Deterministic wave prediction model for irregular long-crested waves with Recurrent Neural Network

doi:10.1016/j.joes.2022.08.002

Abstract

Abstract:

Real-time predicting of stochastic waves is crucial in marine engineering. In this paper, a deep learning wave prediction (Deep-WP) model based on the â€˜probabilistic' strategy is designed for the short-term prediction of stochastic waves. The Deep-WP model employs the long short-term memory (LSTM) unit to collect pertinent information from the wave elevation time series. Five irregular long-crested waves generated in the deepwater offshore basin at Shanghai Jiao Tong University are used to validate and optimize the Deep-WP model. When the prediction duration is 1.92s, 2.56s, and, 3.84s, respectively, the predicted results are almost identical with the ground truth. As the prediction duration is increased to 7.68s or 15.36s, the Deep-WP model's error increases, but it still maintains a high level of accuracy during the first few seconds. The introduction of covariates will improve the Deep-WP model's performance, with the absolute position and timestamp being particularly advantageous for wave prediction. Furthermore, the Deep-WP model is applicable to predict waves with different energy components. The proposed Deep-WP model shows a feasible ability to predict nonlinear stochastic waves in real-time.

Highlights

● A deep learning wave prediction (Deep-WP) model is proposed for stochastic waves.

● The model is based on an effective 'probabilistic' strategy.

● Three covariates are introduced, successfully improving the prediction accuracy.

● The model's performance is validated by experimental measurements.

Key words: Real-time wave prediction, Probability, LSTM, Covariate

Yue Liu, Xiantao Zhang, Gang Chen, Qing Dong, Xiaoxian Guo, Xinliang Tian, Wenyue Lu, Tao Peng. Deterministic wave prediction model for irregular long-crested waves with Recurrent Neural Network[J]. Journal of Ocean Engineering and Science, 2024, 9(3): 251-263.

Yue Liu, Xiantao Zhang, Gang Chen, Qing Dong, Xiaoxian Guo, Xinliang Tian, Wenyue Lu, Tao Peng. [J]. Journal of Ocean Engineering and Science, 2024, 9(3): 251-263.

/ / Recommend / Download Citations

URL: https://www.qk.sjtu.edu.cn/joes/EN/10.1016/j.joes.2022.08.002

https://www.qk.sjtu.edu.cn/joes/EN/Y2024/V9/I3/251

Figures/Tables 18

Fig. 1. Schematic diagram of the structure of the Deep-WP model.

Fig. 2. Basic structure of an LSTM unit.

Fig. 3. Probabilities for a Gaussian distribution, the percentage represents probability.

Table 1. Network structure and training method used for the present model.

Deep-WP
Number of LSTM layers	3
Dropout ratio	0.1
Number of FC layers	1
Affine function	Softplus
Likelihood model	Gaussian
Optimizer	Adam
Maximum epoch size	1000
Learning rate schedule	ReduceLROnPlateau
Early stopping	True
Number of best-trained models kept	5

Table 2. Wave parameters corresponding to the prototype and model.

Sea State	$H_{s}$ (m)	$T_{p}$ (s)	$H_{s}$ (m)	$T_{p}$ (s)
	Prototype		Model
WC01	6	14.85	0.094	1.856
WC02	8	14.85	0.125	1.856
WC03	10	14.85	0.156	1.856
WC04	12	14.85	0.188	1.856
WC05	10	13.58	0.156	1.698

Table 2. Wave parameters corresponding to the prototype and model.

Sea State	$H_{s}$ (m)	$T_{p}$ (s)	$H_{s}$ (m)	$T_{p}$ (s)
	Prototype		Model
WC01	6	14.85	0.094	1.856
WC02	8	14.85	0.125	1.856
WC03	10	14.85	0.156	1.856
WC04	12	14.85	0.188	1.856
WC05	10	13.58	0.156	1.698

Fig. 4. Distribution of wave probes in the basin during the experiment (in model scale).

Fig. 5. Wave elevation time series and probability density distribution for the sea state WC04 with

H_{s} = 12

m and

T_{p} = 14.85

Fig. 5. Wave elevation time series and probability density distribution for the sea state WC04 with $H_{s} = 12$ m and $T_{p} = 14.85$ s.

