductor, and results were demonstrated by scanning more than 79,000 half-Heusler entries in the AFLOWLIB database [41].

#### 2.2.5. Deep Learning Algorithms

#### Convolutional Neural Network (CNN)

#### Recurrent Neural Network (RNN)

#### 2.2.6. Model Validation Methods

#### 2.3. ML Applications in Materials Science

#### 2.3.1. Material Property Prediction

#### Shallow Learning Applications

^{2}values exceeding 0.98 and error rates below 4%. Furthermore, various ANN-based shallow learning schemes were employed in material analysis tasks, including the detection of metal corrosion, asphalt pavement cracking, and the determination of concrete strength [46,47,48,49].

#### Ensemble Learning Applications

^{2}) were utilized.

#### DL Applications

#### 2.3.2. New Materials Discovery

#### Shallow Learning Applications

^{2}value of 0.815, while a 5-fold cross-validation procedure yielded an R

^{2}value of 0.742. These quantitative structure–property relationship (QSPR) models were based on counter propagation neural networks (CPG NNs), which learned the connections between the structural profile of guanidinium cations (represented by 92 descriptors) and the melting point of the corresponding salts with 1 of 4 possible anions. The models were validated through an independent test set, a 5-fold cross-validation and y-randomization, demonstrating their ability to provide accurate predictions.

#### Ensemble Learning Applications

_{2}C adopting Heusler structures, relying solely on composition-based descriptors [39]. This model achieved a high true positive rate of 0.94 and successfully predicted 12 novel gallides, namely MRu

_{2}Ga and RuM

_{2}Ga (M = Ti − Co), as Heusler compounds. The random forest algorithm was utilized to train a model using experimentally reported compounds to predict the stability of half-Heusler compounds [40]. The model retrieved 71,178 compositions and yielded 30 results, predominantly matching half-Heusler compounds, for further exploration. Another similar study focused on the identification of low-thermal-conductivity half-Heusler semiconductors [41]. Here, the random forest algorithm was employed to screen over 79,000 half-Heusler entries in the AFLOWLIB database. Potential half-Heusler compounds were considered from all nonradioactive combinations of elements in the periodic table.

#### 2.3.3. ML Approach for Thermal Conductivity Evaluation

## 3. Prediction of Thermal Conductivity of TBC Using ML

- (1)
- Polynomial Regression;
- (2)
- Neural Network;
- (3)
- Gradient Boosting Regressor.

#### 3.1. Data Collection

_{2}O

_{3}partially stabilized ZrO

_{2}, they are widely used since they have a low thermal conductivity. The effect of YSZ particle size, the stabilized materials and other additives that affect the thermal conductivity of coatings on the performance of the coating have been studied in the last years [73,74,75,76,77,78,79].

#### 3.1.1. Basic Information Gathering

_{2}–8wt.%, and EB-PVD PYSZ coatings with 7 wt.% Y

_{2}O

_{3}-stabilized ZrO

_{2}. Details of measurements using the laser flash method were obtained from Section 2.2. Two types of samples were studied in their research: (1). free-standing APS and EB-PVD coating samples with a diameter of 12.7 mm and a thickness of about 300 μm; and (2). two-layer samples that had an EB-PVD coating deposited on bond coated (50 μm) nickel-base super alloy IN625 substrates (0.5 mm).

#### 3.1.2. Data Extracting

- Step 1: Importing plot;
- Step 2: Calibrating x- and y-axis;
- Step 3: Digitizing dataset points;
- Step 4: Exporting dataset.

#### 3.1.3. Dataset Used in the Present Study

- Input variables for the prediction of thermal conductivity:
- Temperature;
- wt.% of Y
_{2}O_{3}; - Thickness of TBC;
- Aging Temperature;
- Aging Time;
- Output: Conductivity.

#### 3.2. Exploratory Data Analysis

#### 3.2.1. Exploratory Graphs

#### 3.2.2. Correlation Analysis and Principal Components Analysis (PCA)

explained = |

29.495 |

21.514 |

15.645 |

13.993 |

10.258 |

9.0946 |

sum(explained) = 100 |

sum(explained(1:4)) = 80.64 |

sum (explained (1:5)) = 90.905 |

#### 3.3. Prediction of Thermal Conductivity Using Polynomial Regression

#### 3.3.1. Polynomial Regression Model

#### 3.3.2. Multistage Predictive Modeling Framework

#### Forward Selection Orthogonal Least Squares Algorithm

_{j}from the remaining (M-j-1) candidates. This procedure is terminated at the pth selection step when

#### Multistage Predictive Modeling Procedure

_{s}= p) selected in this step is mainly determined by the computational considerations for the wrapper method. Sometimes, Step 3 can be skipped if the number of inputs available from Step 2 is already small, and one can proceed to Step 4 directly. As noted before, since the forward selection OLS algorithm can solve the combined model structure search and parameter estimation problem effectively, sometimes it is not possible for Step 3 to reduce the number of inputs or model terms to a manageable size and it is required to come up with the model structure search space in such a way that Step 3 can work with Step 5 directly by skipping Step 4 altogether..

^{2}), adjusted R

^{2}, mean squared error (MSE), mean absolute error (MAE) and maximum absolute error (MAXE) [105]. Some of these evaluation criteria are described later in Model Performance Evaluation.

#### Model Performance Evaluation

- Coefficient of Determination (R
^{2})

^{2}) is a metric and is used to assess the goodness of fit of regression predictions to the actual data points. An R

^{2}value of 1 indicates that the regression predictions perfectly align with the observed data. The most comprehensive definition of the coefficient of determination is as follows:

_{i}” refers to the real data points and “f

_{i}” represents the corresponding predicted values. The values close to 1 for the coefficient of determination indicate a strong predictive power of the selected inputs for the output variable, while values close to 0 indicate a poor fit between the predicted and actual data.

- 2.
- Mean Squared Error (MSE)

- 3.
- Maximum Absolute Error (MAXE)

#### 3.3.3. Polynomial Regression Modeling Results and Discussion

^{2}values obtained for these models using the OLS algorithm on all data samples are given in Table 4. The outputs of this algorithm depicting the ERR and the corresponding selected input indices along with the accumulated sum of ERR are given for polynomial degrees one and two in Table 5 and Table 6, respectively. It can be noted that the last entry in the last column of Table 5 and Table 6 represent the corresponding R

^{2}values (based on all the data samples) for the polynomial regression models as listed in Table 7. This observation also implies that for any polynomial regression model, which is constructed using the subset of model terms by sequentially adding terms from the first row onwards from the 3rd column of Table 5 and Table 6, one can obtain R

^{2}values from the corresponding row in the last column of Table 5 and Table 6.

^{2}performance for the 3rd, 4th, 5th and 6th degree polynomial regression models when trained on all data samples. It is clear from this figure that a fraction of available model terms is sufficient for achieving the R

^{2}performance close to the maximum possible, as given in Table 4. Adding more model terms than necessary will contribute to the overfitting of models as will be seen below.

