Robust Prediction of Internal Wave-Affected Multi-Scale Sound Speed Distribution Using Lightweight Kolmogorov-Arnold Networks with Hybrid Basis Functions

Wei Huang; Junpeng Lu; Tianhe Xu; Hao Zhang; Feng Yin

eess.SPOct 7, 2026

Robust Prediction of Internal Wave-Affected Multi-Scale Sound Speed Distribution Using Lightweight Kolmogorov-Arnold Networks with Hybrid Basis Functions

Authors: Wei Huang, Junpeng Lu, Tianhe Xu, Hao Zhang, Feng Yin

Organizations: Faculty of Information Science and Engineering, Ocean University of China, Qingdao, Shandong 266404, China · School of Space Science and Technology, Shandong University at Weihai, Weihai, Shandong 264200, China · School of Artificial Intelligence, The Chinese University of Hong Kong (Shenzhen), Shenzhen, Guangdong 518172, China

Abstract

The underwater sound speed distribution directly governs acoustic propagation paths, rendering it critically important for underwater acoustic communication and target localization. Conventional sound speed profile (SSP) prediction methods provide a good way to estimate the underwater sound speed distribution without on-site data measurement, thus breaking through the coverage area constraints of sonar observation equipment and making the model universal in most marine areas. However, underwater sound speed exhibits multi-scale variations, such as diurnal, quarterly, and intermittent fluctuations caused by ocean processes such as internal waves. This makes it difficult for the fixed structure models in existing methods to have good generalization ability for multi-scale sound speed distribution patterns. To tackle this problem, we proposed a lightweight hybrid basis-function empowered Kolmogorov-Arnold network (LHBF-KAN) model for multi-scale sound speed prediction. We aim to construct a multi-branch representation layer in which different basis functions respond to distinct temporal patterns, from slowly varying background trends to rapid fluctuations induced by dynamic ocean processes, allowing the model to naturally accommodate the inherently multi-scale evolution of sound speed at different depths. To prevent the multi-branch structure from increasing model size, a pruning strategy is further introduced to suppress branches with consistently low contribution during training, yielding a compact architecture, suitable for deployment on resource constrained underwater platforms.

Figures & tables

Fig. 2: Heat map of SW06 temperature data.

Dataset	Sampling interval	Main variation characteristic	Evaluation objective
Argo	One month	Seasonal and slowly varying evolution	Long-term trend forecasting
SW06	30 s	Internal-wave-induced rapid fluctuations	Fast-varying robustness
Ocean Experiment	Approximately 2 h	Real observations with limited samples	Few-shot field-data forecasting

TABLE I: Summary of the multi-scale SSP forecasting scenarios.

Basics	Parameters	Value
1	Training epochs	100
2	Backtracking window	12
3	Hidden layer	1
4	Hidden layer neurons	256
5	Learning rate	0.0001
B-spline	Parameters	Value

TABLE II: Parameter Setting of LHBF-KAN

Dataset	Model	RMSE (m/s)	MAE (m/s)	$R^{2}$
Argo	LHBF-KAN	0.728	0.516	0.9989
	Bi-LSTM	1.163	0.790	0.9974
	Transformer	0.843	0.571	0.9986
	PatchTST	0.763	0.535	0.9988
SW06	LHBF-KAN	0.397	0.199	0.9990
	Bi-LSTM	0.953	0.511	0.9944

TABLE III: Overall forecasting performance across the three multi-scale datasets. The best result is shown in bold and the second-best result is underlined.

Panel of Figure 7 — Fig. 4: Comparison figure between actual and predicted (Argo dataset). (a) May. (b) August.

Panel of Figure 8 — Fig. 5: Error distributions (Argo dataset). (a) May. (b) August.

Panel of Figure 9 — Fig. 6: Error heatmap (Argo dataset).

Fig. 7: Comparison figure between actual and predicted (SW06 dataset).

Fig. 8: Error distributions (SW06 dataset).

Panel of Figure 12 — Fig. 9: Error heatmap (SW06 dataset).

Fig. 10: Collected data (Ocean Experiment).

Panel of Figure 14 — Fig. 11: Comparison figure between actual and predicted (Ocean Experiment). (a) SSP1. (b) SSP2.

