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Mix-training physics-informed neural networks for high-order rogue waves of cmKdV equation

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Abstract

Recently, a neural networks method: physics-informed neural networks (PINNs) is proposed to solve nonlinear partial differential equations (NPDEs); this method obtained the predicted solution by minimizing the sum of mean square of initial error, boundary error and residual of equation. It provides a new approach for solving NPDEs by neural networks. The complex modified Korteweg–de Vries (cmKdV) equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas. However, due to its complexity, it is difficult to solve the high-order rogue waves of them by PINNs, so how to solve the high-order rogue waves of complex equation by neural networks method has become a hot research direction. In this paper, based on PINNs, we propose two neural networks methods: mix-training physics-informed neural networks (MTPINNs) and prior information mix-training physics-informed neural networks (PMTPINNs). Numerical experiments have shown that the original PINNs are completely unable to simulate the high-order rogue waves of the cmKdV equation, but our proposed models not only simulate these high-order rogue waves, but also significantly improve the simulation ability, increasing prediction accuracy by three to four orders of magnitude. The inverse problem of these models is tested by some noise data, which also proves that these models have good robustness. The above results validate the superiority of our proposed models in simulating high-order rogue waves of cmKdV equation, and these models provide us some new insights for studying the dynamic characteristics of high-order rogue waves using neural networks methods.

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Acknowledgements

This work is supported by National Natural Science Foundation of China under Grant Nos. 12175111 and 12235007 and K.C. Wong Magna Fund in Ningbo University.

Funding

The funding was provided by National Natural Science Foundation of China under Grant Nos. 12175111 and 12235007 and K.C. Wong Magna Fund in Ningbo University.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, People’s Republic of China

    Shifang Tian, Zhenjie Niu & Biao Li

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  1. Shifang Tian
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  2. Zhenjie Niu
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  3. Biao Li
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Correspondence to Biao Li.

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Appendices

Appendix A: (B(x, t) and C(x, t) of second-order rogue waves)

$$\begin{aligned} B(x,t)= & {} 2239488 a^{4} c^{14} t^{6}+46656 a^{12} c^{6} t^{6}\\{} & {} +2985984 c^{18} t^{6}\\{} & {} +\,559872 a^{8} c^{10} t^{6}+746496 a^{6} c^{10} t^{5} x\\{} & {} +\,93312 a^{10} c^{6} t^{5} x -1492992 a^{4} c^{12} t^{5} x\\{} & {} -\,186624 a^{8} c^{8} t^{5}x+1492992 a^{2} c^{14} t^{5} x\\{} & {} -\,2985984 c^{16} t^{5} x+1244160 c^{14} t^{4} x^{2}\\{} & {} -248832 a^{6} c^{8} t^{4} x^{2} -\,995328 a^{2} c^{12} t^{4} x^{2}\\{} & {} +\,622080 a^{4} c^{10} t^{4} x^{2}+77760 a^{8} c^{6} t^{4} x^{2}\\{} & {} -\,276480 c^{12} t^{3} x^{3}+34560 a^{6} c^{6} t^{3} x^{3}\\{} & {} -\,124416 a^{4} c^{8} t^{3} x^{3}+248832 a^{2} c^{10} t^{3} x^{3}\\{} & {} +\,34560 c^{10} t^{2} x^{4}+8640 a^{4} c^{6} t^{2} x^{4}\\{} & {} -\,27648 a^{2} c^{8} t^{2} x^{4}+1152 a^{2} c^{6} t x^{5}\\{} & {} -2304 c^{8} t x^{5}+64 c^{6} x^{6} -\,331776 c^{6} a^{6} t^{4}\\{} & {} -\,518400 c^{12} t^{4}+155520 c^{8} a^{4} t^{4}\\{} & {} -11664 c^{4} a^{8} t^{4}\\{} & {} +\,290304 c^{10} t^{3} x-15552 c^{4} a^{6} t^{3} x\\{} & {} +\,103680 c^{8} a^{2} t^{3} x-176256 c^{6} a^{4} t^{3} x \\{} & {} -58752 c^{8} t^{2} x^{2}\\{} & {} -\,6912 c^{6} a^{2} t^{2} x^{2}-7776 c^{4} a^{4} t^{2} x^{2}\\{} & {} +\,4992 c^{6} t x^{3}-1728 c^{4} a^{2} t x^{3}-144 c^{4} x^{4} \\{} & {} +\,124416 a c^{10} s_{1} t^{3}+31104 a^{5} c^{6} s_{1} t^{3}\\{} & {} -\,165888 a^{3} c^{8} s_{1} t^{3}+20736 a^{3} c^{6} s_{1} t^{2} x\\{} & {} -41472 a c^{8} s_{1} t^{2} x\\{} & {} +\,3456 a c^{6} s_{1} t x^{2} -18000 c^{6} t^{2}\\{} & {} -\,1620 c^{2} a^{4} t^{2}-1080 c^{2} a^{2} t x\\{} & {} +5616 c^{4} t x-180 c^{2} x^{2}\\{} & {} +\,864 c^{4} a s_{1} t+144 c^{4} s_{1}^{2}\\{} & {} +\,45+i(2985984 a c^{14} t^{5} +1492992 a^{5} c^{10} t^{5}\\{} & {} +\,186624 a^{9} c^{6} t^{5}-497664 a^{5} c^{8} t^{4} x\\{} & {} +\,995328 a^{3} c^{10} t^{4} x-1990656 a c^{12} t^{4} x\\{} & {} +\,248832 a^{7} c^{6} t^{4} x-331776 a^{3} c^{8} t^{3} x^{2}\\{} & {} +\,124416 a^{5} c^{6} t^{3} x^{2}+497664 a c^{10} t^{3} x^{2}\\{} & {} -\,55296 a c^{8} t^{2} x^{3}+27648 a^{3} c^{6} t^{2} x^{3}\\{} & {} +\,2304 a c^{6} t x^{4} +207360 c^{8} a t^{3}31104 c^{4} a^{5} t^{3}\\{} & {} -\,13824 c^{6} a t^{2} x-20736 c^{4} a^{3} t^{2} x\\{} & {} -\,3456 c^{4} a t x^{2}+20736 c^{8} s_{1} t^{2} \\{} & {} -\,41472 a^{2} c^{6} s_{1} t^{2}+5184 a^{4} c^{4} s_{1} t^{2}\\{} & {} +\,3456 a^{2} c^{4} s_{1} t x-6912 c^{6} s_{1} t x+576 c^{4} s_{1} x^{2}\\{} & {} -\,2160 a c^{2} t+144 s_{1} c^{2}) \end{aligned}$$

