Due to the dynamic characteristics of instantaneity and steepness, employing domain decomposition techniques for simulating rogue wave solutions is highly appropriate. Wherein, the backward compatible physics-informed neural network (bc-PINN) is a temporally sequential scheme to solve PDEs over successive time segments while satisfying all previously obtained solutions. In this work, we propose improvements to the original bc-PINN algorithm in two aspects based on the characteristics of error propagation. One is to modify the loss term for ensuring backward compatibility by selecting the earliest learned solution for each sub-domain as pseudo-reference solution. The other is to adopt the concatenation of solutions obtained from individual subnetworks as the final form of the predicted solution. The improved backward compatible PINN (Ibc-PINN) is applied to study data-driven higher-order rogue waves for the nonlinear Schrödinger (NLS) equation and the AB system to demonstrate the effectiveness and advantages. Transfer learning and initial condition guided learning (ICGL) techniques are also utilized to accelerate the training. Moreover, the error analysis is conducted on each sub-domain, and it turns out that the slowdown of Ibc-PINN in error accumulation speed can yield greater advantages in accuracy. In short, numerical results fully indicate that Ibc-PINN significantly outperforms bc-PINN in terms of accuracy and stability without sacrificing efficiency.

1.
L.
Draper
, “
‘Freak’ ocean waves
,”
Weather
21
(
1
),
2
4
(
1966
).
2.
D. R.
Solli
,
C.
Ropers
,
P.
Koonath
, and
B.
Jalali
, “
Optical rogue waves
,”
Nature
450
(
7172
),
1054
1057
(
2007
).
3.
Y. V.
Bludov
,
V. V.
Konotop
, and
N.
Akhmediev
, “
Matter rogue waves
,”
Phys. Rev. A
80
,
033610
(
2009
).
4.
W. M.
Moslem
,
P. K.
Shukla
, and
B.
Eliasson
, “
Surface plasma rogue waves
,”
Euro. Phys. Lett.
96
,
25002
(
2011
).
5.
V. B.
Efimov
,
A. N.
Ganshin
,
G. V.
Kolmakov
,
P. V. E.
McClintock
, and
L. P.
Mezhov-Deglin
, “
Rogue waves in superfluid helium
,”
Eur. Phys. J. Spec. Top.
185
,
181
193
(
2010
).
6.
L.
Stenflo
and
M.
Marklund
, “
Rogue waves in the atmosphere
,”
J. Plasma Phys.
76
(
3-4
),
293
295
(
2010
).
7.
Z. Y.
Yan
, “
Financial rogue waves
,”
Commun. Theor. Phys.
54
(
5
),
947
(
2010
).
8.
D. H.
Peregrine
, “
Water waves, nonlinear Schrödinger equations and their solutions
,”
J. Australian Math. Soc. Ser. B App. Math.
25
(
1
),
16
43
(
1983
).
9.
Y.
Ohta
and
J. K.
Yang
, “
Rogue waves in the Davey-Stewartson I equation
,”
Phys. Rev. E
86
,
036604
(
2012
).
10.
J. C.
Chen
,
Y.
Chen
,
B. F.
Feng
,
K.
Maruno
, and
Y.
Ohta
, “
General high-order rogue waves of the (1+1)-dimensional Yajima-Oikawa system
,”
J. Phys. Soc. Jpn.
87
(
9
),
094007
(
2018
).
11.
B. L.
Guo
,
L. M.
Ling
, and
Q. P.
Liu
, “
Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions
,”
Phys. Rev. E
85
(
2
),
026607
(
2012
).
12.
X. E.
Zhang
and
Y.
Chen
, “
Inverse scattering transformation for generalized nonlinear Schrödinger equation
,”
Appl. Math. Lett.
98
,
306
313
(
2019
).
13.
M.
Raissi
,
P.
Perdikaris
, and
G. E.
Karniadakis
, “
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
,”
J. Comput. Phys.
378
,
686
707
(
2019
).
14.
G. F.
Pang
,
L.
Lu
, and
G. E.
Karniadakis
, “
fPINNs: Fractional physics-informed neural networks
,”
SIAM J. Sci. Comput.