^{2}, MAXE and MSE performance measures. To validate the models, a holdout cross-validation approach is employed by randomly splitting the data into two sets: 80% for training and 20% for testing. This holdout cross-validation process is repeated 100 times to ensure robustness. The training and testing results from each iteration are averaged to evaluate all the model candidates accurately. Figure 9 and Figure 10 show R

^{2}values for a subset of model candidates generated from 3rd, 4th, 5th and 6th degree polynomial regression models and evaluated on the training, testing and all datasets. Top ten models ranked based on the average R

^{2}values generated on the test dataset for 3rd, 4th, 5th and 6th degree polynomial regression models are shown in Table 7, Table 8, Table 9 and Table 10, respectively.

^{2}, MAXE and MSE performance based on all data samples (with 100 times repeat) for the top 10 models selected from 3rd, 4th, 5th and 6th degree polynomial regression models. The TC prediction performance for the best model (6th degree polynomial regression with 100 terms) from Table 10 is displayed in Figure 12.

#### 3.4. Prediction of Thermal Conductivity using Neural Networks

#### 3.4.1. Neural Network Description

#### Basics of Neural Networks

#### 3.4.2. Training Algorithms

_{i}in the training dataset can be expressed as follows:

_{D}is the mean sum of squares of the network error, D is the training set with input–target pairs and M is the number of neural networks.

#### Bayesian Regularization

_{inc}until the change shown above results in a reduced performance value. The change is then made to the network, and m

_{u}is decreased by µ

_{dec}.

- The maximum number of epochs (repetitions) is reached;
- The maximum amount of time is exceeded;
- Performance is minimized to the goal;
- The performance gradient falls below the minimum threshold;
- µ exceeds µ
_{max}.

#### Levenberg–Marquardt Algorithm

- The maximum number of epochs (repetitions) is reached;
- The maximum amount of time allocated for training is exceeded;
- The performance of the network is minimized to a predefined goal;
- The performance gradient falls below a minimum threshold (min grad);
- The value of µ exceeds a specified maximum (µ max);
- The validation performance (if used) has increased more than a certain number of times (max fail) since the last time it decreased.

#### 3.4.3. Neural Network Training Topology and Details

#### Dataset Splitting

#### Training Parameters Setting

#### Exporting Training Results

#### 3.4.4. Neural Network (NN) Training Results and Discussion

#### Results Using Single-Layer NN

^{2}values, smaller MAE and MSE, which proves that a better prediction model for TC is obtained by using the BR algorithm. It also can be seen that the value of R

^{2}does not increase consistently as the number of nodes increases. In addition, a larger repeating validation value does not mean better prediction results. For the LM algorithm, the best prediction model is obtained when the nn value is 20 and the repeating time is 10. Similarly, the best prediction model is obtained when the nn value is 30 and the repeating time is 10, for the BR algorithm.

#### Results Using Two-Layer NN

^{2}values, smaller MAE and MSE, which proves that a better prediction model for TC is obtained by using the BR algorithm. It can be seen that for LM two-layer networks, the value of R

^{2}shows an increasing trend as the number of nodes increases. In addition, in the LM algorithm with a two-layer neuron, the best prediction model is obtained when a larger nn value is adopted. For the BR algorithm, the two-layer and single-layer neuron generate very similar prediction results.

#### 3.5. Prediction of Thermal Conductivity Using Gradient Boosting Regression

#### 3.5.1. Basics of Gradient Boosting Regression (GBR)

- A loss function to be optimized.

- 2.
- A weak-learner or base-learner model to make prediction.

- 3.
- An addictive model to add base-learners to minimize the loss function.

#### 3.5.2. Gradient Boosting Regression (GBR) Topology and Details

- data= pd.read_excel(“xy_data_NRC.xlsx”, header = 0)
- X0 = data.iloc[range(705),1:6]
- y = data.iloc[range(705),0]
- sc = StandardScaler()
- Xn = sc.fit_transform(X0)
- from sklearn.preprocessing import PolynomialFeatures
- poly_features = PolynomialFeatures(degree = 2, include_bias = False)
- X = poly_features.fit_transform(Xn)
- X, y = shuffle(X, y, random_state = 13)
- X = X.astype(np.float32)

- params= {‘n_estimators’: 500, ‘max_depth’: 4, ‘min_samples_split’: 2, ‘learning_rate’: 0.01, ‘loss’: ‘ls’}
- clf = gbr(**params)
- clf = clf.fit(X_train,y_train)
- scoring = [‘r2’]

#### 3.5.3. Prediction Results Using GBR

^{2}are all above 0.85. By comparing Figure 22, it can be seen that the distribution of predicted values and the prediction error show very similar trends for the neural network and GBR approaches.

#### 3.6. Summary of Prediction of TC Using ML

## 4. Conclusions

- This state-of-the-art review covers areas of AI as applied to materials design, characterization, and development, including big data, available algorithms for both ML and DL, NN, and SVM approaches, and various algorithms.
- This paper has also undertaken the prediction of thermal conductivity (TC) in 6–8 wt% YSZ TBCs using ML models. Recent studies have found the improved capability of ML in predicting TC of TBCs. Various ML models and algorithms have been researched, namely support vector regression (SVR), Gaussian process regression (GPR) and convolution neural network (CNN) regression algorithms.
- A large volume of experimental thermal conductivity (TC) data for YSZ (Yttria-Stabilized Zirconia) thermal barrier coatings (TBCs) has been compiled from the existing literature. This dataset serves as the basis for training, testing and validating ML models. The TC data is strongly influenced by five key factors, which have been identified and considered in this analysis. After collecting the TC data, several preprocessing steps such as sorting, filtering, extracting and exploratory analysis were conducted on the dataset. Three different approaches, namely polynomial regression, NN and GBR, were employed for predicting the thermal conductivity. The training, testing and prediction results obtained from these approaches were carefully analyzed, presented and discussed. Based on the results, it was observed that the NN model using the Bayesian regularization (BR) technique and the GBR approach exhibited better prediction capabilities compared to polynomial regression.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Name and Category | Website and References | Description |
---|---|---|

AFLOWLIB Computational | aflowlib.org [118] | Online computational platform for determining thermodynamic stability, electronic band structures, vibrational dispersions and thermomechanical properties of various inorganic compounds. |

Computational Materials Repository Computational | cmr.fysik.dtu.dk [114] | Material database system supporting a variety of tools for collecting, storing, grouping, searching, retrieving and analyzing electronic structure calculations generated by many modern electronic-structure simulators. |

Crystallography open database Crystallography | crystallography.net [119] | Online database that provides information on a variety of known atomic coordinates of crystal structures of organic, inorganic, metal-organic compounds and minerals collected from several research publications. |

MARVEL NCCR Computational | nccr-marvel.ch [120] | Material informatics platform focusing on the design and discovery of new materials via data driven, high performance quantum mechanical simulations. Research tools, computational data and simulation software accessible through the materials cloud platform. |

Materials Project Computational | materialsproject.org [121] | Online platform that provides access to density functional theory (DFT) calculations on a large number of metallic compounds, energy materials and also mechanical properties of many materials. |

MatNavi(NIMS) General Materials data | mits.nims.go.jp/ index_en.html [122] | Integrated material database system comprising structures and properties for various materials including polymers and inorganic substances. |

Organic materials database Computational | omdb.mathub.io [123] | Electronic structure database of three-dimensional organic crystals that also provides tools for search queries. |

Open quantum materials database Computational | oqmd.org [124] | A high throughput database comprising the thermodynamic and structural properties of the known crystalline solids which are calculated using the density functional theory computation technique. |

