Communications in Theoretical Physics ›› 2022, Vol. 74 ›› Issue (7): 075007. doi: 10.1088/1572-9494/ac7202
• Mathematical Physics • Previous Articles Next Articles
Shikun Cui1, Zhen Wang1,2,3,(), Jiaqi Han1, Xinyu Cui1, Qicheng Meng2
Received:
2022-01-26
Revised:
2022-05-22
Accepted:
2022-05-23
Published:
2022-07-01
Contact:
Zhen Wang
E-mail:wangzhen@dlut.edu.cn
Shikun Cui, Zhen Wang, Jiaqi Han, Xinyu Cui, Qicheng Meng, Commun. Theor. Phys. 74 (2022) 075007.
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Table 1.
The results of the one-soliton solution for the Boussinesq equation calculated by the deep learning method."
${k}_{1}$ | 0.8 | 0.9 | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 |
---|---|---|---|---|---|---|---|
${L}^{2}$ error | $2.79\times {10}^{-2}$ | $3.36\times {10}^{-2}$ | $2.03\times {10}^{-2}$ | $1.9\times {10}^{-2}$ | $1.67\times {10}^{-2}$ | $1.60\times {10}^{-2}$ | $1.62\times {10}^{-2}$ |
Time (s) | 224 | 191 | 152 | 160 | 294 | 284 | 480 |
Iterations | 141 | 155 | 170 | 195 | 576 | 416 | 753 |
Figure 5.
(a), (b) The comparison of the exact solution and the predicted solution for the colliding-soliton. (c) The absolute error between the exact solution and the predicted solution. (d)–(f) The detailed comparison of the exact solution and the predicted spatiotemporal solution at the specific time."
Table 2.
The results of different colliding-soliton solutions for the Boussinesq equation calculated by the deep learning method."
${k}_{1}$ | 0.8 | 0.9 | 1.0 | 1.1 | 1.2 |
---|---|---|---|---|---|
${k}_{2}$ | 0.8 | 0.9 | 1.0 | 1.1 | 1.2 |
${L}^{2}$ error | $8.55\times {10}^{-2}$ | $8.10\times {10}^{-2}$ | $1.16\times {10}^{-1}$ | $7.63\times {10}^{-2}$ | $7.64\times {10}^{-2}$ |
Time (s) | 358 | 551 | 642 | 430 | 1065 |
Iterations | 320 | 635 | 801 | 578 | 755 |
Table 3.
The results of different activations function of the one-soliton solution for the Boussinesq equation calculated by the deep learning method."
Activation function | tanh | cos | sin | Sigmoid | Relu |
---|---|---|---|---|---|
${L}^{2}$ error | $1.97\times {10}^{-2}$ | $1.86\times {10}^{-1}$ | $1.27\times {10}^{-1}$ | $9.22\times {10}^{-1}$ | $8.37\times {10}^{-1}$ |
Time (s) | 160 | 1670 | 1531 | 105 | 16 |
Iterations | 195 | 4722 | 3080 | 0 | 4 |
Table 4.
The results of different one-soliton solutions for the fifth-order KdV equation calculated by the deep learning method."
${k}_{1}$ | 0.9 | 0.95 | 1.0 | 1.05 | 1.1 |
---|---|---|---|---|---|
${L}^{2}$ error | $1.47\times {10}^{-2}$ | $2.57\times {10}^{-2}$ | $1.36\times {10}^{-2}$ | $4.64\times {10}^{-2}$ | $2.36\times {10}^{-2}$ |
Time(s) | 687 | 680 | 537 | 600 | 1166 |
Iterations | 300 | 803 | 803 | 883 | 1604 |
Table 5.
The results of different activation functions of the two-soliton solution for the fifth-order KdV equation calculated by the deep learning method."
Activation function | tanh | cos | sin | Sigmoid | Relu |
---|---|---|---|---|---|
${L}^{2}$ error | $1.46\times {10}^{-1}$ | $7.22\times {10}^{-2}$ | $1.27\times {10}^{-1}$ | $8.18\times {10}^{-1}$ | $6.13\times {10}^{-1}$ |
Time (s) | 623 | 1740 | 1531 | 237 | 48 |
Iterations | 1390 | 3600 | 3080 | 0 | 5 |
Figure 9.
(a), (b) The comparison of the one-soliton exact solution and the predicted solution for the fifth-order KdV equation. (c) The absolute error between the exact solution and the predicted solution. (d)–(f) The detailed comparison of the exact solution and the predicted solution at the specific time."
Figure 12.
(a), (b) The comparison of the two-soliton exact solution and the predicted solution for the fifth-order KdV equation. (c) The absolute error between the exact solution and the predicted solution. (d)–(f) The detailed comparison of the exact solution and the predicted spatiotemporal solution at the specific time."
Table 6.
The results of different two-soliton solutions for the fifth-order KdV equation calculated by the deep learning method."
${k}_{1}$ | 1.0 | 0.99 | 1.01 | 1.0 | 1.0 |
---|---|---|---|---|---|
${k}_{2}$ | 0.8 | 0.8 | 0.8 | 0.79 | 0.81 |
${L}^{2}$ error | $7.22\times {10}^{-2}$ | $8.27\times {10}^{-2}$ | $1.05\times {10}^{-1}$ | $9.05\times {10}^{-2}$ | $1.0\times {10}^{-1}$ |
Time (s) | 1774 | 913 | 1011 | 1000 | 657 |
Iterations | 3606 | 1654 | 1738 | 2224 | 1240 |
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[1] | Jiaheng Li,Biao Li. Solving forward and inverse problems of the nonlinear Schrödinger equation with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential via PINN deep learning [J]. Commun. Theor. Phys. 73 (2021) 125001. |
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