# InfPolyn, a Nonparametric Bayesian Characterization for Composition-Dependent Interdiffusion Coefficients

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## Abstract

**:**

## 1. Introduction

- InfPolyn does not require assumptions for the particular functional form of the interdiffusion coefficient; it is robust against overfitting and underfitting.
- InfPolyn does not require a significant number of training data.
- Prior knowledge of the interdiffusion system can be added easily in the framework of InfPolyn.

## 2. Statement of the Problem

## 3. Boltzmann–Matano Polynomial Interdiffusion Coefficients

**Remark**

**1.**

#### Optimization for Polynomial Fitting

## 4. InfPolyn for Interdiffusion Coefficients

#### 4.1. Infinite Order Polynomial Model

#### 4.2. Kernel Formulation

#### 4.3. Ghost Interdiffusion Coefficients

**h**

_{j}and ${\mathbf{Z}}_{j}$ are latent variables that need to be integrated out during the model training and predictions.

#### 4.4. Diagonal-Dominating Prior

#### 4.5. Joint Model Training

**f**and the observed flux

**u**. More specifically, the log marginal likelihood can be computed by

#### 4.6. Interdiffusion Coefficients Predictions

## 5. Results

#### 5.1. Case Study 1: Polynomial Diffusion Coefficients

#### 5.2. Case Study 2: Exponential Diffusion Coefficients

#### 5.3. Case Study 3: Uncertainty Quantification Analysis

#### 5.4. Case Study 4: Experiment Verification

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. A Gaussian Processs and Its Predicted Posterior

#### Appendix A.1. Solving a Ternary System Using Boltzmann–Matano Inverse Method

#### Appendix A.2. Experimental Details

**Table A1.**The polynomial coefficients in the random ternary interdiffusion system, where ${a}_{ij}$ represents the entries in the coefficient matrix on position $\{i,j\},i=1,\phantom{\rule{4pt}{0ex}}2,\phantom{\rule{4pt}{0ex}}j=1,\phantom{\rule{4pt}{0ex}}2$.

A | ${\mathit{a}}_{11}$ | ${\mathit{a}}_{21}$ | ${\mathit{a}}_{22}$ |
---|---|---|---|

${A}_{0}$ | $6.05\times {10}^{-5}$ | $4.81\times {10}^{-6}$ | $5.12\times {10}^{-5}$ |

${A}_{1}^{1}$ | $3.08\times {10}^{-6}$ | $4.18\times {10}^{-7}$ | $3.28\times {10}^{-6}$ |

${A}_{2}^{1}$ | $1.82\times {10}^{-6}$ | $6.42\times {10}^{-7}$ | $2.86\times {10}^{-6}$ |

${A}_{1}^{2}$ | $8.83\times {10}^{-7}$ | $6.07\times {10}^{-8}$ | $1.07\times {10}^{-7}$ |

${A}_{2}^{2}$ | $2.96\times {10}^{-7}$ | $4.21\times {10}^{-8}$ | $9.63\times {10}^{-7}$ |

${A}_{1}^{3}$ | $4.09\times {10}^{-8}$ | $2.82\times {10}^{-9}$ | $1.26\times {10}^{-8}$ |

${A}_{2}^{3}$ | $1.19\times {10}^{-8}$ | $7.29\times {10}^{-9}$ | $1.10\times {10}^{-8}$ |

${A}_{1}^{4}$ | $1.26\times {10}^{-8}$ | $5.02\times {10}^{-10}$ | $1.66\times {10}^{-9}$ |

${A}_{2}^{4}$ | $6.04\times {10}^{-9}$ | $6.34\times {10}^{-10}$ | $1.09\times {10}^{-8}$ |

**Table A2.**The polynomial coefficients in the random quaternary interdiffusion system, where ${a}_{ij}$ represents the entries in the coefficient matrix on position $\{i,j\},i=1,\phantom{\rule{4pt}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}3;\phantom{\rule{4pt}{0ex}}j=1,\phantom{\rule{4pt}{0ex}}2,\phantom{\rule{4pt}{0ex}}3$.

A | ${\mathit{a}}_{11}$ | ${\mathit{a}}_{12}$ | ${\mathit{a}}_{13}$ | ${\mathit{a}}_{22}$ | ${\mathit{a}}_{23}$ | ${\mathit{a}}_{33}$ |
---|---|---|---|---|---|---|

${A}_{0}$ | $2.03\times {10}^{-5}$ | $6.50\times {10}^{-6}$ | $5.15\times {10}^{-6}$ | $7.13\times {10}^{-5}$ | $3.58\times {10}^{-7}$ | $3.27\times {10}^{-5}$ |

${A}_{1}^{1}$ | $9.15\times {10}^{-6}$ | $3.13\times {10}^{-7}$ | $6.87\times {10}^{-7}$ | $9.46\times {10}^{-6}$ | $1.35\times {10}^{-7}$ | $1.76\times {10}^{-6}$ |

