Deep Learning Method Based on Physics-Informed Neural Network for 3D Anisotropic Steady-State Heat Conduction Problems
Abstract
:1. Introduction
2. Equations of 3D Anisotropic Steady-State Heat Conduction Problems
3. The PINN for 3D Anisotropic Steady-State Heat Conduction Problem
Algorithm 1 Algorithmic Procedure | |
Input: Internal training data, (xi); boundary training data, (xj); | |
Output: Prediction of DNN, ; | |
1: | Initialize the parameters of DNN; |
2: | Define the loss function: loss = lossf + lossb; |
3: | for epoch = 1:numEpochs |
4: | U1 = ξ (xi; θ), ξ (xj; θ); |
5: | compute loss; |
6: | obtain gradients by automatic differential; |
7: | minimize the loss by Adam method; |
8: | end; |
9: | Obtain the prediction U = ξ (xk; θ); |
10: | return U. |
4. Numerical Examples
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Xing, Z.; Cheng, H.; Cheng, J. Deep Learning Method Based on Physics-Informed Neural Network for 3D Anisotropic Steady-State Heat Conduction Problems. Mathematics 2023, 11, 4049. https://doi.org/10.3390/math11194049
Xing Z, Cheng H, Cheng J. Deep Learning Method Based on Physics-Informed Neural Network for 3D Anisotropic Steady-State Heat Conduction Problems. Mathematics. 2023; 11(19):4049. https://doi.org/10.3390/math11194049
Chicago/Turabian StyleXing, Zebin, Heng Cheng, and Jing Cheng. 2023. "Deep Learning Method Based on Physics-Informed Neural Network for 3D Anisotropic Steady-State Heat Conduction Problems" Mathematics 11, no. 19: 4049. https://doi.org/10.3390/math11194049
APA StyleXing, Z., Cheng, H., & Cheng, J. (2023). Deep Learning Method Based on Physics-Informed Neural Network for 3D Anisotropic Steady-State Heat Conduction Problems. Mathematics, 11(19), 4049. https://doi.org/10.3390/math11194049