and

$$\begin{aligned} C(x,t)= & {} 2239488 a^{4} c^{14} t^{6}+2985984 c^{18} t^{6}\\{} & {} +\,46656 a^{12} c^{6} t^{6}\\{} & {} +\,559872 a^{8} c^{10} t^{6}+746496 a^{6} c^{10} t^{5} x\\{} & {} -\,1492992 a^{4} c^{12} t^{5} x -186624 a^{8} c^{8} t^{5} x\\{} & {} -\,2985984 c^{16} t^{5} x+93312 a^{10} c^{6} t^{5} x\\{} & {} +\,1492992 a^{2} c^{14} t^{5} x+1244160 c^{14} t^{4} x^{2}\\{} & {} -\,995328 a^{2} c^{12} t^{4} x^{2} +622080 a^{4} c^{10} t^{4} x^{2}\\{} & {} +\,77760 a^{8} c^{6} t^{4} x^{2}-248832 a^{6} c^{8} t^{4} x^{2}\\{} & {} +\,248832 a^{2} c^{10} t^{3} x^{3}+34560 a^{6} c^{6} t^{3} x^{3}\\{} & {} -\,124416 a^{4} c^{8} t^{3} x^{3}-276480 c^{12} t^{3} x^{3}\\{} & {} -\,27648 a^{2} c^{8} t^{2} x^{4}+8640 a^{4} c^{6} t^{2} x^{4}\\{} & {} +\,34560 c^{10} t^{2} x^{4}-2304 c^{8} t x^{5}\\{} & {} +\,1152 a^{2} c^{6} t x^{5}+64 c^{6} x^{6}\\{} & {} +995328 c^{10} a^{2} t^{4} +\,279936 c^{8} a^{4} t^{4}\\{} & {} -82944 c^{6} a^{6} t^{4}+3888 c^{4} a^{8} t^{4}\\{} & {} -\,269568 c^{12} t^{4}+124416 c^{10} t^{3} x\\{} & {} +\,5184 c^{4} a^{6} t^{3} x -51840 c^{6} a^{4} t^{3} x\\{} & {} -\,145152 c^{8} a^{2} t^{3} x\\{} & {} -\,17280 c^{8} t^{2} x^{2}-6912 c^{6} a^{2} t^{2} x^{2}\\{} & {} +\,2592 c^{4} a^{4} t^{2} x^{2}+384 c^{6} t x^{3}+576 c^{4} a^{2} t x^{3} \\{} & {} +\,48 c^{4} x^{4}-165888 a^{3} c^{8} s_{1} t^{3}\\{} & {} +\,124416 a c^{10} s_{1} t^{3}+31104 a^{5} c^{6} s_{1} t^{3}\\{} & {} +\,20736 a^{3} c^{6} s_{1} t^{2} x\\{} & {} -\,41472 a c^{8} s_{1} t^{2} x\\{} & {} +\,3456 a c^{6} s_{1} t x^{2} +20016 c^{6} t^{2}\\{} & {} +\,972 c^{2} a^{4} t^{2}+6912 c^{4} a^{2} t^{2}\\{} & {} +\,648 c^{2} a^{2} t x-2448 c^{4} t x\\{} & {} +\,108 c^{2} x^{2}-2592 c^{4} a s_{1} t+144 c^{4} s_{1}^{2}+9. \end{aligned}$$

