Neural Network Based Deep Learning Method for Multi-Dimensional Neutron Diffusion Problems with Novel Treatment to Boundary
<p>A flowchart of PIDL.</p> "> Figure 2
<p>A schematic diagram of the PIDL method, including (<b>a</b>) training set in the computation domain, (<b>b</b>) local loss function, (<b>c</b>) trial function and (<b>d</b>) ANN.</p> "> Figure 3
<p>Dirichlet BC for two common geometries, including (<b>a</b>) Square boundary (SB) and (<b>b</b>) Circular boundary (CB).</p> "> Figure 4
<p>Dirichlet and Neumann BCs with symmetry applied, including (<b>a</b>) SB and (<b>b</b>) CB.</p> "> Figure 5
<p>Comparisons of BDM and BIM results with analytical solution of Case 1 at <span class="html-italic">t</span> = 0.5 s, including (<b>a</b>) BIM solution, (<b>b</b>) BDM solution, (<b>c</b>) analytical solution and (<b>d</b>) comparisons between different solutions.</p> "> Figure 6
<p>Comparisons of calculation accuracy and efficiency of BDM and BIM for Case 1, including (<b>a</b>) the time-averaged APEs and (<b>b</b>) the training time.</p> "> Figure 7
<p>Comparisons of calculation accuracy and efficiency using <span class="html-italic">tanh</span> and <span class="html-italic">sigmoid</span> as activation function of Case 2, including (<b>a</b>) time-averaged APEs and (<b>b</b>) training time.</p> "> Figure 8
<p>Average value of APE under different hyperparameters.</p> "> Figure 9
<p>Average of training time (s) under different hyperparameters.</p> "> Figure 10
<p>Success rate of training under different hyperparameters.</p> "> Figure 11
<p>Geometric conditions of Case 4.</p> "> Figure 12
<p>Comparisons of BDM and COMSOL solution of Case 4, including (<b>a</b>) BDM solution, (<b>b</b>) COMSOL solution, comparisons between BDM and COMSOL along (<b>c</b>) lines <span class="html-italic">y</span> = 50 cm and (<b>d</b>) <span class="html-italic">x</span> = 40 cm and the PDE residuals along (<b>e</b>) lines <span class="html-italic">y</span> = 50 cm and (<b>f</b>) <span class="html-italic">x</span> = 40 cm.</p> ">
Abstract
:1. Introduction
2. Methodology
2.1. Dimensionless Neutron Diffusion Equation
2.2. BDM and BIM
2.3. Trial Functions for Special BCs in BDM
3. Results and Discussion
3.1. Case 1—Comparison of BDM and BIM
3.2. Case 2—Choice of Activation Function
3.3. Case 3—Impact of Hyperparameters
3.4. Case 4—Application in Complex Geometry
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A
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v (cm/s) | D (cm) | ϕ1 (n·cm−2·s−1) | τ (s) | l (cm) | Σa (cm−1) | |
---|---|---|---|---|---|---|
Value | 1.0 | 0.001 | 1.0 | 1.0 | 1.0 | 0 |
D (cm) | l (cm) | Σa (cm−1) | Q1 (n·cm−3·s−1) | |
---|---|---|---|---|
Value | 2/3 | 100 | 0.5 | 1.0 |
Area No. | D (cm) | Σa (cm−1) | Q (n·cm−3·s−1) |
---|---|---|---|
1 | 0.5556 | 0.07 | 0.79 |
3 | 0.4762 | 0.04 | 0.43 |
5 | 0.3704 | 0.01 | 0 |
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Xie, Y.; Wang, Y.; Ma, Y.; Wu, Z. Neural Network Based Deep Learning Method for Multi-Dimensional Neutron Diffusion Problems with Novel Treatment to Boundary. J. Nucl. Eng. 2021, 2, 533-552. https://doi.org/10.3390/jne2040036
Xie Y, Wang Y, Ma Y, Wu Z. Neural Network Based Deep Learning Method for Multi-Dimensional Neutron Diffusion Problems with Novel Treatment to Boundary. Journal of Nuclear Engineering. 2021; 2(4):533-552. https://doi.org/10.3390/jne2040036
Chicago/Turabian StyleXie, Yuchen, Yahui Wang, Yu Ma, and Zeyun Wu. 2021. "Neural Network Based Deep Learning Method for Multi-Dimensional Neutron Diffusion Problems with Novel Treatment to Boundary" Journal of Nuclear Engineering 2, no. 4: 533-552. https://doi.org/10.3390/jne2040036