Open materials database Computational | openmaterialsdb.se [125] | A high throughput computational database which is based on structures from the Crystallography open database and provides information on the properties of various materials. |

SUNCAT/CatApp Catalysts | suncat.stanford.edu/catapp [126] | Materials informatics center focusing on catalyst and materials design for next-generation energy solutions. Computational results for thousands of surface reactions and online tools accessible at catalysis-hub.org. |

Chemspider Chemical data | chemspider.com [127] | Chemical structure database containing information on physio-chemical properties, interactive spectra, links to chemical vendor’s catalogs, literature references and patents collected from a wide range of data sources. |

Citrination General Materials Data | citrination.com [128] | Materials informatics platform containing information on the computed and experimental properties of various materials and chemicals. |

NIST Materials Data Repository (DSpace) General Materials Data | materialsdata.nist.gov/dspace/xmlui [129] | File repository that accepts materials data in any format related to specific research publications. The repository is implemented using a technology called Dspace. |

NanoHUB Nanomaterials | nanohub.org [130] | Premier online resource that offers course materials, lectures, seminars, tutorials, professional networking and interactive simulation tools for nanotechnology. |

Nanomaterials Registry Nanomaterials | nanomaterialregistry.org [131] | A central web-based repository that provides links to associated journals and publications, interactive simulation tools, computational results and information such as physio-chemical characteristics, and biological and environmental study data for different nanomaterials. |

NIST Interatomic Potentials Repository Computational | ctcms.nist.gov/potentials [132] | A reliable source for interatomic potentials and related files for various metals. Evaluation tools to help researchers judge the quality and applicability of their interatomic models are also available. |

PubChem Chemical data | pubchem.ncbi.nlm.nih.gov [133] | A database that contains information on chemical substances and their biological activities. |

TEDesignLab Thermoelectrics | tedesignlab.org [134] | A virtual platform that contains raw experimental and computational thermoelectric data and a suite of interactive web-based tools that help in the design of new thermoelectric material. |

UCSB-MRL thermoelectric database Thermoelectrics | mrl.ucsb.edu:8080/datamine/thermoelectric.jsp [135] | A large repository created by extracting thermoelectric materials data from several publications. |

Name and Category | Website and References | Description |
---|---|---|

Inorganic Crystal Structure Database Crystallography | cds.dl.ac.uk/cds/datasets/ crys/icsd/llicsd.html [136] | Repository providing information of various inorganic crystal structures. |

Cambridge Crystallographic Data Centre Crystallography | ccdc.cam.ac.uk/pages/Home.aspx [137] | Non-profit organization that compiles and maintains the Cambridge Structural Database, which contains information of various organic and metal organic small molecule crystal structures. |

NIST Standard Reference Data General Materials Data | nist.gov/srd/dblistpcdatabases.cfm [138] | Generic material property data that provides measurable quantitative information related to physical, chemical or biological properties of known substances. |

CALPHAD databases (e.g., Thermocalc SGTE) Thermodynamics | thermocalc.com/products-services/ databases/thermodynamic [139] | Journal publishing the experimental and theoretical information on phase equilibria and thermochemical properties of various materials. |

ASM Alloy Center Database Alloys | mio.asminternational.org/ac [140] | Database for researching accurate materials data of compositions, properties, performance details and processing guidelines from authoritative sources for specific metals and alloys. |

ASM Phase Diagrams Thermodynamics | asminternational.org/AsmEnterprise/APD [141] | Online repository that provides information related to binary and ternary alloy phase diagrams and associated crystal data for many alloy systems. |

MatDat General Materials Data | matdat.com [142] | Online database that provides information on published design-relevant material data to the industrial, academic and research community. |

Pauling File General Materials Data | paulingfile.com [143] | Online database that includes information on the crystal structures, physical properties and phase diagrams for various non-organic solid-state materials. |

Springer Materials General Materials Data | materials.springer.com [144] | Materials research platform that provides curated data for identifying material properties and a set of advanced functionalities for data analysis and visualization of materials properties. |

Total Materia General Materials Data | totalmateria.com [145] | Online materials database that includes search and cross-reference tools, chemical composition, properties and specifications for various metals, polymers, ceramics and composites. |

Year | Author | Material | Research Topic | |
---|---|---|---|---|

1 | 1998 | Taylor [146] | Al_{2}O_{3} and ZrO_{2} and of four and eight alternating layers of Al_{2}O_{3}–ZrO_{2} | TC vs. temp and different thickness |

2 | 1998 | Raghavan [147] | 5.8 wt.% yttria YSZ | TC vs. temp and densities (% of theoretical) and grain diameters (in nm) |

3 | 1999 | An [86] | Al_{2}O_{3} and 8YSZ | TC vs. temp |

4 | 2000 | Zhu [148] | EB-PVD. ZrO2-8 wt.%Y_{2}O_{3} (8YSZ) | TC vs. time for different thickness |

5 | 2002 | Nicholls [87] | EB-PVD TBCs 7YSZ | TC vs. Yttia (wt%), TC vs. T and grain size; thermal conductivities of dopant modified EB-PVD TBCs at 4 mol% addition and 250 mm thickness; data measured at room temperature |

6 | 2002 | Zhu [149] | YSZ-Nd-Yb and YSZ-Gd-Yb; 8YSZ | TC vs. temp and time; TC vs. total dopant concentration |

7 | 2004 | Cernuschi [150] | 8Y_{2}O_{3}ZrO_{2}, 22 wt.%MgO–ZrO_{2}, and 25 wt.%CeO_{2}–2.5Y_{2}O_{3}–ZrO_{2} | TC vs. temp for different cycles |

8 | 2004 | Jang [88] | EB-PVD ZrO_{2}-4 mol% Y_{2}O_{3} | TC vs. substrate thickness (areal thermal diffusion time) |

9 | 2004 | Singh [89] | EB-PVD 8YSZ, ZrO_{2}–8% Y_{2}O_{3} HfO_{2}-40% wtZrO_{2}-27 wt%Y_{2}O_{3} | TC vs. time and number of layers |

10 | 2004 | Matsumoto [90] | ZrO_{2}–Y_{2}O_{3}–La_{2}O_{3} | TC vs. La_{2}O_{3} content % |

11 | 2005 | Wolfe [151] | ZrO_{2}– 8 wt.% Y_{2}O_{3} | TC vs. time and number of layers |

12 | 2006 | Renteria [76] | three morphologically different EB-PVD PYSZ TBC | TC vs. temp and time |

13 | 2006 | Rätzer-Scheibe [91] | EB-PVD PYSZ | TC vs. temp and thickness |

14 | 2006 | Ma [152] | SPPS-7YSZ and SPPS LK-Zr | TC vs. temp and time |

15 | 2007 | Almeida [92] | EB-PVD 2O_{3}–ZrO_{2} | TC vs. temp |

16 | 2007 | Rätzer-Scheibe [84] | EB-PVD ZrO_{2}–7wt.%Y_{2}O_{3} | TC vs. temp and heat treatment time and thickness |

17 | 2007 | Schulz [93] | EB-PVD (Three types) FeCrAlY; PYSZ | TC vs. temp; aging time |

18 | 2008 | Jang [94] | EB-PVD ZrO_{2}–4 mol% Y_{2}O_{3} | TC vs. number of layers, porosity |

