A Sound Velocity Field Reconstruction Method Based on Fully Connected Tensor Network Decomposition and Regularization

Abstract

This invention discloses a sound velocity field reconstruction method based on fully connected tensor network decomposition and regularization, belonging to the field of marine three-dimensional sound velocity field reconstruction technology. It is used for marine three-dimensional sound velocity field reconstruction, including obtaining sparse three-dimensional sound velocity field observations, inputting them into FCTN-T for processing, and obtaining the reconstructed marine three-dimensional sound velocity field. An adaptive rank-increasing strategy is added to FCTN-T, including setting the initial value of the rank of the FCTN model, calculating the maximum rank of the FCTN model, iteratively updating parameters, and obtaining the reconstructed marine three-dimensional sound velocity field after reaching the convergence condition, greatly accelerating the reconstruction efficiency. This invention fully utilizes the spatial correlation and local smoothness characteristics of the three-dimensional sound velocity field, achieving the best reconstruction accuracy in both overall and small-scale details, especially in sparsely sampled regions. The rank-increasing strategy also significantly improves the model's running efficiency, thus simultaneously balancing reconstruction accuracy and efficiency.

Description

Technical Field

[0001] This invention discloses a sound velocity field reconstruction method based on fully connected tensor network decomposition and regularization, belonging to the field of marine three-dimensional sound velocity field reconstruction technology. Background Technology

[0002] Three-dimensional ocean sound velocity fields characterize the propagation characteristics of sound waves in the marine environment and are a key element of ocean observation, crucial for acoustic applications such as underwater positioning, navigation, timing, and communication. Since the vertical variation of sound velocity within the same spatial scale is much greater than its horizontal variation, sound velocity profiles are commonly used to describe the sound velocity field distribution in a region. However, acquiring sound velocity data from the vast ocean remains a significant challenge: limited human and material resources, and the difficulty in achieving sufficiently accurate spatiotemporal sampling, prevent the acquisition of all sound velocity values ​​within a region through sonar observations. Therefore, measured data are often spatially sparse, containing only isolated profiles, far from meeting the needs of complex underwater operational systems. To obtain large-scale, high-resolution three-dimensional sound velocity fields, it is urgent to develop methods capable of reconstructing regional three-dimensional sound velocity fields from sparse sound velocity profiles.

[0003] Reconstructing the three-dimensional sound velocity field of the ocean from sparse sound velocity profiles inherently involves inferring missing values, which constitutes an ill-conditioned, high-dimensional inverse problem. To address this challenge, a common practice is to incorporate prior information into the reconstruction process. The three-dimensional sound velocity field exhibits strong spatial correlation because the fundamental physical processes controlling sound velocity variations are spatially continuous. This property can be mathematically described as a low-rank structure. Furthermore, since ocean parameters such as temperature, salinity, and pressure typically vary slowly on local spatial scales, the sound velocity field also exhibits local smoothness, with small differences between adjacent locations. These intrinsic properties allow for accurate representation of the three-dimensional sound velocity field using relatively few parameters, thus facilitating efficient and reliable reconstruction from limited observational data.

[0004] Currently, numerous studies address this problem using several main approaches. One approach utilizes feature learning methods. Among these, the Empirical Orthogonal Function (EOF) is a classic and widely used method for ocean sound velocity field modeling. As an effective basis function method, EOF accurately characterizes the three-dimensional sound velocity structure by capturing the main spatial patterns of the sound velocity field from sparse observation data. With only a small number of EOF patterns and a suitable optimization algorithm, it can efficiently reconstruct the sound velocity field within a given region. Another approach utilizes interpolation. The basic idea is to assume that the sound velocity at nearby points in space is smooth, meaning that physically close locations have similar sound velocity values. Therefore, the function values ​​at unknown points are estimated based on the function values ​​at known points. A common method is Inverse Distance Weighted Interpolation (IDW), which infers the sound velocity value at an unknown point based on the distance from nearby known points to the unknown point. The influence of these known points weakens with increasing distance to the reconstruction location, thus achieving the reconstruction of the three-dimensional SSF.

[0005] However, a key limitation of the two methods mentioned above is their reliance on the matrix expansion of the three-dimensional sound velocity field, which ignores the multidimensional correlations between sound velocities. Specifically, two points that were originally closely adjacent along the y-axis become significantly separated after expansion. This operation fails to capture the horizontal variability of sound velocity, thus destroying the spatial information of the three-dimensional sound velocity field. Therefore, the inherent three-dimensional structure of the sound velocity field is not preserved, leading to data loss and degraded representation performance.