${A}_{2}^{1}$ | $9.56\times {10}^{-6}$ | $8.08\times {10}^{-7}$ | $3.01\times {10}^{-7}$ | $1.61\times {10}^{-6}$ | $4.89\times {10}^{-7}$ | $1.18\times {10}^{-6}$ |

${A}_{3}^{1}$ | $2.01\times {10}^{-6}$ | $9.05\times {10}^{-7}$ | $2.47\times {10}^{-7}$ | $5.48\times {10}^{-6}$ | $7.41\times {10}^{-7}$ | $2.95\times {10}^{-6}$ |

${A}_{1}^{2}$ | $2.61\times {10}^{-7}$ | $7.28\times {10}^{-8}$ | $6.95\times {10}^{-8}$ | $2.18\times {10}^{-7}$ | $5.91\times {10}^{-8}$ | $1.58\times {10}^{-7}$ |

${A}_{2}^{2}$ | $5.90\times {10}^{-7}$ | $2.23\times {10}^{-8}$ | $4.55\times {10}^{-8}$ | $7.08\times {10}^{-7}$ | $5.79\times {10}^{-8}$ | $3.94\times {10}^{-7}$ |

${A}_{3}^{2}$ | $2.17\times {10}^{-9}$ | $2.96\times {10}^{-8}$ | $5.60\times {10}^{-8}$ | $8.11\times {10}^{-7}$ | $3.82\times {10}^{-8}$ | $6.70\times {10}^{-8}$ |

${A}_{1}^{3}$ | $9.00\times {10}^{-8}$ | $2.45\times {10}^{-9}$ | $2.52\times {10}^{-10}$ | $9.00\times {10}^{-8}$ | $5.90\times {10}^{-9}$ | $2.48\times {10}^{-8}$ |

${A}_{2}^{3}$ | $2.70\times {10}^{-8}$ | $2.57\times {10}^{-9}$ | $2.11\times {10}^{-9}$ | $6.50\times {10}^{-10}$ | $6.48\times {10}^{-9}$ | $6.01\times {10}^{-9}$ |

${A}_{3}^{3}$ | $8.04\times {10}^{-8}$ | $2.55\times {10}^{-9}$ | $3.11\times {10}^{-9}$ | $8.91\times {10}^{-8}$ | $2.76\times {10}^{-9}$ | $1.03\times {10}^{-8}$ |

${A}_{1}^{4}$ | $4.82\times {10}^{-9}$ | $3.18\times {10}^{-10}$ | $1.87\times {10}^{-10}$ | $9.07\times {10}^{-9}$ | $2.45\times {10}^{-10}$ | $3.91\times {10}^{-9}$ |

${A}_{2}^{4}$ | $6.58\times {10}^{-9}$ | $9.28\times {10}^{-10}$ | $2.44\times {10}^{-10}$ | $2.05\times {10}^{-9}$ | $6.07\times {10}^{-10}$ | $7.94\times {10}^{-9}$ |

${A}_{3}^{4}$ | $9.38\times {10}^{-9}$ | $1.30\times {10}^{-10}$ | $9.50\times {10}^{-10}$ | $9.17\times {10}^{-9}$ | $6.05\times {10}^{-10}$ | $1.46\times {10}^{-9}$ |

**Table A3.**The table shows the setting for coefficient matrix of polynomials function and ${A}_{i}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}{A}_{i}^{{}^{\prime}}$ represents the entries in each coefficient matrix on position $\left\{i\right\},i=1,\phantom{\rule{4pt}{0ex}}2;\phantom{\rule{4pt}{0ex}}j=1,\phantom{\rule{4pt}{0ex}}2$.

A | ${\mathit{a}}_{11}$ | ${\mathit{a}}_{21}$ | ${\mathit{a}}_{22}$ |
---|---|---|---|

${A}_{0}$ | $6.15\times {10}^{-5}$ | $5.08\times {10}^{-7}$ | $5.46\times {10}^{-5}$ |

${A}_{1}$ | $1.25\times {10}^{-7}$ | $3.77\times {10}^{-7}$ | $8.50\times {10}^{-8}$ |

${A}_{2}$ | $9.09\times {10}^{-7}$ | $3.51\times {10}^{-7}$ | $9.08\times {10}^{-7}$ |

${A}_{1}^{{}^{\prime}}$ | $4.85\times {10}^{-7}$ | $2.85\times {10}^{-8}$ | $7.39\times {10}^{-7}$ |

${A}_{2}^{{}^{\prime}}$ | $4.67\times {10}^{-7}$ | $7.29\times {10}^{-7}$ | $8.27\times {10}^{-7}$ |

**Table A4.**The table shows the setting for coefficient matrix of exponential function and ${A}_{i}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}{A}_{i}^{{}^{\prime}}$ represents the entries in each coefficient matrix on position $\left\{i\right\},i=1,\phantom{\rule{4pt}{0ex}}2,\phantom{\rule{4pt}{0ex}},3;\phantom{\rule{4pt}{0ex}}j=1,\phantom{\rule{4pt}{0ex}}2,\phantom{\rule{4pt}{0ex}}3$.

