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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

arXiv:2610.10877v1eess.SPcs.SD

TL;DR

SSP prediction must capture variations across temporal scales and depths, while existing models can be limited to single-region or single-scale settings. LHBF-KAN combines multiple basis families and prunes low-contribution branches; the paper reports improved forecasting across datasets and reduced parameter counts on two datasets.

  • Problem

    Many sound-speed prediction models focus on single-region tasks and a single scale of change, limiting their ability to predict multi-scale SSPs.

    Source: C. Temporal Prediction Methods · p. 7

  • Method

    LHBF-KAN combines basis functions with different scale sensitivities in a unified representation and prunes branches with persistently low impact.

    Source: §I · p. 4

  • Results

    LHBF-KAN achieves the best aggregate forecasting performance across monthly, 30-s, and approximately two-hourly observations, and pruning reduces parameters by more than 60% on SW06 and ocean-experiment datasets.

    Source: G. Summary of Experimental Findings · pp. 37–38

  • Takeaways & Limitations

    On the evaluated datasets, improved SSP forecasts are associated with lower localization errors.

    Source: G. Summary of Experimental Findings · p. 38

Abstract

from arXiv · show

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.

I. INTRODUCTION

Underwater sound-speed prediction must handle variability across temporal scales and depths, yet existing approaches have limited generalization under dynamic ocean conditions. The paper introduces LHBF-KAN, combining multiple basis families with pruning, and reports improvements across several datasets.

I. INTRODUCTION

  • Instrument-based SSP measurement and inversion approaches rely on on-site operations or acoustic-field observations, constraining their applicability across randomly given sea areas.Source: §I · p. 2

II. RELATED WORK

  • SST-based CNN and SOM methods have limited deep information, while temporal prediction methods remain region-specific and generalize insufficiently under complex ocean dynamics.Source: §I · pp. 2–3

I. INTRODUCTION

  • Seasonal, diurnal, and abrupt internal-wave-driven variations coexist, while fixed homogeneous or single-basis representations can underfit or overfit different water-column layers.Source: §I · p. 3
  • LHBF-KAN combines basis families responsive to different temporal scales and depths, then prunes persistently low-impact branches to keep the model compact for constrained platforms.Source: §I · p. 4
  • The paper analyzes time-domain sequence fitting and frequency-domain information fitting, and proves a frequency-domain approximation theorem to guide model design.Source: §I · p. 4
  • Experiments on Argo, South China Sea daily observations, and internal-wave-affected ocean measurements show consistent improvements over state-of-the-art SSP prediction models.Source: §I · p. 4

A. SSP Inversion Methods

  • SSP inversion research spans computationally complex MFP, CS methods with accuracy constrained by first-order Taylor approximation, and neural approaches including noise-aware features and few-shot meta-learning.Source: A. SSP Inversion Methods · p. 5

B. Spatial Construction Methods

The section reviews spatial and temporal SSP construction and prediction methods, then introduces LHBF-KAN’s evaluation data and preprocessing. Existing methods face challenges with nonlinear marine variability and multi-scale prediction.

B. Spatial Construction Methods

  • IDW is computationally efficient but cannot capture nonlinear marine variability; CNNs and SOMs model spatial patterns, with stated depth and robustness constraints.Wu et al. fuse SST, location, and historical SSP components; Xu et al. use SOMs, whose performance depends on initialization and environmental conditions.Source: B. Spatial Construction Methods · pp. 5–6

C. Temporal Prediction Methods

  • Historical SSP evolution supports forecasts without on-site measurements, while prior approaches include RBF prediction, LSTM, and convolutional bidirectional LSTM with attention.Yu et al. predict temperature and salinity before deriving sound speed, but the method misses fine-scale SSP variation and tends toward average values; later approaches include Lu et al. and Liu et al. and Zhang et al..Source: C. Temporal Prediction Methods · pp. 6–7
  • Existing prediction models largely target single-region, single-scale changes, motivating a framework that adapts to diverse temporal behaviors across depths and sparse historical observations.The proposed LHBF-KAN is intended to represent diverse temporal patterns within a unified learning framework.Source: C. Temporal Prediction Methods · p. 7, §III · p. 7

III. DATA AND METHOD

  • LHBF-KAN is evaluated using Argo, SW06, and an hourly South China Sea dataset to assess generalization across datasets.The framework is designed to adapt to different temporal behaviors across depths while maintaining robustness where historical observations are sparse.Source: A. Data Source · p. 7, §III · p. 7

A. Data Source

  • Argo provides monthly gridded sound-speed data, SW06 supplies 30-second temperature samples converted to sound speed, and South China Sea profiles were collected about every two hours.The study uses Argo data at (16.5°N, 168.5°E), SW06 data from seven depths interpolated to 1-meter intervals, and CTD/XCTD profiles at (17.3°N, 116.2°E).Source: A. Data Source · pp. 7–8

B. Data Preprocessing

  • Datasets were split 8:2 and standardized separately by depth using only training-set means and standard deviations; evaluation uses MSE, RMSE, MAE, and R2.The training objective is MSE, and Z denotes maximum depth while s̄ denotes the mean of true values.Source: B. Data Preprocessing · pp. 9–10

C. Frequency-Domain Motivation

The section motivates frequency-domain design by relating temporal-signal reconstruction to spectral approximation. LHBF-KAN combines basis-function mixing with fully connected and output layers, using residual connections for training stability.