η

in the sub-time series, the timestamp refers to the moment in global time when wave elevation

η

occurs, and the category W relates to the sea state to which wave elevation

η

belongs.

Fig. 6. The wave elevation at each time point and the corresponding covariates information. Using a wave elevation sub-time series with 6 points as an example, the absolute position P corresponds to the local position of wave elevation $η$ in the sub-time series, the timestamp refers to the moment in global time when wave elevation $η$ occurs, and the category W relates to the sea state to which wave elevation $η$ belongs.

Table 3. Dimension of the covariates used in the Deep-WP model.

Covariate	Dimension
Absolute Position	20
Timestamp	3
Category	20

Fig. 7. The NDRMSE and training time for each mini-batch size

{64, 128, 256}

and learning rate

{0.001, 0.0005, 0.0001}

under different dimension of the hidden state for the sea state WC04.

Fig. 7. The NDRMSE and training time for each mini-batch size ${64, 128, 256}$ and learning rate ${0.001, 0.0005, 0.0001}$ under different dimension of the hidden state for the sea state WC04.

Fig. 8. Statistical boxplots of NDRMSE on the test dataset for sea states WC01-WC05 with different forecasting horizons

{6, 8, 12, 24, 48}

Fig. 8. Statistical boxplots of NDRMSE on the test dataset for sea states WC01-WC05 with different forecasting horizons ${6, 8, 12, 24, 48}$ .

Fig. 11. Predicted wave elevation and confidence interval for the sea state WC04 on the test data set with different forecasting horizons

{24, 48}

. The prediction at each point is the mean (

μ

), whereas the shaded areas (red, blue, and green) correspond to confidence intervals of standard deviations (1

σ

, 2

σ

, and 3

σ

). For each subplot title, the integer represents the identifier of sub-time series in the test set. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 11. Predicted wave elevation and confidence interval for the sea state WC04 on the test data set with different forecasting horizons ${24, 48}$ . The prediction at each point is the mean ( $μ$ ), whereas the shaded areas (red, blue, and green) correspond to confidence intervals of standard deviations (1 $σ$ , 2 $σ$ , and 3 $σ$ ). For each subplot title, the integer represents the identifier of sub-time series in the test set. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 4. Input data for the current model.

Dataset	Information
Elevation	elevation
Position	elevation + absolute position
Category	elevation + category
Timestamp	elevation + timestamp
Global	elevation + absolute position + category + timestamp

Table 5. The NDRMSE of the Deep-WP model under different covariates and prediction lengths for sea states WC01-WC05. Lower is better, the best-performing dataset is highlighted in bold. The count is the total number of prediction errors that contain covariates that are lower than or equal to the prediction errors using elevation alone for different prediction lengths.

Metric		NDRMSE
Length		6	8	12	24	48
WC01	Elevation	0.018	0.032	0.063	0.144	0.209
	Position	0.019	0.031	0.062	0.123	0.189
	Category	0.020	0.040	0.068	0.144	0.198
	Timestamp	0.019	0.028	0.067	0.139	0.199
	Global	0.020	0.033	0.064	0.134	0.190
WC02	Elevation	0.014	0.021	0.043	0.093	0.156
	Position	0.013	0.021	0.038	0.093	0.154
	Category	0.015	0.023	0.052	0.101	0.162
	Timestamp	0.013	0.022	0.045	0.101	0.161
	Global	0.016	0.024	0.044	0.088	0.155
WC03	Elevation	0.011	0.018	0.033	0.089	0.137
	Position	0.011	0.016	0.031	0.076	0.146
	Category	0.013	0.021	0.037	0.102	0.169
	Timestamp	0.010	0.016	0.034	0.074	0.153
	Global	0.013	0.019	0.032	0.081	0.140
WC04	Elevation	0.014	0.025	0.048	0.097	0.164
	Position	0.016	0.022	0.041	0.099	0.159
	Category	0.017	0.024	0.053	0.120	0.172
	Timestamp	0.014	0.024	0.044	0.102	0.169
	Global	0.015	0.021	0.043	0.098	0.169
WC05	Elevation	0.018	0.027	0.064	0.139	0.179
	Position	0.016	0.029	0.064	0.129	0.172
	Category	0.018	0.031	0.074	0.140	0.181
	Timestamp	0.017	0.030	0.068	0.141	0.175
	Global	0.019	0.031	0.063	0.131	0.171
Count		8	9	9	11	10

Fig. 12. Training time compared to NDRMSE on the test set for sea states WC01-WC05 under different covariates and prediction lengths, where Elevation is the baseline.