41
(
4
),
A2603
A2626
(
2019
).
15.
L.
Lu
,
R.
Pestourie
,
W. J.
Yao
,
Z. C.
Wang
,
F.
Verdugo
, and
S. G.
Johnson
, “
Physics-informed neural networks with hard constraints for inverse design
,”
SIAM J. Sci. Comput.
43
(
6
),
B1105
B1132
(
2021
).
16.
L.
Yang
,
X. H.
Meng
, and
G. E.
Karniadakis
, “
B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data
,”
J. Comput. Phys.
425
,
109913
(
2021
).
17.
D.
Zhang
,
L.
Lu
,
L.
Guo
, and
G. E.
Karniadakis
, “
Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems
,”
J. Comput. Phys.
397
,
108850
(
2019
).
18.
S. N.
Lin
and
Y.
Chen
, “
A two-stage physics-informed neural network method based on con- served quantities and applications in localized wave solutions
,”
J. Comput. Phys.
457
,
111053
(
2022
).
19.
S. N.
Lin
and
Y.
Chen
, “
Physics-informed neural network methods based on Miura transformations and discovery of new localized wave solutions
,”
Phys. D
445
,
133629
(
2023
).
20.
Z. Y.
Zhang
,
H.
Zhang
,
L. S.
Zhang
, and
L. L.
Guo
, “
Enforcing continuous symmetries in physics-informed neural network for solving forward and inverse problems of partial differential equations
,”
J. Comput. Phys.
492
,
112415
(
2023
).
21.
A. D.
Jagtap
,
E.
Kharazmi
, and
G. E.
Karniadakis
, “
Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems
,”
Comput. Methods Appl. Mech. Eng.
365
,
113028
(
2020
).
22.
A. D.
Jagtap
and
G. E.
Karniadakis
, “
Extended physics-informed neural networks (XPINNs): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations
,”
Commun. Comput. Phys.
28
,
2002
2041
(
2020
).
23.
K.
Shukla
,
A. D.
Jagtap
, and
G. E.
Karniadakis
, “
Parallel physics-informed neural networks via domain decomposition
,”
J. Comput. Phys.
447
,
110683
(
2021
).
24.
B.
Moseley
,
A.
Markham
, and
T.
Nissen-Meyer
, “
Finite basis physics-informed neural networks (FBPINNs): A scalable domain decomposition approach for solving differential equations
,”
Adv. Comput. Math.
49
(
4
),
62
(
2023
).
25.
E.
Kharazmi
,
Z.
Zhang
, and
G. E.
Karniadakis
, “
hp-VPINNs: Variational physics-informed neural networks with domain decomposition
,”
Comput. Methods Appl. Mech. Eng.
374
,
113547
(
2021
).
26.
P.
Stiller
,
F.
Bethke
,
M.
Böhme
,
R.
Pausch
,
S.
Torge
,
A.
Debus
,
J.
Vorberger
,
M.
Bussmann
, and
N.
Hoffmann
, “Large-scale neural solvers for partial differential equations, driving scientific and engineering discoveries through the convergence of HPC, big data and AI,” in 17th Smoky Mountains Computational Sciences and Engineering Conference, SMC 2020, Oak Ridge, TN, USA, August 26–28, 2020, Revised Selected Papers 17 (Springer International Publishing, 2020), pp. 20–34.
27.
N.
Shazeer
,
A.
Mirhoseini
,
K.
Maziarz
,
A.
Davis
,
Q.
Le
,
G.
Hinton
, and
J.
Dean
, “Outrageously large neural networks: The sparsely-gated mixture-of-experts layer,” arXiv:1701.06538 (2017).
28.
C. L.
Wight
and
J.
Zhao
, “Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks,” arXiv:2007.04542 (2020).
29.
R.
Mattey
and
S.
Ghosh
, “
A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations
,”
Comput. Methods Appl. Mech. Eng.
390
,
114474
(
2022
).
30.
S. F.
Wang
,
S.
Sankaran
, and
P.