19 | 2009 | Matsumoto [95] | EB-PVD YSZ, La_{2}O_{3} and HfO_{2} | TC vs. annealing time |

20 | 2010 | Yu [153] | plasma sprayed Sm_{2}Zr_{2}O_{7} | TC vs. temp and different heat-treating temperature |

21 | 2011 | Jang [96] | EB-PVD ZrO_{2}–4 mol% Y_{2}O_{3} | TC vs. coating thickness |

22 | 2011 | Liu [97] | EB-PVD 7wt% Y_{2}O_{3} (7YSZ) | TC vs. substrate rotation speed |

23 | 2012 | Limarga [154] | EB-PVD 3wt% Y_{2}O_{3} (3YSZ) | TC vs. temp and different heat-treating temperature and time |

24 | 2012 | Łatka [155] | ZrO_{2}+8 wt.%Y_{2}O_{3} (8YSZ) | TC vs. temp |

25 | 2012 | Zhang [156] | (La_{0.95}Mg_{0.05})2Ce_{2}O_{6.95} (La_{0.95}Mg_{0.05})2Ce_{2}O_{6.95} La_{2}Ce_{2}O_{7} | TC vs. temp |

26 | 2013 | Jang [157] | ZrO_{2}–4 mol.%Y_{2}O_{3} (TZ4Y) | TC vs. temp and different sintered temp and different GD_{2}O_{3} percentile |

27 | 2013 | Bobzin [98] | EB–PVD 7YSZ, La_{2}Zr_{2}O_{7}, 7YSZ + Gd_{2}Zr_{2}O_{7} DCL, Gd_{2}Zr_{2}O_{7}, 7YSZ + Gd_{2}Zr_{2}O_{7} | TC vs. temp |

28 | 2013 | Sun [158] | Yb_{2}O_{3}–Y_{2}O_{3}–ZrO_{2} | TC vs. temp |

29 | 2013 | Zhao [159] | EB-PVD ZrO_{2} Y_{2}O_{3} (8YSZ), 4TiYSZ, to 16TiYSZ | TC vs. temp |

30 | 2014 | Jordan [160] | SPPS YSZ TBCs with IPBs | TC for different trials |

31 | 2014 | Lu [161] | LSMZATO, La_{1−x}Sr_{x}Mg_{1−x}Zn_{x}Al_{11} xTixO_{19} | TC vs. temp |

32 | 2014 | Wang [162] | YSZ/NiCoCrAlY | TC vs. temp (numerical) |

33 | 2015 | Rai [163] | YSZ and GZO | TC for different layer and thickness |

34 | 2016 | Guo [164] | 1RE_{1}Yb–YSZ 1La_{1}Yb–YSZ | TC vs. temp |

35 | 2016 | Arai [165] | YSZ (0, 5, 10, 15 wt%) | TC vs. porosity, width of pore, at 570 K |

36 | 2016 | Guo [166] | La_{2}Zr_{2}O_{7} | TC vs. temp |

37 | 2016 | Wang [167] | 8YSZ | Numerical work (mathematic model) TC vs. porosity and pores size |

38 | 2016 | Zhang [168] | La_{2}(Ce_{0.3}Zr_{0.7})_{2}O_{7}-3 wt.%Y_{2}O_{3} | TC vs. temp (deposited at 5, 15 and 25 RPM) |

39 | 2017 | Meng [169] | (a) La_{2}Zr_{2}O_{7}; (b) Nd_{2}Zr_{2}O_{7}; (c) Sm_{2}Zr_{2}O_{7}; (d) Gd_{2}Zr_{2}O_{7}. | TC vs. temp, concentration increase in oxygen vacancies |

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**Figure 5.**Scatterplot and histogram for TC vs. the five important inputs: (

**A**) TC vs. Temp; (

**B**) TC vs. Material; (

**C**) TC vs. Thickness; (

**D**) TC vs. AgingTemp; and (

**E**) TC vs. AgingTime.

**Figure 6.**Distribution of TC and five inputs: (

**A**) TC; (

**B**) Temp; (

**C**) Material; (

**D**) Thickness; (

**E**) AgingTemp; and (

**F**) AgingTime.

**Figure 8.**Effect of number of model terms on the R

^{2}performance for 3rd, 4th, 5th and 6th degree polynomial regression models trained with the OLS algorithm on all the data samples.

**Figure 9.**Effect of number of model terms on the average R

^{2}performance using the OLS algorithm with 100 times repeat for 3rd and 4th degree polynomial regression models.

**Figure 10.**Effect of number of model terms on the average R

^{2}performance using the OLS algorithm with 100 times repeat for 5th and 6th degree polynomial regression models.

**Figure 11.**Average R

^{2}, MAXE and MSE performance of the top 10 models (with 100 times repeat) selected from 3rd, 4th, 5th and 6th degree polynomial regression model candidates.

**Figure 12.**TC prediction performance using the best model from Table 10 (6th degree polynomial regression model with 100 terms, and R

^{2}= 0.8296, MAXE = 0.792 and MSE = 0.019).

**Figure 17.**Effect of node number on prediction using single-layer NN model with LM and BR algorithms.

**Figure 18.**Histogram plots of prediction error using single-layer NN model with LM and BR algorithms.

**Figure 19.**Comparison of results using single-layer NN model with LM and BR algorithms, x-axis is the used ample number, y-axis is TC value in the unit of W/(m∙K). (

**A**) LM algorithm (number of neurons = 30, repeating 10). (

**B**) BR algorithm (number of neurons = 100, repeating 10, R

^{2}= 0.8346).

**Figure 21.**Comparison of results using two-layer NN model with LM and BR algorithms, x-axis is the used sample number, y-axis is TC value in the unit of W/(m∙K). (

**A**) LM algorithm (number of neurons = 10 + 8, repeating 10, R

^{2}= 0.7338). (

**B**) BR algorithm (number of neurons = 8 + 6, repeating 10, R

^{2}= 0.8462).

**Figure 22.**Comparison of results using the GBR approach, x-asis is the used sample number, y-axis is TC value in the unit of W/(m∙K). (

**A**) 1 degree inputs. (

**B**) 2 degree inputs.

Paper No. | Year | Author | wt.% of Y_{2}O_{3} | Sample No. |
---|---|---|---|---|

[86] | 1999 | An | 8 | 18 |

[87] | 2002 | Nichols | 7 | 1 |

[88] | 2004 | Jang | 7 | 10 |

[89] | 2004 | Singh | 8 | 11 |

[90] | 2004 | Matsumoto | 7.1 | 3 |

[76] | 2006 | Renteria | 7.5 | 128 |

[91] | 2006 | Scheibe | 7 | 152 |

[92] | 2007 | Almeida | 8 | 7 |

[84] | 2007 | Scheibe | 7 | 187 |

[93] | 2007 | Schulz | 7 | 146 |

[94] | 2008 | Jang | 7 | 10 |

[95] | 2009 | Matsumoto | 7 | 6 |

[96] | 2011 | Jang | 8 | 18 |

[97] | 2011 | Liu | 7 | 4 |

[98] | 2013 | Bobzin | 7 | 4 |

Number of Total Samples | 705 |

Variables | Unit | Description | Functions |
---|---|---|---|

TC | W/(m∙K) | Thermal conductivity of TBC layer | Output |

Temp | °C | Temperature during measurement | Inputs |

Material | NA | wt.% of Y_{2}O_{3} | |

Thickness | mm | Thickness of the top layer of the TBC | |

AgingTemp | ℃ | Temperature of heat treatment | |

AgingTime | Hour | Time of heat treatment |

TC | Temp | Material | Thickness | AgingTemp | AgingTime | |
---|---|---|---|---|---|---|