Appendix B: (L1(x, t) and L2(x, t) of third-order rogue waves)

$$\begin{aligned} L_{1}(x,t)= & {} -4939273445868140625 t^{12}\\{} & {} -\,545023276785450000 t^{11} x\\{} & {} -\,388407392651700000 t^{10} x^{2} \\{} & {} -\,34025850934560000 t^{9} x^{3}\\{} & {} -\,12374529519456000 t^{8} x^{4}\\{} & {} -\,841946352721920 t^{7} x^{5} \\{} & {} -\,204871837925376 t^{6} x^{6}\\{} & {} -10322713903104 t^{5} x^{7}\\{} & {} -\,1860148592640 t^{4} x^{8}\\{} & {} -\,62710087680 t^{3} x^{9}-8776581120 t^{2} x^{10}\\{} & {} -\,150994944 t x^{11}-16777216 x^{12}\\{} & {} +\,2300237292280725000 t^{10} \\{} & {} +\,500080291038360000 t^{9} x\\{} & {} +\,178207653254544000 t^{8} x^{2}\\{} & {} +\,20160748631347200 t^{7} x^{3}\\{} & {} +\,4231975119851520 t^{6} x^{4}\\{} & {} +\,253682249269248 t^{5} x^{5}\\{} & {} +\,40317552230400 t^{4} x^{6}\\{} & {} +880347709440 t^{3} x^{7} \\{} & {} +\,141203865600 t^{2} x^{8}-1447034880 t x^{9}\\{} & {} +\,75497472 x^{10}+257120426548112400 t^{8}\\{} & {} -\,88521031030049280 t^{7} x\\{} & {} -\,1841385225323520 t^{6} x^{2}\\{} & {} -617799343104000 t^{5} x^{3} \\{} & {} -\,88747774156800 t^{4} x^{4}\\{} & {} -\,8252622766080 t^{3} x^{5}\\{} & {} -200693514240 t^{2} x^{6}\\{} & {} +\,1415577600 t x^{7}+235929600 x^{8}\\{} & {} +\,12647412412496640 t^{6}\\{} & {} +42148769126400 t^{5} x\\{} & {} -\,2996161228800 t^{4} x^{2} \\{} & {} -\,15902996889600 t^{3} x^{3}\\{} & {} +313860096000 t^{2} x^{4}\\{} & {} -\,8139571200 t x^{5}+707788800 x^{6}\\{} & {} -\,149676507590400 t^{4}\\{} & {} +2148738969600 t^{3} x\\{} & {} -\,1622998425600 t^{2} x^{2}+31186944000 t^{3} x \\{} & {} -\,928972800 x^{4}-95215564800 t^{2}\\{} & {} +21598617600 t x\\{} & {} -\,464486400 x^{2}+58060800 \\{} & {} +\,i(-6540279321425400000 t^{11}\\{} & {} -\,601404995073600000 t^{10} x\\{} & {} -\,423057306879360000 t^{9} x^{2}\\{} & {} -29901007761408000 t^{8} x^{3}\\{} & {} -\,10648770814771200 t^{7} x^{4}\\{} & {} -\,552411244265472 t^{6} x^{5}\\{} & {} -130559642173440 t^{5} x^{6}\\{} & {} -\,4494741995520 t^{4} x^{7}\\{} & {} -\,779700142080 t^{3} x^{8}-13589544960 t^{2} x^{9}\\{} & {} -\,1811939328 t x^{10}\\{} & {} -303419904173280000 t^{9}\\{} & {} +\,263517097430016000 t^{8} x \\{} & {} +29562711599923200 t^{7} x^{2}\\{} & {} +\,7820253397647360 t^{6} x^{3}\\{} & {} +\,799710056939520 t^{5} x^{4}\\{} & {} +58500443013120 t^{4} x^{5}\\{} & {} +\,5243865661440 t^{3} x^{6}\\{} & {} +\,40768634880 t^{2} x^{7}+6794772480 t x^{8}\\{} & {} +\,10822374648023040 t^{7}\\{} & {} -\,15805156998758400 