Perdikaris
, ‘Respecting causality is all you need for training physics-informed neural networks,” arXiv:2203.07404 (2022).
31.
K.
Haitsiukevich
and
A.
Ilin
, “Improved training of physics-informed neural networks with model ensembles,” in 2023 International Joint Conference on Neural Networks (IJCNN) (IEEE, 2023), pp. 1–8.
32.
J.
Li
and
Y.
Chen
, “
Solving second-order nonlinear evolution partial differential equations using deep learning
,”
Commun. Theor. Phys.
72
(
10
),
105005
(
2020
).
33.
J.
Li
and
Y.
Chen
, “
A deep learning method for solving third-order nonlinear evolution equations
,”
Commun. Theor. Phys.
72
(
11
),
115003
(
2020
).
34.
J. C.
Pu
,
J.
Li
, and
Y.
Chen
, “
Soliton, breather and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints
,”
Chin. Phys. B
30
(
6
),
060202
(
2021
).
35.
W. Q.
Peng
,
J. C.
Pu
, and
Y.
Chen
, “
PINN deep learning for the Chen-Lee-Liu equation: Rogue wave on the periodic background
,”
Commun. Nonlinear Sci. Numer. Simul.
105
,
106067
(
2022
).
36.
Z. W.
Miao
and
Y.
Chen
, “
Physics-informed neural network method in high-dimensional integrable systems
,”
Mod. Phys. Lett. B
36
(
1
),
2150531
(
2022
).
37.
J. C.
Pu
and
Y.
Chen
, “
Data-driven vector localized waves and parameters discovery for Manakov system using deep learning approach
,”
Chaos, Solitons Fractals
160
,
112182
(
2022
).
38.
J. C.
Pu
and
Y.
Chen
, “
Data-driven forward-inverse problems for Yajima-Oikawa system using deep learning with parameter regularization
,”
Commun. Nonlinear Sci. Numer. Simul.
118
,
107051
(
2023
).
39.
L.
Wang
and
Z. Y.
Yan
, “
Data-driven rogue waves and parameter discovery in the defocusing nonlinear Schrödinger equation with a potential using the PINN deep learning
,”
Phys. Lett. A
404
,
127408
(
2021
).
40.
M.
Zhong
,
S.
Gong
,
S. F.
Tian
, and
Z. Y.
Yan
, “
Data-driven rogue waves and parameters discovery in nearly integrable PT-symmetric Gross-Pitaevskii equations via PINNs deep learning
,”
Phys. D
439
,
133430
(
2022
).
41.
J. C.
Chen
,
J.
Song
,
Z. J.
Zhou
, and
Z. Y.
Yan
, “
Data-driven localized waves and parameter discovery in the massive Thirring model via extended physics-informed neural networks with interface zones
,”
Chaos, Solitons Fractals
176
,
114090
(
2023
).
42.
J. H.
Li
and
B.
Li
, “
Mix-training physics-informed neural networks for the rogue waves of nonlinear Schrödinger equation
,”
Chaos, Solitons Fractals
164
,
112712
(
2022
).
43.
Y. B.
Zhang
,
H. Y.
Liu
,
L.
Wang
, and
W. R.
Sun
, “
The line rogue wave solutions of the nonlocal Davey-Stewartson I equation with PT symmetry based on the improved physics-informed neural network
,”
Chaos
33
(
1
),
013118
(
2023
).
44.
J.
Pedlosky
, “
Finite amplitude baroclinic wave packets
,”
J. Atmos. Sci.
29
,
680
686
(
1972
).
45.
M. L.
Stein
, “
Large sample properties of simulations using latin hypercube sampling
,”
Technometrics
29
(
2
),
143
151
(
1987
).
46.
A. G.
Baydin
,
B. A.
Pearlmutter
,
A. A.
Radul
, and
J. M.
Siskind
, “
Automatic differentiation in machine learning: A survey
,”
J. Mach. Learn. Res.
18
,
1
43
(
2018
).
47.
X.
Glorot
and
Y.
