A Preliminary Study on the Resolution of Electro-Thermal Multi-Physics Coupling Problem Using Physics-Informed Neural Network (PINN)
Abstract
:1. Introduction
2. Materials and Methods
2.1. DNN
2.2. PINN
2.3. PINNs for the Electro-Thermal Coupling Problem
3. Experiments and Results
3.1. Electro-Thermal Coupling Problem
A Rectangle Electro-Thermal Coupling Problem
3.2. The Comparison between FEM and PINN
3.3. Empirical Properties of the PINN
3.3.1. The Effect of Depth and Width of the Network on the Training Time
3.3.2. Effect of Number of Samples on the Prediction Error and Training Time
4. Discussions
- PINN embeds physical constraints into the loss function of the neural network by using automatic differentiation. The imposition of the “hard” boundary makes the approximate solution automatically meet the Dirichlet boundary, which accelerates the convergence speed and improves the prediction accuracy [1].
- In addition to the advantage of being mesh-free, PINN can also generalize the construction process of various PDEs. On top of that, classical methods, such as FEM, can only obtain the solution on discrete points, while further interpolation is required for other points. This property makes the solution of the state variables at the interpolated points less accurate. For PINN, when considering the points that do not appear in the training set, there is no need to conduct an interpolation scheme to obtain the solutions.
- With the help of automatic differentiation, the derivative of each state variable can be easily calculated. Hence, the derivative distribution is smooth. However, for the first-order FEM, the first derivative of the state variable inside of an element is constant, which makes the derivative distribution discontinuous.
- Based on the experimental results, the convergence speed of PINN gradually increases as the size of the neural network goes up at the beginning. However, when it reaches a certain point, the improvement in computation time stagnates or even becomes worse. In addition, when studying the number of training samples influence on the convergence speed, except the case where the sampling points are too few, the convergence speed increases as the number of training samples increases; however, when it reaches a certain number, the improvement in convergence speed stagnates or even becomes worse.
- According to our experiments, although the PINN offers unique advantages for solving PDEs, its computational efficiency is an obvious disadvantage compared to FEM. Therefore, figuring out how to accelerate the training of PINN is an important research topic. Ref. [40] introduced an efficient approach based on adaptive sampling strategy, which speeds up the computation of the PINN. In [19], parallel calculation of the PINN was successfully implemented, which can easily handle any complex regional problems, but the improvement in computing time is still on the way. Huang et al. [41] realized speeding up convergence for high-frequency wavefields solutions by using the information from a pre-trained model instead of initializing the PINN randomly.
- PINN can be conveniently utilized to generate a surrogate model in the parametric analysis. In [42], the authors conducted a sensitivity analysis experiment. They first trained the PINN with merely a few values of a specific parameter and then utilized the trained neural network to predict the solution to this parameter within a range. The result is less accurate but still useful for the specific condition, which shows the possibility of PINN in tackling this kind of issue; we will work on this subject in our future study.
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Property | Steady Electric Field | Thermal Field |
---|---|---|
Lowest Loss | 9.97 × | 9.84 × |
Time(s) | 101 | 109 |
Depth | 1 | 2 | 4 | 6 | |
---|---|---|---|---|---|
Width | |||||
5 | 20,000 | 8539 | 5126 | 5193 | |
10 | 13,120 | 4832 | 2634 | 3046 | |
20 | 8791 | 2239 | 1430 | 1591 | |
30 | 7493 | 1993 | 1146 | 1584 | |
60 | 5958 | 1161 | 1343 | 2936 |
Depth | 1 | 2 | 4 | 6 | |
---|---|---|---|---|---|
Width | |||||
5 | 1129 | 547 | 410 | 497 | |
10 | 739 | 311 | 211 | 293 | |
20 | 495 | 144 | 115 | 153 | |
30 | 422 | 128 | 92 | 152 | |
60 | 335 | 75 | 108 | 286 |
Number | 10 | 100 | 500 | 1000 | 5000 | 10,000 | |
---|---|---|---|---|---|---|---|
Property | |||||||
Time (s) | 35 | 97 | 91 | 74 | 97 | 101 | |
RMSE | 1.59 | 5.60 | 1.09 | 5.14 × | 7.07 × | 5.48 × | |
Epoch | 551 | 1508 | 1123 | 1066 | 1480 | 1539 |
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Ma, Y.; Xu, X.; Yan, S.; Ren, Z. A Preliminary Study on the Resolution of Electro-Thermal Multi-Physics Coupling Problem Using Physics-Informed Neural Network (PINN). Algorithms 2022, 15, 53. https://doi.org/10.3390/a15020053
Ma Y, Xu X, Yan S, Ren Z. A Preliminary Study on the Resolution of Electro-Thermal Multi-Physics Coupling Problem Using Physics-Informed Neural Network (PINN). Algorithms. 2022; 15(2):53. https://doi.org/10.3390/a15020053
Chicago/Turabian StyleMa, Yaoyao, Xiaoyu Xu, Shuai Yan, and Zhuoxiang Ren. 2022. "A Preliminary Study on the Resolution of Electro-Thermal Multi-Physics Coupling Problem Using Physics-Informed Neural Network (PINN)" Algorithms 15, no. 2: 53. https://doi.org/10.3390/a15020053
APA StyleMa, Y., Xu, X., Yan, S., & Ren, Z. (2022). A Preliminary Study on the Resolution of Electro-Thermal Multi-Physics Coupling Problem Using Physics-Informed Neural Network (PINN). Algorithms, 15(2), 53. https://doi.org/10.3390/a15020053