TC | 1 | −0.21871 | 0.071464 | −0.14227 | 0.175485 | 0.317287 |

Temp | −0.21871 | 1 | 0.2324 | −0.09883 | 0.147777 | 0.076015 |

Material | 0.071464 | 0.2324 | 1 | −0.18194 | 0.137038 | 0.234873 |

Thickness | −0.14227 | −0.09883 | −0.18194 | 1 | −0.19923 | −0.05417 |

AgingTemp | 0.175485 | 0.147777 | 0.137038 | −0.19923 | 1 | 0.290966 |

AgingTime | 0.317287 | 0.076015 | 0.234873 | −0.05417 | 0.290966 | 1 |

Polynomial Degree | Number of Terms | R^{2} (on All Data) |
---|---|---|

1 | 5 | 0.19114 |

2 | 20 | 0.40414 |

3 | 55 | 0.62683 |

4 | 125 | 0.78005 |

5 | 251 | 0.84011 |

6 | 461 | 0.88241 |

Selection Step | ERR | Input/Model Term Index | Sum of ERR |
---|---|---|---|

1 | 0.10067 | 5 | 0.10067 |

2 | 0.059308 | 1 | 0.15998 |

3 | 0.022232 | 3 | 0.18221 |

4 | 0.0089309 | 4 | 0.19114 |

5 | 1.22 × 10^{−6} | 2 | 0.19114 |

Selection Step | ERR | Input/Model Term Index | Sum of ERR |
---|---|---|---|

1 | 0.1019 | 17 | 0.1019 |

2 | 0.086333 | 7 | 0.18823 |

3 | 0.029454 | 16 | 0.21768 |

4 | 0.027596 | 18 | 0.24528 |

5 | 0.056016 | 13 | 0.3013 |

6 | 0.023156 | 20 | 0.32445 |

7 | 0.013976 | 12 | 0.33843 |

8 | 0.015815 | 8 | 0.35424 |

9 | 0.010708 | 6 | 0.36495 |

10 | 0.013737 | 3 | 0.37869 |

11 | 0.0049433 | 2 | 0.38363 |

12 | 0.0061258 | 1 | 0.38976 |

13 | 0.0026508 | 11 | 0.39241 |

14 | 0.0029095 | 10 | 0.39532 |

15 | 0.0033999 | 5 | 0.39872 |

16 | 0.0010177 | 14 | 0.39974 |

17 | 0.00084935 | 9 | 0.40058 |

18 | 0.00050989 | 19 | 0.40109 |

19 | 0.0028143 | 15 | 0.40391 |

20 | 0.00023171 | 4 | 0.40414 |

**Table 7.**Top 10 performing 3rd degree PRM models ranked based on test data R

^{2}and selected using the OLS algorithm (100 times repeat).

Train Data (100 Times) | Test Data (100 Times) | All Data (100 Times) | |||||||
---|---|---|---|---|---|---|---|---|---|

# of Terms | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE |

45 | 0.62749 | 1.1053 | 0.041667 | 0.55114 | 0.94892 | 0.050533 | 0.61107 | 1.1367 | 0.04344 |

46 | 0.62768 | 1.1072 | 0.041603 | 0.54925 | 0.9571 | 0.050934 | 0.61082 | 1.1384 | 0.043469 |

32 | 0.57957 | 1.1156 | 0.04701 | 0.53946 | 0.96682 | 0.051537 | 0.57073 | 1.167 | 0.047916 |

31 | 0.57842 | 1.0936 | 0.047111 | 0.53765 | 0.97857 | 0.052283 | 0.56895 | 1.1664 | 0.048145 |

47 | 0.62909 | 1.1043 | 0.041448 | 0.53262 | 1.03 | 0.053746 | 0.60734 | 1.1774 | 0.043908 |

38 | 0.6083 | 1.0821 | 0.043814 | 0.53212 | 0.93576 | 0.053089 | 0.59125 | 1.1371 | 0.045669 |

29 | 0.56916 | 1.1071 | 0.04812 | 0.5316 | 0.94384 | 0.05256 | 0.56089 | 1.1414 | 0.049008 |

39 | 0.60982 | 1.0894 | 0.043646 | 0.53089 | 0.93726 | 0.053284 | 0.59215 | 1.1444 | 0.045573 |

48 | 0.62975 | 1.0957 | 0.041373 | 0.53083 | 1.0299 | 0.053977 | 0.60747 | 1.1686 | 0.043894 |

35 | 0.59576 | 1.1181 | 0.045229 | 0.53067 | 0.95151 | 0.052893 | 0.58136 | 1.1678 | 0.046762 |

**Table 8.**Top 10 performing 4th degree PRM models ranked based on test data R

^{2}and selected using the OLS Algorithm (100 times repeat).

Train Data (100 Times) | Test Data (100 Times) | All Data (100 Times) | |||||||
---|---|---|---|---|---|---|---|---|---|

# of Terms | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE |

63 | 0.76379 | 0.87438 | 0.026429 | 0.64568 | 1.0605 | 0.044038 | 0.73468 | 1.1313 | 0.02995 |

62 | 0.75617 | 0.88107 | 0.027252 | 0.62731 | 1.2072 | 0.057195 | 0.71697 | 1.292 | 0.033241 |

69 | 0.76814 | 0.88605 | 0.025935 | 0.61991 | 1.2078 | 0.055133 | 0.72555 | 1.2849 | 0.031774 |

67 | 0.76705 | 0.88869 | 0.02606 | 0.61712 | 1.2599 | 0.053943 | 0.72542 | 1.3401 | 0.031637 |

61 | 0.75483 | 0.88853 | 0.027398 | 0.61692 | 1.3323 | 0.062347 | 0.71083 | 1.4122 | 0.034388 |

70 | 0.76839 | 0.88623 | 0.025899 | 0.6164 | 1.2297 | 0.055927 | 0.72479 | 1.3009 | 0.031905 |

57 | 0.71268 | 1.1149 | 0.032142 | 0.61543 | 1.0718 | 0.047358 | 0.68768 | 1.282 | 0.035185 |

68 | 0.76739 | 0.88766 | 0.026022 | 0.61537 | 1.2589 | 0.055661 | 0.72449 | 1.3246 | 0.03195 |

64 | 0.7652 | 0.87173 | 0.026248 | 0.6124 | 1.2677 | 0.055151 | 0.72249 | 1.3347 | 0.032029 |

66 | 0.76697 | 0.89022 | 0.026063 | 0.61239 | 1.2683 | 0.054678 | 0.72414 | 1.3462 | 0.031786 |

**Table 9.**Top 10 performing 5th degree PRM models ranked based on test data R

^{2}and selected using the OLS algorithm (100 times repeat).