Bengio
, “
Understanding the difficulty of training deep feedforward neural networks
,”
J. Mach. Learn. Res.
9
,
249
256
(
2010
).
48.
K. M.
He
,
X. Y.
Zhang
,
S. Q.
Ren
, and
J.
Sun
, “Delving deep into rectifiers: Surpassing human-level performance on imagenet classification,” in Proceedings of the IEEE International Conference on Computer Vision (ICCV) (ICCV, 2015), pp. 1026–1034.
49.
D. C.
Liu
and
J.
Nocedal
, “
On the limited memory BFGS method for large scale optimization
,”
Math. Program.
45
(
1
),
503
528
(
1989
).
50.
E.
Schrödinger
, “
An undulatory theory of the mechanics of atoms and molecules
,”
Phys. Rev.
28
(
6
),
1049
(
1926
).
51.
A.
Hasegawa
and
F.
Tappert
, “
Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion
,”
Appl. Phys. Lett.
23
,
142
144
(
1973
).
52.
L.
Kavitha
and
M.
Daniel
, “
Integrability and soliton in a classical one-dimensional site-dependent biquadratic Heisenberg spin chain and the effect of nonlinear inhomogeneity
,”
J. Phys. A
36
(
42
),
10471
10492
(
2003
).
53.
V. B.
Matveev
and
M. A.
Salle
,
Darboux Transformations and Solitons
(
Springer
,
Berlin
,
1991
).
54.
J. K.
Yang
, Nonlinear Waves in Integrable and Nonintegrable Systems (Society for Industrial and Applied Mathematics,
2010
).
55.
X.
Wang
,
B.
Yang
,
Y.
Chen
, and
Y. Q.
Yang
, “
Higher-order rogue wave solutions of the Kundu-Eckhaus equation
,”
Phys. Scr.
89
(
9
),
095210
(
2014
).
56.
S. J.
Pan
and
Q.
Yang
, “
A survey on transfer learning
,”
IEEE Trans. Knowl. Data Eng.
22
(
10
),
1345
1359
(
2010
).
57.
A.
Kundu
, “
Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger-type equations
,”
J. Math. Phys.
25
,
3433
3438
(
1984
).
58.
F.
Calogero
and
W.
Eckhaus
, “
Nonlinear evolution equations, rescalings, model PDEs and their integrability: I
,”
Inverse Probl.
3
,
229
262
(
1987
).
59.
I.
Moroz
, “
Slowly modulated baroclinic waves in a three-layer model
,”
J. Atmos. Sci.
38
,
600
608
(
1981
).
60.
I. M.
Moroz
and
J.
Brindley
, “
Evolution of baroclinic wave packets in a flow with continuous shear and stratification
,”
Proc. Roy. Soc. London A
377
,
397
404
(
1981
).
61.
R.
Dodd
,
J.
Eilbeck
,
J.
Gibbon
, and
H.
Morris
,
Solitons and Nonlinear Wave Equations
(
Academic Press
,
London
,
1982
), p.
630
62.
B.
Tan
and
J. P.
Boyd
, “
Envelope solitary waves and periodic waves in the AB equations
,”
Stud. Appl. Math.
109
,
67
87
(
2002
).
63.
A. M.
Kamchatnov
and
M. V.
Pavlov
, “
Periodic solutions and Whitham equations for the AB system
,”
J. Phys. A, Math. Gen.
28
,
3279
3288
(
1995
).
64.
R.
Guo
,
H. Q.
Hao
, and
L. L.
Zhang
, “
Dynamic behaviors of the breather solutions for the AB system in fluid mechanics
,”
Nonlinear Dyn.
74
,
701
709
(
2013
).
65.
J. Y.
Zhu
and
X. G.
Geng
, “
The AB equations and the ¯-dressing method in semi-characteristic coordinates
,”
Math. Phys. Anal. Geom.
17
,
49
65
(
2014
).
66.
X.
Wang
,
Y. Q.
Li
,
F.
Huang
, and
Y.
Chen
, “
Rogue wave solutions of AB system
,”
Commun. Nonlinear Sci. Numer. Simulat.
20
,
434
442
(
2015
).
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