Fig. 13. The wave elevation time series generated by the different filter frequencies for sea state WC04. In order to display the differences in waves after filtering at different frequencies, only a portion of the wave time series (approximately 1000s) is depicted in the figure, and the time duration of the wave time series corresponding to each filtering frequency during training is 4096s.

Table 6. Remaining energy after wave filtration.

Frequency	Energy ratio (%)
1.5 $f_{p}$	76.055
2.0 $f_{p}$	89.324
2.5 $f_{p}$	94.642
3.0 $f_{p}$	97.059
4.5 $f_{p}$	99.419

Table 6. Remaining energy after wave filtration.

Frequency	Energy ratio (%)
1.5 $f_{p}$	76.055
2.0 $f_{p}$	89.324
2.5 $f_{p}$	94.642
3.0 $f_{p}$	97.059
4.5 $f_{p}$	99.419

Fig. 14. The Deep-WP model’s prediction error on the test set with different filtering frequencies

{1.5 f_{p}, 2.0 f_{p}, 2.5 f_{p}, 3.0 f_{p}, 4.5 f_{p}}

and different covariates for different prediction lengths

{6, 8, 12, 24, 48}

. The E, P, C, T, and G correspondingly represent Elevation, Position, Category, Timestamp, and Global.

Fig. 14. The Deep-WP model’s prediction error on the test set with different filtering frequencies ${1.5 f_{p}, 2.0 f_{p}, 2.5 f_{p}, 3.0 f_{p}, 4.5 f_{p}}$ and different covariates for different prediction lengths ${6, 8, 12, 24, 48}$ . The E, P, C, T, and G correspondingly represent Elevation, Position, Category, Timestamp, and Global.