[0006] Third, deep learning methods are utilized. The speed of sound in the marine environment typically exhibits complex nonlinear variations due to various environmental factors. Deep learning technology, with its superior ability to model nonlinear relationships, offers significant advantages in constructing three-dimensional sound speed fields. However, deep learning methods are inherently data-driven, requiring large amounts of training data to ensure satisfactory performance, and their reconstruction efficiency is low.

[0007] Finally, the idea of ​​tensor decomposition is used to reconstruct the sound velocity field. Tensor decomposition aims to decompose a high-order objective tensor into a set of factor tensors by defining different factor structures and related multilinear operations. This method can extract the dominant physical features from high-dimensional data using relatively few parameters, thereby promoting accurate data reconstruction. Therefore, tensor decomposition provides a powerful means to capture the inherent spatial characteristics of high-dimensional data, effectively represent the complex three-dimensional variations of sound velocity, and provide a unique perspective for SSF (sound velocity field) construction. However, previous tensor decomposition methods are still insufficient in fully exploiting the inherent global correlations of high-dimensional data, leading to a significant loss of accuracy.

[0008] The Fully Connected Tensor Network (FCTN) decomposition framework explicitly models the interconnections between any pair of latent factors, effectively addressing the aforementioned problems. Unlike traditional methods, FCTN decomposes the N-order target tensor into a set of interconnected N-order factor tensors, with each pair of factors linked by explicit edges. A key advantage of FCTN is its ability to overcome the structural constraints of traditional tensor decomposition, enabling global modeling of correlations across all modes, thus excelling at capturing the latent structure of high-order data. This has led to its outstanding performance in tensor completion tasks such as high-dimensional image restoration. Since the ocean's three-dimensional sound velocity field (SSF) also exhibits a globally low-rank structure similar to that of images, FCTN decomposition is introduced into SSF reconstruction to characterize the global correlations of the sound velocity field. However, relying solely on global correlations limits the model's ability to represent highly nonlinear variations caused by small-scale ocean processes such as internal waves and eddies. Therefore, there is still room for improvement in reconstructing local structural details. The ocean's three-dimensional sound velocity field also possesses local continuity, and factor regularization has proven to play a crucial role in recovering local details. Therefore, this invention introduces factor-based Tikhonov regularization into the FCTN decomposition framework to construct the FCTN-T model. This model, while fully capturing the spatial correlation of the three-dimensional sound velocity field, further characterizes the detailed changes in the sound velocity field, and can construct a complete three-dimensional sound velocity field from sparse sound velocity profiles. Summary of the Invention

[0009] The purpose of this invention is to provide a sound velocity field reconstruction method based on fully connected tensor network decomposition and regularization, so as to solve the problem of insufficient accuracy in the reconstruction of three-dimensional sound velocity fields in the ocean in the prior art.

[0010] A sound velocity field reconstruction method based on fully connected tensor network decomposition and regularization includes obtaining sparse three-dimensional sound velocity field observations. The input is processed by the FCTN-T ocean three-dimensional sound velocity field model based on tensor decomposition and factor regularization to obtain the reconstructed ocean three-dimensional sound velocity field.

[0011] FCTN-T includes:

[0012] ;

[0013] ;

[0014] ;

[0015] ;

[0016] In the formula, It is the Nth-order tensor composed of the three-dimensional sound velocity field of the ocean, given We obtain the initial value and then iterate to solve for it. It is the FCTN factor tensor All possible values, including , , , It is a matrix factorization factor All possible values, including , , , It is a matrix factorization factor All possible values, including , , , This is FCTN model processing, subscript yes Norm, It is a regularization parameter. It is a first-order difference matrix. It is a factor tensor model - Expand the matrix. It is a projection operator. It is an index set of known sound speed locations; the projection operator will... lie in The element at a given position projects onto itself, located at... The mapping for elements outside the specified positions is 0.

[0017] FCTN includes:

[0018] ;

[0019] ;

[0020] In the formula, It is the first Physical indexes in each dimension It is the rank of the FCTN model. yes and The join index between them, the maximum value of the join index is , middle The maximum value is .

[0021] An adaptive rank-increasing strategy is incorporated into FCTN-T, including:

[0022] S1. Set the initial value of the rank of the FCTN model;

[0023] S2. Set the maximum rank of the FCTN model;

[0024] S3, Iterative Update , , and ;

[0025] S4. After the convergence condition is met, the reconstructed three-dimensional sound velocity field of the ocean is obtained.

[0026] The initial rank of the FCTN model is set to... .

[0027] The maximum rank of the FCTN model is:

[0028] ;

[0029] In the formula, It is the maximum rank of the FCTN model. , and This corresponds to the three ranks in the three-dimensional sound velocity field.

