Communications in Theoretical Physics ›› 2020, Vol. 72 ›› Issue (11): 115003. doi: 10.1088/1572-9494/abb7c8
• Mathematical Physics • Previous Articles Next Articles
Jun Li(李军)1,Yong Chen(陈勇)2,3,4,(
)
Received: 2020-05-15
Revised: 2020-07-29
Accepted: 2020-07-29
Published: 2020-11-01
Contact:
Yong Chen(陈勇)
E-mail:ychen@sei.ecnu.edu.cn
Jun Li(李军),Yong Chen(陈勇), Commun. Theor. Phys. 72 (2020) 115003.
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Figure 1.
The KdV equation. Top: a one-soliton solution to the KdV equation (left panel) is compared to the corresponding predicted solution to the learned equation (right panel). The network correctly captures the dynamics behavior and accurately reproduces the soliton solution with a relative ${{\mathbb{L}}}_{2}$ error of 3.44 × 10–3 . Bottom: the comparison of the predicted and exact soliton solutions which correspond to the three temporal snapshots depicted by the white vertical lines in the top panel is presented."
Figure 2.
The spatiotemporal behavior of a one-soliton solution to the learned KdV equation."
Figure 3.
The KdV equation. Top: a two-soliton solution to the KdV equation is compared to the corresponding predicted solution to the learned equation (right panel). The model correctly exhibits the dynamics behavior and accurately reproduces the solution with a relative ${{\mathbb{L}}}_{2}$ error of 7.39 × 10−2 . Bottom: the comparison of the predicted solutions and exact solutions which correspond to the three temporal snapshots is presented."
Figure 4.
The spatiotemporal behavior of a two-soliton solution to the learned KdV equation."
Figure 5.
The KdV equation. Top: another two-soliton solution to the KdV equation (left panel) is compared to the predicted solution to the learned equation. The model correctly exhibits the dynamics behavior and accurately reproduces the solution with a relative ${{\mathbb{L}}}_{2}$ error of 2.53 × 10−2 . Bottom: the comparison of the predicted solutions and exact solutions is presented. The model training took about 7.5 min."
Figure 6.
The mKdV equation. Top: a one-soliton solution to the mKdV equation (left panel) is compared to the predicted solution to the learned equation. The model correctly exhibits the dynamics behavior and accurately reproduces the solution with a relative ${{\mathbb{L}}}_{2}$ error of 4.57 × 10−2 . Bottom: the comparison of the predicted solutions and exact solutions corresponding to the three temporal snapshots is given."
Figure 7.
The mKdV equation. Top: a breather solution to the mKdV equation (left panel) is compared to the predicted solution to the learned equation. The model correctly exhibits the dynamics behavior and accurately reproduces the solution with a relative ${{\mathbb{L}}}_{2}$ error of 1.05 × 10−2 . Bottom: the comparison of the predicted solutions and exact solutions is presented."
Figure 8.
The KdV–Burgers equation. Top: a one-kink solution to the KdVB equation (left panel) is compared to the predicted solution to the learned equation. The model correctly exhibits the dynamics behavior and accurately reproduces the solution with a relative ${{\mathbb{L}}}_{2}$ error of 8.08 × 10−3 . Bottom: the comparison of the predicted solutions and exact solutions is presented."
Figure 9.
The soliton fusion phenomenon of the STO equation. Top: a solution to the STO equation (left panel) is compared to the predicted solution to the learned equation. The model correctly exhibits the dynamics behavior and accurately reproduces the solution with a relative ${{\mathbb{L}}}_{2}$ error of 1.61 × 10−2 . Middle: the comparison of the predicted solutions and exact solutions is presented. Bottom: the comparison of the corresponding predicted solutions and exact solutions of the potential −u x is also given."
Figure 10.
The soliton fusion pattern of the STO equation. (a) The spatiotemporal behavior of the reconstructed solution; (b) the spatiotemporal dynamics of the corresponding potential."
Figure 11.
The soliton fission phenomenon of the STO equation. Top: a solution to the STO equation (left panel) is compared to the predicted solution to the learned equation. The model approximately exhibits the dynamics behavior and reproduces the solution with a relative ${{\mathbb{L}}}_{2}$ error of 2.41 × 10−2 . Middle: the comparison of the predicted solutions and exact solutions is presented. Bottom: the comparison of the corresponding predicted solutions and exact solutions of the potential is also given."
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