References 35

[1]	G. Wu,Direct simulation and deterministic prediction of large-scale nonlinear ocean wave-field, Massachusetts Institute of Technology (2004) Ph.D. thesis
[2]	WAMDI. J Phys Oceanogr, 18 (12) (1988), pp. 1775-1810,
[3]	N. Booij, R.C. Ris, L.H. Holthuijsen, Journal of Geophysical Research: Oceans, 104 (C4) (1999), pp. 7649-7666
[4]	H.L. Tolman, et al.. Technical note, MMAB Contribution, 276 (220) (2009)
[5]	E.L. Morris, H.K. Zienkiewicz, M.R. Belmont. Int, Shipbuild. Prog., 45 (444) (1998), pp. 383-400
[6]	E. Blondel, F. Bonnefoy, P. Ferrant. Ocean Eng., 37 (10) (2010), pp. 913-926, .com/science/article/pii/S0029801810000661
[7]	E. Blondel-Couprie, F. Bonnefoy, P. Ferrant, et al.. Ocean Eng., 58 (2013), pp. 284-292, .com/science/article/pii/S0029801812003988
[8]	E. Blondel-Couprie, Reconstruction et prévision déterministe de houle à partir de données mesurées, Ecole Centrale de Nantes (ECN) (2009) .fr/tel-00449343
[9]	Y. Perignon, Modélisation déterministe des états de mer-Application à la rétrodiffusion d'ondes radar, Ecole Centrale de Nantes (ECN) (2011) .fr/tel-00624645
[10]	D.G. Dommermuth, D.K.P. Yue. J Fluid Mech, 184 (1987), pp. 267-288
[11]	Z. Dai, Z. Yang, Y. Yang, J. Carbonell, Q. Le, R. Salakhutdinov,Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, Association for Computational Linguistics, Florence, Italy (2019), pp. 2978-2988,
[12]	Z. Liu, Y. Lin, Y. Cao, H. Hu, Y. Wei, Z. Zhang, S. Lin, B. Guo,Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV) (2021), pp. 10012-10022
[13]	A. Sharma, D. Bhuriya, U. Singh,2017 International conference of Electronics, Communication and Aerospace Technology (ICECA), volume 2 (2017), pp. 506-509
[14]	I. Livieris, E. Pintelas, P. Pintelas, Neural Computing and Applications, 32 (2020),
[15]	P. Jain, M.C. Deo. Appl, Ocean Res., 29 (1) (2007), pp. 72-79, .com/science/article/pii/S0141118707000417
[16]	S. Asma, A. Sezer, O. Ozdemir, Computers and Geosciences, 49 (2012), pp. 231-237, .com/science/article/pii/S0098300412001951
[17]	J. BerbiÄ‡, E. Ocvirk, D. CareviÄ‡, G. LonÄar. Oceanologia, 59 (3) (2017), pp. 331-349, .com/science/article/pii/S0078323417300271
[18]	H. Kagemoto, Ocean Eng., 207 (2020), p. 107380 .com/science/article/pii/S0029801820304108
[19]	S.C. James, Y. Zhang, F. O'Donncha. Coastal Eng., 137 (2018), pp. 1-10, .com/science/article/pii/S0378383917304969
[20]	Y.Z. Law, H. Santo, K.Y. Lim, E.S. Chan,Ocean Eng., 195 (2020), p. .com/science/article/pii/S0029801819308339
[21]	W. Duan, X. Ma, L. Huang, Y. Liu,S. Duan. Comput Methods Appl Mech Eng, 372 (2020), p. .com/science/article/pii/S0045782520305351
[22]	X. Ma, L. Huang, W. Duan, Y. Jing,Q. Zheng. Ocean Eng., 239 (2021), p.
[23]	D. Salinas, V. Flunkert, J. Gasthaus, T. Januschowski. Int J Forecast, 36 (3) (2020), pp. 1181-1191,.com/science/article/pii/S0169207019301888
[24]	X. Guo, X. Zhang, X. Tian, W. Lu,X. Li. Ocean Eng., 247 (2022), p. .com/science/article/pii/S0029801822000506
[25]	J.L. Elman, Cogn Sci, 14 (2) (1990), pp. 179-211
[26]	S. Hochreiter, J. Schmidhuber. Neural Comput, 9 (8) (1997), pp. 1735-1780
[27]	K. Cho, B. van Merrienboer, C. Gulcehre, F. Bougares, H. Schwenk, Y. Bengio,Conference on Empirical Methods in Natural Language Processing (EMNLP 2014) (2014)
[28]	J. Koutnik, K. Greff, F. Gomez, J. Schmidhuber, E.P. Xing, T. Jebara,Proceedings of the 31st International Conference on Machine Learning, Proceedings of Machine Learning Research, volume 32, PMLR, Bejing, China (2014), pp. 1863-1871
[29]	M. Schuster, K.K. Paliwal, IEEE Trans. Signal Process., 45 (11) (1997), pp. 2673-2681,
[30]	O. Sprangers, S. Schelter, M. de Rijke, Int J Forecast (2022), .com/science/article/pii/S0169207021001850
[31]	H. Hewamalage, C. Bergmeir, K. Bandara. Int J Forecast, 37 (1) (2021), pp. 388-427, DOI:
	.com/science/article/pii/S0169207020300996
[32]	D.P. Kingma, J. Ba,International Conference on Learning Representations (ICLR) (2015)
[33]	F.R. Ramos, Data Science na ModelaÃ§Ã£o e PrevisÃ£o de SÃ©ries EconÃ³mico-financeiras: das Metodologias Clssicas ao Deep Learning (2021) Ph.D. thesis
[34]	H. Zhou, S. Zhang, J. Peng, S. Zhang, J. Li, H. Xiong, W. Zhang, The Thirty-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Virtual Conference, volume 35, AAAI Press (2021), pp. 11106-11115
[35]	S. Yang, Z. Deng, X. Li, C. Zheng, L. Xi, J. Zhuang, Z. Zhang, Z. Zhang. Renew Energy, 173 (2021), pp. 531-543,

Please choose a citation manager

Content to export