[0030] Iterative update , , and If the following conditions are met:

[0031] ;

[0032] ;

[0033] ;

[0034] but , Continue iterating;

[0035] In the formula, It is the first The next iteration , It is the convergence threshold. They are two different random numbers that are positive integers.

[0036] The convergence condition is:

[0037] .

[0038] Compared with existing technologies, the present invention has the following advantages: it makes full use of the spatial correlation and local smoothing characteristics of the three-dimensional sound velocity field, and achieves the best reconstruction accuracy in both overall and small-scale details, especially in sparsely sampled regions; the rank-increasing strategy also greatly improves the model's running efficiency, that is, it takes into account both reconstruction accuracy and reconstruction efficiency. Attached Figure Description

[0039] Figure 1 It is the third-order representation of FCTN decomposition.

[0040] Figure 2 This is the overall reconstruction process of the present invention.

[0041] Figure 3 It represents all the sound velocity profiles in the experimental area.

[0042] Figure 4 It studies how the standard deviation of sound velocity varies with depth at each depth level in the research area.

[0043] Figure 5 These are the reconstructed longitudes of six models under different missing rates.

[0044] Figure 6 It shows the reconstruction time of six models under different missing rates in a single trial. Detailed Implementation

[0045] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention are described clearly and completely below. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0046] A sound velocity field reconstruction method based on fully connected tensor network decomposition and regularization includes obtaining sparse three-dimensional sound velocity field observations. The input is processed by the FCTN-T ocean three-dimensional sound velocity field model based on tensor decomposition and factor regularization to obtain the reconstructed ocean three-dimensional sound velocity field.

[0047] FCTN-T includes:

[0048] ;

[0049] ;

[0050] ;

[0051] ;

[0052] In the formula, It is the Nth-order tensor composed of the three-dimensional sound velocity field of the ocean, given We obtain the initial value and then iterate to solve for it. It is the FCTN factor tensor All possible values, including , , , It is a matrix factorization factor All possible values, including , , , It is a matrix factorization factor All possible values, including , , , This is FCTN model processing, subscript yes Norm, It is a regularization parameter. It is a first-order difference matrix. It is a factor tensor model - Expand the matrix. It is a projection operator. It is an index set of known sound speed locations; the projection operator will... lie in The element at a given position projects onto itself, located at... The mapping for elements outside the specified positions is 0.

[0053] FCTN includes:

[0054] ;

[0055] ;

[0056] In the formula, It is the first Physical indexes in each dimension It is the rank of the FCTN model. yes and The join index between them, the maximum value of the join index is , middle The maximum value is .

[0057] An adaptive rank-increasing strategy is incorporated into FCTN-T, including:

[0058] S1. Set the initial value of the rank of the FCTN model;

[0059] S2. Set the maximum rank of the FCTN model;

[0060] S3, Iterative Update , , and ;

[0061] S4. After the convergence condition is met, the reconstructed three-dimensional sound velocity field of the ocean is obtained.

[0062] The initial rank of the FCTN model is set to... .

[0063] The maximum rank of the FCTN model is:

[0064] ;

[0065] In the formula, It is the maximum rank of the FCTN model. , and This corresponds to the three ranks in the three-dimensional sound velocity field.

[0066] Iterative update , , and If the following conditions are met:

[0067] ;

[0068] ;

[0069] ;

[0070] but , Continue iterating;

[0071] In the formula, It is the first The next iteration , It is the convergence threshold. They are two different random numbers that are positive integers.

[0072] The convergence condition is:

[0073] .

[0074] This invention ensures the global low-rank structure of the sound velocity field through FCTN decomposition. The following section introduces how Tikhonov regularization enhances the local continuity of the sound velocity field.

[0075] First, according to We can obtain:

[0076] .

[0077] This means minimizing and A smaller one can be obtained .

[0078] because It is a difference matrix, therefore Local continuity within each column can be improved.

[0079] With all A linear combination of columns can be expressed as a matrix. Each column, i.e. Each column can be represented as A linear combination of all columns. Therefore, it promotes Local continuity within each column can be enhanced. The local continuity within each column, that is... The Local continuity in each dimension.