t^{6} x\\{} & {} -366298181959680 t^{5} x^{2} \\{} & {} +\,157459297075200 t^{4} x^{3}\\{} & {} -\,6736379904000 t^{3} x^{4}-249707888640 t^{2} x^{5}\\{} & {} +\,16986931200 t x^{6} \\{} & {} +\,2321279745269760 t^{5}\\{} & {} -164322282700800 t^{4} x\\{} & {} +\,7849554739200 t^{3} x^{2}\\{} & {} -\,700710912000 t^{2} x^{3}+38220595200 t x^{4}\\{} & {} +4992863846400 t^{3}\\{} & {} -\,967458816000 t^{2} x\\{} & {} -\,33443020800 t x^{2}-8360755200 t) \end{aligned}$$
$$\begin{aligned} L_{2}(x,t)= & {} 4939273445868140625 t^{12}\\{} & {} +\,545023276785450000 t^{11} x\\{} & {} +388407392651700000 t^{10} x^{2} \\{} & {} +\,34025850934560000 t^{9} x^{3}\\{} & {} +\,12374529519456000 t^{8} x^{4}\\{} & {} +841946352721920 t^{7} x^{5} \\{} & {} +\,204871837925376 t^{6} x^{6}\\{} & {} +\,10322713903104 t^{5} x^{7}\\{} & {} +1860148592640 t^{4} x^{8} \\{} & {} +\,62710087680 t^{3} x^{9}+8776581120 t^{2} x^{10}\\{} & {} +\,150994944 t x^{11}+16777216 x^{12}\\{} & {} +\,1671541529351175000 t^{10}\\{} & {} -201221180459640000 t^{9} x\\{} & {} +\,29583374214960000 t^{8} x^{2} \\{} & {} -\,8646206984294400 t^{7} x^{3}\\{} & {} -205872145551360 t^{6} x^{4}\\{} & {} -\,102185078390784 t^{5} x^{5} \\{} & {} -\,5361951375360 t^{4} x^{6}-141416202240 t^{3} x^{7}\\{} & {} -\,16349921280 t^{2} x^{8}+2202009600 t x^{9} \\{} & {} +\,25165824 x^{10}+214192109547903600 t^{8}\\{} & {} +\,4346367735790080 t^{7} x \\{} & {} +\,3783154346910720 t^{6} x^{2}\\{} & {} -1075635551969280 t^{5} x^{3}\\{} & {} +\,40942170316800 t^{4} x^{4} \\{} & {} +\,1840480911360 t^{3} x^{5}+84333035520 t^{2} x^{6}\\{} & {} +\,849346560 t x^{7}+141557760 x^{8} \\{} & {} +\,3537485306138880 t^{6}\\{} & {} +789853361080320 t^{5} x\\{} & {} +\,130057245388800 t^{4} x^{2} \\{} & {} -\,11222478028800 t^{3} x^{3}\\{} & {} +402245222400 t^{2} x^{4}\\{} & {} -\,4034396160 t x^{5}+613416960 x^{6} \\{} & {} +\,90779142700800 t^{4}-6216180019200 t^{3} x\\{} & {} -\,269186457600 t^{2} x^{2}+11988172800 t x^{3} \\{} & {} +\,221184000 x^{4}+213178521600 t^{2}\\{} & {} -\,5009817600 t x+199065600x^{2}\\{} & {} +8294400 \end{aligned}$$

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Tian, S., Niu, Z. & Li, B. Mix-training physics-informed neural networks for high-order rogue waves of cmKdV equation. Nonlinear Dyn 111, 16467–16482 (2023). https://doi.org/10.1007/s11071-023-08712-3

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  • DOI: https://doi.org/10.1007/s11071-023-08712-3

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