A | ${\mathit{a}}_{11}$ | ${\mathit{a}}_{12}$ | ${\mathit{a}}_{13}$ | ${\mathit{a}}_{22}$ | ${\mathit{a}}_{23}$ | ${\mathit{a}}_{33}$ |
---|---|---|---|---|---|---|

${A}_{0}$ | $4.82\times {10}^{-5}$ | $4.61\times {10}^{-6}$ | $4.17\times {10}^{-6}$ | $4.26\times {10}^{-5}$ | $4.34\times {10}^{-6}$ | $6.27\times {10}^{-5}$ |

${A}_{1}$ | $8.77\times {10}^{-8}$ | $8.40\times {10}^{-8}$ | $1.07\times {10}^{-8}$ | $9.65\times {10}^{-8}$ | $8.20\times {10}^{-8}$ | $2.62\times {10}^{-8}$ |

${A}_{2}$ | $5.15\times {10}^{-8}$ | $8.39\times {10}^{-9}$ | $1.87\times {10}^{-8}$ | $9.39\times {10}^{-8}$ | $1.55\times {10}^{-8}$ | $9.78\times {10}^{-8}$ |

${A}_{3}$ | $6.81\times {10}^{-8}$ | $2.67\times {10}^{-8}$ | $2.23\times {10}^{-8}$ | $7.68\times {10}^{-9}$ | $5.41\times {10}^{-9}$ | $3.78\times {10}^{-8}$ |

${A}_{1}^{{}^{\prime}}$ | $3.45\times {10}^{-8}$ | $3.79\times {10}^{-8}$ | $3.48\times {10}^{-8}$ | $3.11\times {10}^{-9}$ | $1.42\times {10}^{-8}$ | $5.00\times {10}^{-8}$ |

${A}_{2}^{{}^{\prime}}$ | $1.21\times {10}^{-9}$ | $3.53\times {10}^{-8}$ | $6.64\times {10}^{-8}$ | $7.51\times {10}^{-8}$ | $2.52\times {10}^{-8}$ | $5.45\times {10}^{-8}$ |

${A}_{3}^{{}^{\prime}}$ | $6.21\times {10}^{-8}$ | $3.77\times {10}^{-8}$ | $6.93\times {10}^{-8}$ | $1.23\times {10}^{-9}$ | $6.45\times {10}^{-8}$ | $1.75\times {10}^{-8}$ |

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**Figure 1.**The relative errors (REs) of predictive diffusion coefficients ${\tilde{D}}_{ij}\left(\mathbf{c}\left(x\right)\right)$ in the center areas $x\in [0.4,0.6]$ for the evaluated methods in a random ternary system.

**Figure 2.**The relative errors (REs) of predictive diffusion coefficients ${\tilde{D}}_{ij}\left(\mathbf{c}\left(x\right)\right)$ in the center areas $x\in [0.4,0.6]$ for the evaluated methods in a quadternary system.

**Figure 3.**The relative errors (REs) of predictive diffusion coefficients ${\tilde{D}}_{ij}\left(\mathbf{c}\left(x\right)\right)$ in the center areas $x\in [0.4,0.6]$ for the evaluated methods in a ternary system.

**Figure 4.**The relative errors (REs) of predictive diffusion coefficients ${\tilde{D}}_{ij}\left(\mathbf{c}\left(x\right)\right)$ in the center areas $x\in [0.4,0.6]$ for the evaluated methods in a quadternary system.

**Figure 5.**The Tukey box plot of average relative error of ${\tilde{D}}_{11}$ (

**top**) and ${\tilde{D}}_{22}$ (

**bottom**) based on computation using concentration profile consisting $\{20,30,40,50\}$ EMPA samples.

**Figure 6.**The actual and predicted diffusion fluxes for the Mg-Al, Mg-AL-Zn, and Mg-Al-Zn-Cu system (from left to right columns) using 100%, 50%, and 25% of all available samples (from top to bottom rows).

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## Share and Cite

**MDPI and ACS Style**

Xing, W.W.; Cheng, M.; Cheng, K.; Zhang, W.; Wang, P.
InfPolyn, a Nonparametric Bayesian Characterization for Composition-Dependent Interdiffusion Coefficients. *Materials* **2021**, *14*, 3635.
https://doi.org/10.3390/ma14133635

**AMA Style**

Xing WW, Cheng M, Cheng K, Zhang W, Wang P.
InfPolyn, a Nonparametric Bayesian Characterization for Composition-Dependent Interdiffusion Coefficients. *Materials*. 2021; 14(13):3635.
https://doi.org/10.3390/ma14133635

**Chicago/Turabian Style**

Xing, Wei W., Ming Cheng, Kaiming Cheng, Wei Zhang, and Peng Wang.
2021. "InfPolyn, a Nonparametric Bayesian Characterization for Composition-Dependent Interdiffusion Coefficients" *Materials* 14, no. 13: 3635.
https://doi.org/10.3390/ma14133635