Panel of Figure 15 — Fig. 12: Error distributions (Ocean Experiment). (a) SSP1. (b) SSP2.

Month	RMSE (m/s)
	B-spline	Jacobi	Taylor	Wavelet	LHBF-KAN	Best Model
1	1.629	0.631	0.788	1.416	0.743	Jacobi
2	1.306	0.470	0.839	1.099	0.544	Jacobi
3	0.828	0.740	0.698	0.707	0.648	LHBF-KAN
4	0.690	1.067	0.974	0.799	0.984	B-spline
5	0.404	0.503	0.445	0.548	0.468	B-spline

TABLE IV: Ablation study of different basis functions on the Argo dataset. where the best result is shown in bold.

RMSE ( $m/s$ )				MAE ( $m/s$ )	$R^{2}$
Model	Max	Min	Average
LHBF-KAN	0.880	0.159	0.397	0.199	0.9990
B-spline	0.936	0.153	0.472	0.229	0.9986
Jacobi	0.802	0.134	0.412	0.207	0.9989
Taylor	0.879	0.126	0.381	0.190	0.9990
Wavelet	0.853	0.139	0.395	0.204	0.9990

TABLE V: RMSE OF SSP Prediction by Different Basis Function(SW06 dataset)

RMSE ( $m/s$ )				MAE ( $m/s$ )	$R^{2}$
Model	SSP1	SSP2	Average
LHBF-KAN	0.200	0.168	0.184	0.112	0.9997
B-spline	0.264	0.109	0.186	0.115	0.9997
Jacobi	0.278	0.217	0.247	0.145	0.9996
Taylor	0.308	0.177	0.242	0.147	0.9996
Wavelet	0.260	0.142	0.201	0.120	0.9997

TABLE VI: RMSE OF SSP Prediction by Different Basis Function(Ocean Experiment)

Panel of Figure 19 — Fig. 13: Comparison figure between actual and predicted of different basis functions (Argo dataset). (a) January. (b) February.

Panel of Figure 20 — Fig. 14: Error distributions of different basis functions (Argo dataset). (a) January. (b) February.

Panel of Figure 21 — Fig. 15: Error heatmap of different basis functions (Argo dataset).

Fig. 16: Comparison figure between actual and predicted of different basis functions (SW06 dataset).

Fig. 17: Error distributions of different basis functions (SW06 dataset).

Panel of Figure 24 — Fig. 18: Error heatmap of different basis functions (SW06 dataset).

Panel of Figure 25 — Fig. 19: Comparison figure between actual and predicted of different basis functions (Ocean Experiment). (a) SSP1. (b) SSP2.

Panel of Figure 26 — Fig. 20: Error distributions of different basis functions (Ocean Experiment). (a) SSP1. (b) SSP2.

Panel of Figure 27 — Fig. 21: Analysis of frequency components of the dataset. (a) Argo. (b) SW06. (c) Ocean Experiment.

Panel of Figure 28 — Fig. 22: Examples of weight evolution visualization. (a-c) Argo. (d-f) SW06. (g-h) Ocean Experiment.

Model		Ours	Bi-LSTM	Transformer	PatchTST
Time(s)	Argo	54.0	77.9	62.2	230.3
	SW06	409.3	740.2	504.2	1134.1
	Ocean Experiment	4.5	8.2	6.9	11.7
Trainable Parameters		6657	5325441	3945985	102019

TABLE VII: Time Complexity and Trainable Parameters of the Models

Dataset	Parameters before	Parameters after	Reduction	RMSE change
Argo	6657	6657	0.0%	$\Delta_{\mathrm{Argo}}$
SW06	6657	2439	63.4%	$\Delta_{\mathrm{SW06}}$
Ocean Experiment	6657	2219	66.7%	$\Delta_{\mathrm{OE}}$

TABLE VIII: Effect of pruning on model complexity and forecasting accuracy.

Fig. 23: Scenario of localization experiments. (Argo dataset).

Model	Ours	BiLSTM	PatchTST	Transformer
SSP Prediction errors (m/s)	0.539	1.017	0.914	0.756
Positioning errors (m)	0.2001	0.7566	0.7850	0.5102

TABLE IX: Average positioning error with the predicted sound speed obtained by different models (Argo dataset).

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Abstract

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