Train Data (100 Times) | Test Data (100 Times) | All Data (100 Times) | |||||||
---|---|---|---|---|---|---|---|---|---|

# of Terms | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE |

66 | 0.79891 | 0.85624 | 0.022456 | 0.70892 | 0.93127 | 0.034216 | 0.77879 | 1.04 | 0.024808 |

65 | 0.79815 | 0.85049 | 0.022541 | 0.70813 | 0.92153 | 0.033986 | 0.77841 | 1.0249 | 0.02483 |

67 | 0.79966 | 0.85581 | 0.022374 | 0.70673 | 0.94112 | 0.034698 | 0.77866 | 1.0485 | 0.024839 |

68 | 0.79883 | 0.85777 | 0.022474 | 0.70504 | 0.94058 | 0.035055 | 0.77754 | 1.0478 | 0.02499 |

69 | 0.79939 | 0.85926 | 0.022415 | 0.70469 | 0.94026 | 0.035019 | 0.77802 | 1.0502 | 0.024936 |

71 | 0.79997 | 0.86136 | 0.022347 | 0.70454 | 0.93829 | 0.035062 | 0.77839 | 1.048 | 0.02489 |

64 | 0.79699 | 0.84963 | 0.022671 | 0.7025 | 0.95808 | 0.035076 | 0.77577 | 1.0511 | 0.025152 |

70 | 0.79967 | 0.86065 | 0.022387 | 0.70069 | 0.97523 | 0.036334 | 0.77647 | 1.077 | 0.025177 |

72 | 0.80123 | 0.86336 | 0.022217 | 0.697 | 1.0077 | 0.037491 | 0.77616 | 1.1076 | 0.025271 |

73 | 0.80134 | 0.86947 | 0.022191 | 0.69149 | 1.0307 | 0.038879 | 0.77416 | 1.1308 | 0.025528 |

**Table 10.**Top 10 performing 6th degree PRM models ranked based on test data R

^{2}and selected using the OLS algorithm (100 times repeat).

Train Data (100 Times) | Test Data (100 Times) | All Data (100 Times) | |||||||
---|---|---|---|---|---|---|---|---|---|

# of Terms | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE |

100 | 0.83527 | 0.77303 | 0.018422 | 0.72653 | 0.85674 | 0.031695 | 0.81188 | 0.93618 | 0.021077 |

97 | 0.8342 | 0.7724 | 0.018524 | 0.72343 | 0.90631 | 0.033052 | 0.80933 | 0.9756 | 0.02143 |

101 | 0.83554 | 0.77389 | 0.018379 | 0.72315 | 0.8829 | 0.033182 | 0.81028 | 0.96055 | 0.021339 |

102 | 0.83581 | 0.77315 | 0.018361 | 0.72308 | 0.8703 | 0.032871 | 0.81081 | 0.94719 | 0.021263 |

98 | 0.83462 | 0.77269 | 0.018476 | 0.72221 | 0.90712 | 0.033362 | 0.80922 | 0.98244 | 0.021454 |

99 | 0.83496 | 0.77386 | 0.018445 | 0.72208 | 0.89571 | 0.033394 | 0.80944 | 0.9735 | 0.021435 |

103 | 0.83607 | 0.77357 | 0.018332 | 0.71998 | 0.89182 | 0.033889 | 0.80967 | 0.96321 | 0.021443 |

94 | 0.83183 | 0.77169 | 0.018791 | 0.71942 | 0.94028 | 0.034201 | 0.80591 | 1.0085 | 0.021873 |

93 | 0.83143 | 0.77165 | 0.018835 | 0.71791 | 0.94227 | 0.034348 | 0.80528 | 1.0088 | 0.021938 |

95 | 0.8321 | 0.77179 | 0.018755 | 0.71573 | 0.96354 | 0.035515 | 0.80437 | 1.0307 | 0.022107 |

LM Algorithm | BR Algorithm | |||
---|---|---|---|---|

Percentage | Number of Data Points | Percentage | Number of Data Points | |

Training | 70 | 493 | 85 | 599 |

Validation | 15 | 106 | 0 | 0 |

Testing | 15 | 106 | 15 | 106 |

Cases | Single-Layer | Two-Layer | |
---|---|---|---|

nn | nn1 | nn2 | |

1 | 10 | 4 | 2 |

2 | 20 | 6 | 2 |

3 | 30 | 6 | 4 |

4 | 40 | 8 | 2 |

5 | 50 | 8 | 4 |

6 | 60 | 8 | 6 |

7 | 70 | 10 | 2 |

8 | 80 | 10 | 4 |

9 | 90 | 10 | 6 |

10 | 100 | 10 | 8 |

Training Parameters | Default Value | Definition |
---|---|---|

net.trainParam.epochs | 1000 | Maximum number of epochs to train |

net.trainParam.goal | 0 | Performance goal |

net.trainParam.mu | 0.005 | Marquardt adjustment parameter |

net.trainParam.mu_dec | 0.1 | Decrease factor for mu |

net.trainParam.mu_inc | 10 | Increase factor for mu |

net.trainParam.mu_max | 1 × 10^{10} | Maximum value for mu |

net.trainParam.max_fail | inf | Maximum validation failures |

net.trainParam.min_grad | 1 × 10^{−7} | Minimum performance gradient |

net.trainParam.show | 25 | Epochs between displays (NaN for no displays) |

net.trainParam.showCommandLine | false | Generate command-line output |

net.trainParam.showWindow | true | Show training GUI |

net.trainParam.time | inf | Maximum time to train in seconds |

LM | 10 Train | 10 Test | 10 All | ||||||
---|---|---|---|---|---|---|---|---|---|

nn | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE |

10 | 0.6826 | 0.9918 | 0.0360 | 0.5748 | 0.8232 | 0.0457 | 0.6617 | 1.0755 | 0.0381 |

20 | 0.7694 | 0.8975 | 0.0257 | 0.6509 | 0.9232 | 0.0406 | 0.7329 | 1.0689 | 0.0299 |

30 | 0.7753 | 0.9422 | 0.0250 | 0.6270 | 0.9293 | 0.0466 | 0.7238 | 1.2181 | 0.0311 |

40 | 0.7609 | 0.9386 | 0.0268 | 0.6162 | 0.8681 | 0.0401 | 0.7235 | 1.0816 | 0.0309 |

50 | 0.7444 | 0.9675 | 0.0283 | 0.6117 | 0.9441 | 0.0451 | 0.6965 | 1.1875 | 0.0344 |

60 | 0.7436 | 1.0142 | 0.0295 | 0.5512 | 1.3672 | 0.0865 | 0.6568 | 1.7214 | 0.0417 |

70 | 0.7974 | 0.8665 | 0.0225 | 0.6489 | 0.8708 | 0.0397 | 0.7470 | 1.1126 | 0.0284 |

LM | 10 Train | 10 Test | 100 All | ||||||

nn | R^{2} | MAE | MSE | R^{2} | MAE | MSE | R^{2} | MAE | MSE |

90 | 0.7954 | 0.9860 | 0.0226 | 0.5319 | 1.4754 | 0.0739 | 0.7004 | 1.7896 | 0.0349 |

100 | 0.8078 | 0.8757 | 0.0212 | 0.6312 | 0.9481 | 0.0472 | 0.7315 | 1.4427 | 0.0317 |

LM | 100 Train | 100 Test | 100 All | ||||||
---|---|---|---|---|---|---|---|---|---|

nn | R^{2} | MAE | MSE | R^{2} | MAE | MSE | R^{2} | MAE | MSE |

10 | 0.6774 | 0.9883 | 0.0363 | 0.5459 | 0.9828 | 0.0529 | 0.6449 | 1.1464 | 0.0398 |