C. Frequency-Domain Motivation

  • For finite-energy signals, minimizing frequency-domain approximation error is equivalent to minimizing time-domain reconstruction error, motivating spectral coverage in the model design.The section states that sound-speed variation is concentrated mainly in medium and low frequency bands, and that the architecture is designed to cover the potential frequency range of these changes.Source: C. Frequency-Domain Motivation · p. 10

D. Structure of LHBF-KAN

  • LHBF-KAN comprises input, basis-function mixing, fully connected, and output layers, with residual connections intended to improve training stability.The architecture is proposed to improve generalization for multi-scale changes.Source: D. Structure of LHBF-KAN · p. 11, Figure 3 · p. 11

1) Input layer:

The input layer segments sound-speed data into shorter sequences sized to the dataset’s temporal scale; the basis-function mixture combines four basis families with complementary frequency responses.

1) Input layer:

  • Input data are segmented into smaller sequences, with sample size chosen to match the dataset’s inherent temporal scale.For the annually periodic Argo dataset, the segmented sample size is 12 months.Source: 1) Input layer · p. 12

2) Basis function mixture layer:

  • The mixture layer combines B-splines, Jacobi polynomials, Shannon wavelets, and Taylor series to represent sound-speed variation across scales.The paper describes their combined frequency coverage as extending from DC to high-frequency bands.Source: 2) Basis function mixture layer · pp. 12, 14, 15
  • Shannon wavelets provide localized band-pass representations suited to medium- and high-frequency variations.Their frequency-domain response is nonzero only within a finite interval.Source: 2) Basis function mixture layer · p. 13
  • Taylor bases mainly capture DC and ultra-low-frequency components, while Jacobi polynomials represent smooth low-frequency structures.The paper attributes Jacobi bases’ low-frequency suitability to polynomial spectral decay.Source: 2) Basis function mixture layer · pp. 13–14
  • The design principle is that combining complementary basis responses enlarges frequency coverage and can improve approximation of multi-scale temporal signals.B-splines mainly respond to low frequencies and are suited to slowly varying components.Source: 2) Basis function mixture layer · pp. 12, 15

3) Fully connected layer:

The third layer is fully connected and restores data from a high-dimensional space to a lower-dimensional space.

3) Fully connected layer:

  • The fully connected third layer restores data from a high-dimensional space to a lower-dimensional space.The supplied description does not specify further layer operations.Source: 3) Fully connected layer · p. 17

4) Pruning method:

The pruning strategy uses temperature annealing to suppress low-contribution basis-function branches and control model parameter scale for underwater deployment.

4) Pruning method:

  • Temperature-annealing-based pruning is used in the basis-function mixture layer to control model parameter scale for underwater-platform deployment.The procedure begins by pretraining basis-function parameters while freezing the branch weights.Source: 4) Pruning method · p. 17
  • Branch importance is estimated from per-branch losses against ground-truth labels using an exponential moving average, with smaller loss indicating greater importance.Importance scores dynamically adjust branch coefficients during soft pruning.Source: 4) Pruning method · p. 18
  • Branches whose coefficients remain below a preset threshold for several rounds are zeroed, and their weights are reallocated among remaining branches.If soft pruning does not select a clear best branch, hard pruning may retain only the branch with the largest coefficient.Source: 4) Pruning method · p. 18

A. Experimental Protocol

The evaluation spans slow, fast, and intermediate sound-speed-profile (SSP) variation, comparing LHBF-KAN with three forecasting baselines. LHBF-KAN achieves the lowest overall RMSE across all three datasets.

1) Datasets and Multi-Scale Evaluation Scenarios:

2) Baselines and Implementation Details:

  • LHBF-KAN was evaluated against Bi-LSTM, PatchTST, and Transformer using RMSE, MAE, and R2; training used back-propagation and Adam.The experiments were implemented on an NVIDIA RTX 5090 GPU.Source: 2) Baselines and Implementation Details · p. 19

B. Prediction Accuracy Across Temporal Scales

C. Robustness to Depth-Dependent and Abrupt Variations

LHBF-KAN maintains stronger, more stable prediction performance under rapid internal-wave-related fluctuations and across depth. Its errors are concentrated more in variable shallow layers than in stable deeper layers.

