DVL-DeepONet: A Physics-Guided Operator Learning for Resilient Underwater Navigation Arup Kumar Sahooa,∗ , Itzik Kleina a The Hatter Department of Marine Technologies, Leon H. Charney School of Marine Sciences, University of
arXiv:2606.23502v1 [cs.RO] 22 Jun 2026
Haifa, Haifa, 3498838, Israel
ARTICLE INFO
Abstract
Keywords: DeepONet Inertial Sensors Doppler Velocity Log Underwater Navigation Least Squares Estimation Autonomous Underwater Vehicles
Autonomous Underwater Vehicles (AUVs) rely heavily on the fusion of inertial sensors and Doppler velocity logs (DVLs) for navigation. In standard autonomous navigation systems, the DVL measures four beam velocities, thereby enabling the estimation of the AUV velocity vector. However, during real-world missions, the DVL may receive noisy or incomplete beam measurements due to marine obstacles, seabed reflections, or environmental disturbances. Furthermore, some low-cost underwater platforms operate without inertial sensors to reduce system complexity and cost. In such cases, reliable estimation of the AUV velocity vector in real-world missing beam scenarios, becomes challenging, leading to degraded navigation solutions. To circumvent these challenges and enable resilient underwater navigation, we propose DVL-DeepONet, a physics-guided deep neural operator framework along with three variants. The proposed models are designed to estimate DVL-based velocity information under multiple operational scenarios, including (i) noise-resilient estimation in coupled inertial/DVL measurements, (ii) DVL-only learning, and (iii) beam measurement recovery. By learning a nonlinear operator that maps temporal inertial/DVL observations directly to vehicle velocity while enforcing DVL measurement physics through a consistency constraint, the proposed approach enables robust velocity estimation even under degraded sensing conditions. The proposed framework is validated using real-world AUV experiments, comprising a cumulative path length of approximately 10,000 m. Experimental results demonstrate that the proposed DVL-DeepONet architectures outperform baseline model-based approaches and learning-based algorithms by 40%.
1. Introduction Autonomous underwater vehicles (AUVs) are increasingly employed in marine missions such as oceanographic surveying, underwater infrastructure inspection, environmental monitoring, and military operations [1, 2]. The success and operational effectiveness of these missions inherently depend on accurate navigation of AUVs. However, this objective remains a significant challenge in deep underwater environments. Since global navigation satellite system signals cannot penetrate underwater, AUVs commonly rely on the fusion of inertial navigation systems (INS) and Doppler velocity logs (DVL) for navigation. The INS provides continuous navigation states using inertial measurements obtained from accelerometers and gyroscopes [3, 4], while the DVL provides velocity measurements via acoustic Doppler sensing [5, 6]. Though INS sensors provide high-rate navigation estimates, their errors accumulate over time due to sensor noise and bias, leading to ∗ Corresponding author
[email protected] (A.K. Sahoo); [email protected] (I. Klein) orcid(s): 0000-0003-4515-7434 (A.K. Sahoo); 0000-0001-7846-0654 (I. Klein)
Page 1 of 15
significant navigation drift [7, 8]. To mitigate this, DVL measurements are commonly integrated into the navigation framework as aiding velocity updates. In standard bottom-lock operation, the DVL transmits four acoustic beams toward the seafloor and estimates the vehicle velocity vector by exploiting the Doppler effect. Due to its high velocity accuracy, the DVL plays a critical role in underwater navigation systems [9]. Over the years, a wide range of velocity estimation strategies have been investigated, including classical model-based approaches such as leastsquares (LS) estimator, Kalman filtering, loosely coupled and tightly coupled INS/DVL integration, factor-graph optimization, and information-aided estimation methods [10, 11, 12]. Despite their success, robust velocity estimation across diverse underwater operating conditions remains an open challenge. In practice, both inertial and DVL measurements are affected by sensor noise, environmental disturbances, and measurement uncertainties, which can significantly degrade estimation accuracy. Furthermore, in certain applications, only DVL measurements may be available, necessitating velocity estimation without inertial data. The problem becomes even more challenging when DVL beam measurements are
DVL-DeepONet
partially unavailable due to acoustic interference, poor bottom-lock conditions, marine obstacles, rough seafloor terrain, temporary sensor degradation, or aggressive AUV maneuvers [13]. In such circumstances, conventional modelbased approaches may experience significant performance degradation, leading to navigation solution drift. To overcome these limitations, recent advances in sensing technologies and computational capabilities have motivated the adoption of data-driven approaches based on machine learning and deep learning. For inertial/DVL-integrated navigation, several learning-based approaches have been investigated as either end-to-end estimators or hybrid frameworks combined with model-based filtering algorithms. BeamsNet [14] introduced a neural-network-based velocity estimator that directly maps DVL beam measurements and inertial observations to vehicle velocity. More recently, ResAlignNet [15], DMIAN [16], and several other hybrid learning-filtering frameworks [17, 18] have