20 | 0.7117 | 0.9841 | 0.0331 | 0.5754 | 0.9967 | 0.0520 | 0.6741 | 1.1705 | 0.0378 |

30 | 0.7544 | 0.9340 | 0.0275 | 0.6123 | 0.9289 | 0.0456 | 0.7122 | 1.1762 | 0.0324 |

40 | 0.7609 | 0.9429 | 0.0270 | 0.5726 | 1.1360 | 0.0556 | 0.7024 | 1.3485 | 0.0339 |

50 | 0.7643 | 0.9531 | 0.0267 | 0.5885 | 1.0932 | 0.0528 | 0.7097 | 1.3440 | 0.0334 |

60 | 0.7777 | 0.9321 | 0.0250 | 0.5993 | 1.1853 | 0.0675 | 0.7146 | 1.4348 | 0.0343 |

70 | 0.7766 | 0.9609 | 0.0254 | 0.5732 | 1.0903 | 0.0543 | 0.7140 | 1.3496 | 0.0330 |

90 | 0.7874 | 0.9396 | 0.0240 | 0.5705 | 1.2272 | 0.0670 | 0.7081 | 1.5107 | 0.0348 |

100 | 0.7991 | 0.9282 | 0.0226 | 0.5795 | 1.1686 | 0.0610 | 0.7242 | 1.4433 | 0.0321 |

BR | 10 Train | 10 Test | 10 All | ||||||
---|---|---|---|---|---|---|---|---|---|

nn | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE |

10 | 0.7643 | 0.8418 | 0.0262 | 0.6202 | 1.1756 | 0.0541 | 0.7321 | 1.2170 | 0.0304 |

20 | 0.8469 | 0.6833 | 0.0170 | 0.6443 | 1.0706 | 0.0528 | 0.8034 | 1.1250 | 0.0224 |

30 | 0.8540 | 0.7328 | 0.0163 | 0.7228 | 0.7228 | 0.7179 | 0.0313 | 0.8339 | 0.0185 |

40 | 0.8255 | 0.7904 | 0.0195 | 0.6588 | 0.9704 | 0.0479 | 0.7895 | 1.0908 | 0.0238 |

50 | 0.8563 | 0.6921 | 0.0159 | 0.6861 | 0.9551 | 0.0398 | 0.8261 | 1.0484 | 0.0195 |

60 | 0.8606 | 0.6599 | 0.0156 | 0.5310 | 1.8826 | 0.1611 | 0.7548 | 1.8988 | 0.0375 |

70 | 0.8275 | 0.7896 | 0.0192 | 0.6733 | 0.8740 | 0.0369 | 0.8040 | 1.0180 | 0.0219 |

80 | 0.8540 | 0.7017 | 0.0161 | 0.6608 | 1.0552 | 0.0464 | 0.8167 | 1.1335 | 0.0207 |

90 | 0.8492 | 0.7088 | 0.0167 | 0.6983 | 1.0233 | 0.0388 | 0.8212 | 1.0845 | 0.0200 |

100 | 0.8595 | 0.6963 | 0.0160 | 0.6992 | 0.6992 | 0.9259 | 0.0328 | 0.8346 | 0.9658 |

BR | 100 Train | 100 Test | 100 All | ||||||
---|---|---|---|---|---|---|---|---|---|

nn | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE |

10 | 0.7856 | 0.8410 | 0.0239 | 0.6554 | 0.9931 | 0.0466 | 0.7595 | 1.1017 | 0.0273 |

20 | 0.8441 | 0.7420 | 0.0173 | 0.6868 | 0.9604 | 0.0388 | 0.8169 | 1.0409 | 0.0205 |

30 | 0.8522 | 0.7239 | 0.0164 | 0.6775 | 0.9916 | 0.0437 | 0.8185 | 1.0678 | 0.0205 |

40 | 0.8466 | 0.7338 | 0.0171 | 0.6510 | 1.1477 | 0.0649 | 0.8047 | 1.2102 | 0.0243 |

50 | 0.8468 | 0.7405 | 0.0170 | 0.6632 | 1.1140 | 0.0493 | 0.8098 | 1.1783 | 0.0219 |

60 | 0.8502 | 0.7383 | 0.0167 | 0.6631 | 1.1337 | 0.0581 | 0.8088 | 1.2029 | 0.0229 |

70 | 0.8518 | 0.7123 | 0.0164 | 0.6303 | 1.3802 | 0.0822 | 0.7916 | 1.4364 | 0.0263 |

80 | 0.8494 | 0.7213 | 0.0167 | 0.6614 | 1.1500 | 0.0585 | 0.8098 | 1.2133 | 0.0230 |

90 | 0.8487 | 0.7289 | 0.0168 | 0.6539 | 1.1900 | 0.0723 | 0.8033 | 1.2615 | 0.0252 |

100 | 0.8503 | 0.7133 | 0.0167 | 0.6499 | 1.1991 | 0.0611 | 0.8051 | 1.2549 | 0.0233 |

10 Train | 10 Test | 10 All | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

nn1 | nn2 | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE |

4 | 2 | 0.4996 | 1.0426 | 0.0556 | 0.4062 | 1.1792 | 0.0786 | 0.4764 | 1.3394 | 0.0592 |

6 | 2 | 0.6660 | 0.9897 | 0.0382 | 0.4469 | 1.1897 | 0.0651 | 0.6178 | 1.3659 | 0.0435 |

6 | 4 | 0.6835 | 0.9355 | 0.0343 | 0.5479 | 1.0531 | 0.0588 | 0.6451 | 1.1437 | 0.0398 |

8 | 2 | 0.6717 | 0.9355 | 0.0358 | 0.5069 | 1.0985 | 0.0573 | 0.6390 | 1.1896 | 0.0404 |

8 | 4 | 0.7320 | 0.9554 | 0.0297 | 0.5834 | 1.1156 | 0.0562 | 0.6833 | 1.3080 | 0.0358 |

8 | 6 | 0.7363 | 0.9673 | 0.0297 | 0.6414 | 0.8477 | 0.0414 | 0.7067 | 1.0266 | 0.0329 |

10 | 2 | 0.7021 | 0.9618 | 0.0326 | 0.5784 | 0.8267 | 0.0501 | 0.6703 | 1.0191 | 0.0368 |

10 | 4 | 0.6482 | 0.9385 | 0.0387 | 0.5421 | 0.9438 | 0.0554 | 0.6119 | 1.0221 | 0.0437 |

10 | 6 | 0.7229 | 0.9049 | 0.0315 | 0.6180 | 0.8010 | 0.0443 | 0.6954 | 1.0116 | 0.0342 |

10 | 8 | 0.7624 | 0.8974 | 0.0263 | 0.6766 | 0.7962 | 0.0380 | 0.7388 | 0.9780 | 0.0292 |

100 Ttrain | 100 Test | 100 All | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

nn1 | nn2 | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE |

4 | 2 | 0.5141 | 1.0472 | 0.0548 | 0.4633 | 0.9372 | 0.0627 | 0.4972 | 1.1178 | 0.0565 |

6 | 2 | 0.6012 | 1.0126 | 0.0449 | 0.4914 | 1.1610 | 0.0867 | 0.5656 | 1.3142 | 0.0519 |