1) Depth-Wise Error Characteristics:

2) Robustness to Internal-Wave-Induced Variations:

  • On 30-second SW06 data, LHBF-KAN most closely matches the original sound-speed profile through abrupt depth changes, with smaller, less variable errors and an advantage over other models in extreme-error magnitude and distribution shape.Source: 2) Robustness to Internal-Wave-Induced… · pp. 22, 25

3) Few-Shot Performance in the Ocean Experiment:

In the field experiment, LHBF-KAN performs well with only 12 training profiles. The basis-function ablation shows that no single basis is consistently best across datasets, whereas the hybrid model remains close to or better than the dataset-specific best.

3) Few-Shot Performance in the Ocean Experiment:

1) Single-Basis Ablation:

  • Jacobi leads on Argo, Taylor on SW06, and B-spline is competitive in the ocean experiment; single-basis performance degrades across differing spectral characteristics.The ablation tests whether one basis remains reliable as the dominant temporal scale changes.Source: 1) Single-Basis Ablation · p. 27
  • LHBF-KAN is within 0.014 m/s of the best basis on Argo and 0.016 m/s on SW06, and achieves the best result on the ocean experiment.Its cross-dataset performance approaches dataset-specific basis selection without requiring that prior selection.Source: 1) Single-Basis Ablation · pp. 28–29

2) Frequency-Domain Interpretation:

The datasets have distinct spectral profiles, matching the observed differences among single-basis results. LHBF-KAN learns branch contributions during training, suppressing less useful branches where possible while retaining a compact representation.

2) Frequency-Domain Interpretation:

  • Argo is dominated by slow components, SW06 has stronger internal-wave-driven short-term fluctuations, and the ocean experiment mixes low, medium, and relatively high frequencies.The spectral analysis is used to interpret ablation results, not to assign each dataset exclusively to one frequency band.Source: 2) Frequency-Domain Interpretation · p. 33, Figure 21 · p. 34

3) Branch Selection and Weight Evolution:

  • Training weights diverge as branches matching the data gain weight; Argo retains several active branches, while SW06 and the ocean experiment suppress branches and yield compact models.The proposed pruning strategy selects branches according to dataset spectral characteristics rather than removing parameters arbitrarily.Source: Figure 22 · p. 34, 3) Branch Selection and Weight Evolution · pp. 34–35

E. Model Complexity and Pruning Effectiveness

LHBF-KAN combines multi-scale forecasting performance with substantial parameter reductions, and the paper evaluates its predicted sound speeds in a localization task. The reported findings span monthly, 30-s, and approximately two-hourly observations, with gains most pronounced on internal-wave-affected SW06.

E. Model Complexity and Pruning Effectiveness

  • Pruning reduces LHBF-KAN from 6,657 parameters to 2,439 on SW06 and 2,219 on the ocean-experiment dataset, with RMSE changing only modestly.These reductions are approximately 63.4% and 66.7%; before pruning, the parameter count is about three orders of magnitude below recurrent and attention-based baselines.Source: Table VIII · p. 36, E. Model Complexity and Pruning… · p. 36

F. Impact on Underwater Localization

  • Localization evaluation assumes ranging error comes only from sound-speed differences and reports an overall trend of smaller positioning errors when sound-speed errors are smaller.The positioning errors are calculated using ray tracing with predicted sound-speed distributions to correct simulated signal-propagation times.Source: F. Impact on Underwater Localization · p. 37, Table IX · p. 37

G. Summary of Experimental Findings

  • LHBF-KAN achieves the best aggregate forecasting performance across monthly, 30-s, and approximately two-hourly SSP observations, with the strongest improvement on internal-wave-affected SW06.The reported error heatmaps show reductions in both the magnitude and spatial extent of large prediction errors.Source: G. Summary of Experimental Findings · pp. 37–38
  • Jacobi, Taylor, B-spline, and Wavelet perform best on different datasets, while LHBF-KAN stays near the dataset-specific optimum without prior basis selection.No branch is removed on Argo when learned branch weights remain distributed across basis functions, which the paper interprets as complementary contributions being preserved.Source: E. Model Complexity and Pruning… · p. 36, G. Summary of Experimental Findings · p. 38

V. CONCLUSION

  • Across Argo, SW06, and real ocean-experiment datasets, LHBF-KAN improves over Transformer, BiLSTM, and PatchTST while maintaining low computational cost.The conclusion presents hybrid basis-function modeling as a potential practical solution for future underwater sensing and prediction systems.Source: §V · p. 38
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