[0080] The solution process for FCTN-T in this invention is as follows, introducing the following indicator functions. :

[0081] ;

[0082] ;

[0083] In the formula, It includes all of them. Above and A consistent set of tensors;

[0084] Rewrite FCTN-T as an unconstrained problem as follows:

[0085] ;

[0086] In the formula, It is a penalty parameter;

[0087] The unconstrained problem is solved by updating within the PAM framework using the following iterative scheme:

[0088] ;

[0089] ;

[0090] ;

[0091] ;

[0092] In the formula, It is a neighboring parameter;

[0093] In one experiment, a three-dimensional SSF of size 20×20×20 was considered, containing a total of 8000 parameters. For simplicity, the rank of the FCTN is usually set to be equal, indicating... If the selected FCTN rank is too low (e.g., The model will only need Using a few parameters to represent the original 8000 parameters leads to reduced computational cost, but at the cost of underfitting and failing to capture the complex structure within the SSF. Conversely, if the FCTN rank is set too high (e.g., The model will use The presence of multiple parameters can lead to slow convergence and unnecessary computational overhead. Therefore, adaptively adjusting the FCTN rank during optimization is crucial, aiming to minimize runtime costs while maintaining sufficient model expressiveness. To this end, a significantly computationally efficient FCTN rank incrementing strategy is proposed. Once the current model is considered sufficiently close to the optimal solution, the FCTN rank is continuously increased by adding random numbers from a uniform distribution between 0 and 1. and The algorithm continues until a model structure that effectively represents the data without unnecessary complexity is obtained, up to the appropriate new size. Under this rank-increasing scheme, the final FCTN rank is determined as the maximum FCTN rank achieved during the optimization process. The strategy proposed in this invention significantly reduces computation time. The principle is that with a smaller initial FCTN rank, the algorithm can quickly capture the global structural information of the target tensor; as the rank increases, finer details are gradually recovered, thereby improving overall efficiency. Subsequent numerical experiments verify the effectiveness of the proposed method.

[0094] The third-order representation of FCTN decomposition is as follows: Figure 1 As shown, The decomposition yields three components. , , . Figure 2 The overall reconstruction process is demonstrated. The true sound velocity field of the region is obtained by using the temperature, salinity and depth data provided by Argo and the empirical formula of sound velocity. This is used as the true value. Then, sparse observation profiles are obtained by random sampling. The observation values ​​are then input into the FCTN-T model, and finally, a high-precision reconstructed sound velocity field is obtained.

[0095] This invention analyzes three-dimensional SSF data from December 21, 2012, utilizing temperature, salinity, and depth data provided by the Hybrid Coordinate Ocean Model (HYCOM). The horizontal resolution of the experimental area is [missing information]. ,about Construct a horizontal dimension of The total coverage area is approximately The mesh. Depth range is It is divided into 37 water layers with unequal intervals from the sea surface to the seabed. Because the depth intervals of the provided grid temperature, salinity, and depth data are not uniform, studies have shown that the variation in sound speed in shallow water is much greater than that in deep water.

[0096] Analyze existing data, Figure 3 It shows 400 selected sound velocity profiles. Figure 4 This demonstrates the standard deviation of sound velocity distribution at different depths, clearly showing that the variation in ocean sound velocity is much greater in shallow waters than in deep waters. Therefore, to better illustrate the effect, this experiment used CTD data from 20 fixed depths in shallow waters: 0m, 10m, 20m, 30m, 40m, 50m, 60m, 70m, 80m, 90m, 100m, 125m, 150m, 200m, 250m, 300m, 350m, 400m, 500m, and 600m.

[0097] Taking into account factors such as ocean complexity, accuracy, and regional characteristics, the Mackenzie empirical formula was selected to convert the temperature, salinity, and depth (TDM) data into sound velocity data for analysis. The sound velocity values ​​at each point were calculated, and the specific calculation formula is shown below:

[0098] ;

[0099] In the formula, This represents the ocean sound speed value. Indicates the seawater temperature value. Indicates the salinity value of seawater. This indicates the depth of the seawater.

[0100] To facilitate the training process, all the obtained data are normalized to intervals. Within this range, it plays a crucial role in enhancing the convergence and stability of the training algorithm. After preprocessing, the final result is of size [size missing]. 3D-SSF data was used as the ground truth for sound velocity. The method for selecting the index location of missing elements was simple uniform random sampling. Since in reality, a sound velocity profiler can only measure bar-shaped sound velocity data of a line profile each time, to more closely approximate the real situation, the randomly sampled data was also set as line profile data, and a subset of the profile was randomly and uniformly selected as the observation tensor. The missing rate (MR) is defined as the ratio of the number of missing elements in the data to the total number of elements. Considering the real situation, the missing rates for this test were set to 70%, 75%, 80%, 85%, and 90%, respectively. First, the sound velocity field was reconstructed using sparse tensors. Then, the reconstructed data was compared with the ground truth, and error analysis was performed to evaluate the expressive power of different methods.