6 | 4 | 0.6426 | 0.9800 | 0.0403 | 0.5168 | 1.0306 | 0.0588 | 0.6082 | 1.1624 | 0.0441 |

8 | 2 | 0.6395 | 0.9831 | 0.0404 | 0.5418 | 0.9678 | 0.0534 | 0.6138 | 1.1124 | 0.0433 |

8 | 4 | 0.7052 | 0.9656 | 0.0330 | 0.5564 | 1.0385 | 0.0540 | 0.6677 | 1.1793 | 0.0374 |

8 | 6 | 0.7162 | 0.9272 | 0.0318 | 0.5812 | 1.0167 | 0.0585 | 0.6754 | 1.1749 | 0.0373 |

10 | 2 | 0.6827 | 0.9705 | 0.0354 | 0.5729 | 0.9255 | 0.0512 | 0.6518 | 1.0926 | 0.0391 |

10 | 4 | 0.7275 | 0.9397 | 0.0304 | 0.5694 | 1.0280 | 0.0511 | 0.6870 | 1.1677 | 0.0352 |

10 | 6 | 0.7158 | 0.9591 | 0.0318 | 0.5830 | 0.9806 | 0.0497 | 0.6797 | 1.1410 | 0.0359 |

10 | 8 | 0.7512 | 0.9049 | 0.0279 | 0.6127 | 0.9300 | 0.0459 | 0.7160 | 1.0583 | 0.0319 |

10 Train | 10 Test | 10 All | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

nn1 | nn2 | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE |

4 | 2 | 0.7152 | 0.9551 | 0.0321 | 0.6615 | 0.7508 | 0.0357 | 0.7071 | 0.9696 | 0.0327 |

6 | 2 | 0.7608 | 0.9108 | 0.0266 | 0.6623 | 0.8873 | 0.0383 | 0.7459 | 0.9839 | 0.0284 |

6 | 4 | 0.8327 | 0.7700 | 0.0185 | 0.6875 | 1.0085 | 0.0441 | 0.8019 | 1.0303 | 0.0224 |

8 | 2 | 0.8046 | 0.8324 | 0.0219 | 0.7244 | 0.7237 | 0.0297 | 0.7932 | 0.8941 | 0.0231 |

8 | 4 | 0.8583 | 0.6963 | 0.0157 | 0.6798 | 1.1972 | 0.0492 | 0.8196 | 1.2427 | 0.0207 |

8 | 6 | 0.8773 | 0.6274 | 0.0138 | 0.6989 | 1.1218 | 0.0372 | 0.8462 | 1.1616 | 0.0173 |

10 | 2 | 0.8607 | 0.6391 | 0.0155 | 0.6848 | 1.0583 | 0.0431 | 0.8259 | 1.1097 | 0.0197 |

10 | 4 | 0.8693 | 0.6572 | 0.0145 | 0.6324 | 1.8664 | 0.1027 | 0.7950 | 1.8878 | 0.0278 |

10 | 6 | 0.8842 | 0.5968 | 0.0130 | 0.6953 | 1.1535 | 0.0426 | 0.8463 | 1.1784 | 0.0174 |

10 | 8 | 0.8890 | 0.5804 | 0.0123 | 0.6124 | 1.8345 | 0.1261 | 0.7963 | 1.8483 | 0.0294 |

100 Train | 100 Test | 100All | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

nn1 | nn2 | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE |

4 | 2 | 0.6881 | 0.9444 | 0.0348 | 0.5918 | 0.9074 | 0.0457 | 0.6735 | 1.0273 | 0.0364 |

6 | 2 | 0.7620 | 0.9166 | 0.0265 | 0.6240 | 1.0206 | 0.0540 | 0.7343 | 1.1326 | 0.0306 |

6 | 4 | 0.8237 | 0.8006 | 0.0196 | 0.6766 | 1.0300 | 0.0433 | 0.7950 | 1.1024 | 0.0232 |

8 | 2 | 0.8111 | 0.8039 | 0.0209 | 0.6781 | 1.0534 | 0.0500 | 0.7819 | 1.1362 | 0.0253 |

8 | 4 | 0.8607 | 0.6759 | 0.0155 | 0.6799 | 1.0881 | 0.0464 | 0.8240 | 1.1322 | 0.0202 |

100 Train | 100 Test | 100All | ||||||||

nn1 | nn2 | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE |

10 | 2 | 0.8429 | 0.7348 | 0.0175 | 0.6450 | 1.2597 | 0.0636 | 0.7967 | 1.3176 | 0.0244 |

10 | 4 | 0.8740 | 0.6275 | 0.0140 | 0.6788 | 1.1092 | 0.0458 | 0.8352 | 1.1381 | 0.0188 |

10 | 6 | 0.8846 | 0.6083 | 0.0128 | 0.6900 | 1.1417 | 0.0468 | 0.8443 | 1.1681 | 0.0179 |

10 | 8 | 0.8886 | 0.6002 | 0.0124 | 0.6392 | 1.5001 | 0.0608 | 0.8321 | 1.5185 | 0.0196 |

Train | Test | All | |||||||
---|---|---|---|---|---|---|---|---|---|

Hyper Parameters | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE | R^{2} | MAXE | MSE |

deg 1 (10 times) | 0.9364 | 0.5969 | 0.0071 | 0.7637 | 0.7723 | 0.0261 | 0.9024 | 0.7723 | 0.0109 |

deg 1 (100 times) | 0.9360 | 0.6022 | 0.0071 | 0.7531 | 0.8268 | 0.0268 | 0.9007 | 0.8493 | 0.0111 |

deg 2 (10 times) | 0.9450 | 0.5881 | 0.0061 | 0.7290 | 0.8203 | 0.0299 | 0.9024 | 0.8203 | 0.0109 |

deg 2 (100 times) | 0.9441 | 0.5953 | 0.0062 | 0.7326 | 0.8265 | 0.0289 | 0.9033 | 0.8379 | 0.0108 |

Approach | Settings | R^{2} | MAXE | MSE |
---|---|---|---|---|

Polynomial Input Selection | OLS Algorithm Repeating 100 Times for 6th degree PRM | 0.8121 | 0.9321 | 0.0211 |

Neural Network | Double Layer 10 + 6 nodes | 0.8443 | 1.1681 | 0.0179 |

Gradient Boosting Regression | 1 Degree Inputs | 0.9024 | 0.7723 | 0.0109 |

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## Share and Cite

**MDPI and ACS Style**

Liu, Y.; Chen, K.; Kumar, A.; Patnaik, P.
Principles of Machine Learning and Its Application to Thermal Barrier Coatings. *Coatings* **2023**, *13*, 1140.
https://doi.org/10.3390/coatings13071140

**AMA Style**

Liu Y, Chen K, Kumar A, Patnaik P.
Principles of Machine Learning and Its Application to Thermal Barrier Coatings. *Coatings*. 2023; 13(7):1140.
https://doi.org/10.3390/coatings13071140

**Chicago/Turabian Style**

Liu, Yuan, Kuiying Chen, Amarnath Kumar, and Prakash Patnaik.
2023. "Principles of Machine Learning and Its Application to Thermal Barrier Coatings" *Coatings* 13, no. 7: 1140.
https://doi.org/10.3390/coatings13071140