Rational Design of FieldEffect Sensors Using Partial Differential Equations, Bayesian Inversion, and Artificial Neural Networks
Abstract
:1. Introduction
2. The Model Equations
3. Parameter Estimation Based on Bayesian Inference
Algorithm 1 The Metropolis–Hastings algorithm. 
Initialization: Start the process with the initial guess ${\chi}^{0}$ and number of samples N. 
while$j<N$ 
1. Propose a new sample according to the proposal density ${\chi}^{*}\sim \mathcal{T}\left({\chi}^{*}\right\phantom{\rule{0.166667em}{0ex}}{\chi}^{j1})$. 
2. Compute the acceptance/rejection ratio
$$\zeta \left({\chi}^{*}\right\phantom{\rule{0.166667em}{0ex}}{\theta}^{j1})=min\left(1,\frac{\pi \left({\chi}^{*}\right\phantom{\rule{0.166667em}{0ex}}m)}{\pi \left({\chi}^{j1}\right\phantom{\rule{0.166667em}{0ex}}m)}\phantom{\rule{0.166667em}{0ex}}\frac{\mathcal{T}\left({\chi}^{j1}\right\phantom{\rule{0.166667em}{0ex}}{\chi}^{*})}{\mathcal{T}\left({\chi}^{*}\right\phantom{\rule{0.166667em}{0ex}}{\chi}^{j1})}\right).$$

3. Sample $\mathcal{R}\sim \mathrm{Uniform}\phantom{\rule{0.166667em}{0ex}}(0,1)$. 
4.$\phantom{\rule{3.33333pt}{0ex}}\mathbf{if}\phantom{\rule{3.33333pt}{0ex}}$$\mathcal{R}<\zeta \phantom{\rule{3.33333pt}{0ex}}$ then 
accept ${\chi}^{*}$ and set ${\chi}^{j}:={\chi}^{*}$ 
else 
reject ${\chi}^{*}$ and set ${\chi}^{j}:={\chi}^{j1}$ 
end if 
5. Set $j=j+1.$ 
Mcmc with EnsembleKalman Filter (EnKFMCMC)
Algorithm 2 Bayesian inference using EnKFMCMC 
Initialization ($j=0$): Start the process with the initial guess ${\chi}^{0}$ and number of samples N. 
while$j<N$ 
1. Estimate the model response with respect to ${\chi}^{j1}$ 
2. Compute the Kalman gain $\mathcal{K}={\mathcal{C}}_{\chi M}{\left({\mathcal{C}}_{MM}+\mathcal{R}\right)}^{1}$ 
3. Produce the new proposal using the shift ${\chi}^{\U0001f7c9}={\chi}^{j1}+\mathcal{K}\left({y}^{j1}+{s}^{j1}\right)$ 
4. Accepted/rejected ${\chi}^{\U0001f7c9}$ 
5. Set $j=j+1$. 
4. Multilayer FeedForward Neural Networks
5. Numerical Experiments
5.1. Model Verification
5.2. Bayesian Inversion
5.3. Machine Learning Based on MFNNs
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Parameter  Min  Max  EnKF (Median)  True Values  Acceptance Rate 

C${}_{\mathrm{dop}}\phantom{\rule{0.166667em}{0ex}}$(cm${}^{3}$)  1 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{15}$  5 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{16}$  9.4 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{15}$  1 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{16}$  91% 
$\rho $ (q/nm${}^{2}$)  −5  1  −1.55  −1.5  86% 
Cases  Inputs  ${\mathit{V}}_{\mathbf{g}}$ [V]  SiO${}_{2}$ [nm]  N${}_{\mathit{W}}$ [nm]  ${\mathit{C}}_{\mathbf{dop}}\phantom{\rule{0.166667em}{0ex}}\left[{\mathbf{cm}}^{3}\right]$  N${}_{\mathbf{H}}$ [nm] 

Case 1  1  $\mathcal{U}(1,5)$  8  100  1$\times \phantom{\rule{0.166667em}{0ex}}{10}^{16}$  50 
Case 2  2  $\mathcal{U}(1,5)$  $\mathcal{U}(5,15)$  100  1$\times \phantom{\rule{0.166667em}{0ex}}{10}^{16}$  50 
Case 3  3  $\mathcal{U}(1,5)$  $\mathcal{U}(5,15)$  $\mathcal{U}(80,120)$  1$\times \phantom{\rule{0.166667em}{0ex}}{10}^{16}$  50 
Case 4  4  $\mathcal{U}(1,5)$  $\mathcal{U}(5,15)$  $\mathcal{U}(80,120)$  $\mathcal{U}(1\times {10}^{15},5\times {10}^{16})$  50 
Case 5  5  $\mathcal{U}(1,5)$  $\mathcal{U}(5,15)$  $\mathcal{U}(80,120)$  $\mathcal{U}(1\times {10}^{15},5\times {10}^{16})$  $\mathcal{U}(40,60)$ 
Case  No. Neurons in 1st Hidden Layer  No. Neurons in 2nd Hidden Layer  MSETrain  MSETest  No. Epochs  $\mathsf{\eta}$ 

1  10  4  0.00057  0.00061  1000  0.1 
2  20  7  0.00147  0.00184  2000  0.2 
3  20  7  0.00181  0.000836  4000  0.2 
4  20  7  0.000842  0.000517  8000  0.2 
5  20  7  0.0011  0.000058  10,000  0.2 
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Khodadadian, A.; Parvizi, M.; Teshnehlab, M.; Heitzinger, C. Rational Design of FieldEffect Sensors Using Partial Differential Equations, Bayesian Inversion, and Artificial Neural Networks. Sensors 2022, 22, 4785. https://doi.org/10.3390/s22134785
Khodadadian A, Parvizi M, Teshnehlab M, Heitzinger C. Rational Design of FieldEffect Sensors Using Partial Differential Equations, Bayesian Inversion, and Artificial Neural Networks. Sensors. 2022; 22(13):4785. https://doi.org/10.3390/s22134785
Chicago/Turabian StyleKhodadadian, Amirreza, Maryam Parvizi, Mohammad Teshnehlab, and Clemens Heitzinger. 2022. "Rational Design of FieldEffect Sensors Using Partial Differential Equations, Bayesian Inversion, and Artificial Neural Networks" Sensors 22, no. 13: 4785. https://doi.org/10.3390/s22134785