[0101] For this test dataset In the proposed FCTN-T decomposition-based imputation method, the maximum FCTN rank needs to be adjusted; for simplicity, they are set to the same value here. The reconstruction performance of different methods under different missing rates is as follows: Figure 5 As shown in the figure, it is evident that the Empirical Orthogonal Function (EOF) and the Inverse Distance Weighted Interpolation (IDW) model perform poorly in constructing 3D SSFs. Conversely, the Tensor Neural Network (TNN) utilizing tensor decomposition, the Fully Connected Tensor Network (FCTN), and the proposed FCTN-T ocean 3D sound velocity field model based on network decomposition and factor regularization exhibit significantly higher accuracy than the first two traditional methods. This is attributed to the fact that tensor decomposition can capture the correlation between different dimensions. More specifically, when the missing rate reaches a high level of 90%, the root mean square error (RMSE) of the FCTN-T method proposed in this invention is 0.66 m / s. Compared with IDW, EOF, TNN, U-net, and FCTN methods, the reconstruction accuracy of the FCTN-T method is improved by 42.6%, 48.4%, 56.3%, 27.5%, and 69.7%, respectively. When the missing data rate is low, reaching 70%, the FCTN-T method achieves an RMSE of 0.32 m / s, representing improvements in accuracy of 62.8%, 73.9%, 42.9%, 27.3%, and 43.9% compared to IDW, EOF, TNN, U-net, and FCTN methods, respectively. Furthermore, when the missing data rate reaches 90%, the reconstruction accuracy of the TNN method drops rapidly, exhibiting the worst performance with an accuracy of 1.51 m / s. This is because deep learning-based methods rely excessively on data-driven approaches, rendering them ineffective when training data is limited. The FCTN-T model consistently maintains the best reconstruction results, fully demonstrating the strong potential of factor-regularized tensor completion models in marine 3D-SSF reconstruction tasks.

[0102] Figure 6 The running time of six different reconstruction methods under varying missing rates is listed. It is evident that the latter four methods require significantly more time than the traditional EOF and IDW methods. This is because traditional methods use closed-form expressions throughout the reconstruction process, making their time negligible. Furthermore, the latter four methods require iterative updates when handling high-dimensional data, which typically leads to complex calculations and inevitably increases computational load. In addition, the U-Net model involves a large number of parameters, totaling 42,424,608, which inevitably results in a significant increase in computation time. Further comparison shows that the FCTN-T method significantly reduces running time compared to TNN and FCTN methods, even by several times at lower missing rates. This is because the rank-increasing FCTN-T method can dynamically adjust the rank during runtime, maximizing efficiency while maintaining model performance. This fully demonstrates the effectiveness of this invention based on the FCTN rank-increasing method.

[0103] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for velocity field reconstruction based on full connected tensor network decomposition and regularization, characterized in that, including obtaining sparse three-dimensional sound speed field observations , inputting a marine three-dimensional sound speed field model FCTN-T based on tensor decomposition and factor regularization for processing to obtain a reconstructed marine three-dimensional sound speed field; The FCTN-T comprises: ； ； ； ； In the formula, It is the Nth-order tensor composed of the three-dimensional sound velocity field of the ocean, given We obtain the initial value and then iterate to solve for it. It is the FCTN factor tensor All possible values, including , , , It is a matrix factorization factor All possible values, including , , , It is a matrix factorization factor All possible values, including , , , This is FCTN model processing, subscript yes Norm, It is a regularization parameter. It is a first-order difference matrix. It is a factor tensor model - Expand the matrix. It is a projection operator. It is an index set of known sound speed locations; the projection operator will... lie in The element at a given position projects onto itself, located at... The mapping for elements outside the specified positions is 0.

2. The method of claim 1, wherein, The FCTN comprises: ； ； wherein is the physical index in the th dimension, is the rank of the FCTN model, is the connection index between and , the maximum value of the connection index is , the maximum value of is .

3. The method of claim 2, wherein, An adaptive rank increasing strategy is added in the FCTN-T, comprising: S1, setting an initial value of the rank of the FCTN model; S2, setting a maximum rank of the FCTN model; S3, iteratively updating , , and ; S4, obtaining a reconstructed marine three-dimensional sound speed field after a convergence condition is reached.

4. The method of claim 3, wherein, The rank of the FCTN model is initially set to .

5. The method of claim 3, wherein, The maximum rank of the FCTN model is: ； wherein is the maximum rank of the FCTN model, , and correspond to three ranks in three dimensions in the three-dimensional sound speed field.

6. The method of claim 5, wherein, iterative update , , and , if ； ； ； then , , continue iteration; wherein is the first iteration of , is a convergence threshold, are two different random numbers belonging to positive integers.

7. The method of claim 6, wherein, The convergence condition is: 。

Images