been explored for inertial/DVL fusion through error state estimation. These methods have demonstrated improved robustness and accuracy over model-based techniques. For DVL-only navigation tasks, data-driven models have been developed to extract navigation information directly from beam measurements without relying on auxiliary inertial sensors. DCNet [19] demonstrated the effectiveness of convolutional neural networks for DVL calibration, while UDON [20] explored learning-based underwater odometry using DVL observations. Nevertheless, learning-based approaches for direct DVL-only velocity estimation remain comparatively limited. Furthermore, under degraded DVL operating conditions, including partial beam availability and beam outages, several learning-based methods have focused on recovering missing information. LiBeamsNet [21] addressed limitedbeam scenarios by reconstructing missing beam measurements using available beams and inertial data. MissBeamNet [22] further investigated learning-based velocity estimation under beam outages, while LAS [13] proposed adaptive strategies for maintaining navigation performance in degraded DVL conditions. More recently, Miao et al. [23] developed a physicsguided long short-term memory (LSTM)-based framework for robust AUV navigation under degraded underwater observations. Despite these advances, most existing learningbased approaches are tailored to specific navigation scenarios [24, 25]. As a result, a unified Page 2 of 15
learning framework capable of operating across diverse conditions within a single paradigm is still lacking. Furthermore, while these methods have demonstrated promising performance, they generally operate as black-box predictors and do not explicitly exploit the underlying DVL observation model that governs the relationship between beam measurements and vehicle velocity. Consequently, achieving physically consistent and robust velocity estimation across a wide range of operating scenarios, while reducing reliance on expensive external sensors and complex multi-sensor integration architectures, remains an open challenge. Meanwhile, operator-learning methods have emerged as a powerful paradigm for learning mappings between functional inputs and outputs. Deep Operator Networks (DeepONets), introduced by Lu et al. [26], have achieved remarkable success in scientific machine learning, surrogate modeling, and dynamical system prediction [27]. Unlike conventional neural networks, DeepONets preserve the physical properties of the system and offer white-box prediction. Additionally, it learns operators rather than pointwise mappings, enabling efficient modeling of complex temporal and spatial dependencies. Although DeepONets have been successfully applied to a variety of engineering problems alongside land vehicle dynamics [28], their application to underwater navigation and DVL-based velocity estimation remains largely unexplored. Motivated by these challenges, this research introduces DVL-DeepONet, a physics-guided operator learning framework for underwater velocity estimation. Unlike conventional DVL navigation methods that rely on explicit matrix inversion, the proposed framework incorporates the DVL observation model solely as a physics-based consistency constraint during training. By avoiding direct inversion of the beam geometry matrix, the framework eliminates the rank requirements associated with LS reconstruction. Consequently, DVL-DeepONet enables robust and accurate velocity estimation under severe beam outages and degraded sensing conditions. Furthermore, the framework is designed to operate under three different sensing conditions. DVL-DeepONet-I addresses noise-resilient estimation using inertial and DVL beam data. DVL-DeepONet-II focuses on DVL only beam-tovelocity operator learning and enables velocity estimation even in the absence of inertial measurements. DeepONet-III handles scenarios involving partial or missing beam configurations. By integrating (a) physical consistency, (b) temporal sensor fusion, and (c) operator learning,
DVL-DeepONet
the proposed approach aims to provide accurate and resilient velocity estimation for real-world underwater navigation. The contributions of this research are outlined as follows: 1. A novel physics-guided DVL-DeepONet framework, uniquely designed to forecast DVL velocity vector. 2. Three complementary DVL-DeepONet architectures to address practical underwater navigation scenarios. 3. Extensive validation on AUV datasets collected from real-world sea trials and competitive baselines, demonstrating improved robustness, velocity estimation accuracy, and navigation performance. 4. To support the reproduction of results and encourage future research and benchmarking, our codebase has been made publicly available on https://github.com/ansfl/DVL-DeepO Net. The proposed framework is validated using real-world AUV datasets collected during sea trials in the Mediterranean Sea, Israel, comprising approximately 10,000 m of underwater navigation trajectories. Extensive experiments are conducted under noisy measurements, partial beam outages, and limited sensing scenarios. The results demonstrate that the proposed DVLDeepONet framework provides accurate and resilient velocity estimation under challenging underwater conditions. Our approach offers an average improvement of 40% over baselines. The remainder of this paper is organized as follows. Section 2 presents the problem formulation and DVL measurement model. Section 3 introduces the DVL-DeepONet framework and network architectures. Section 4 presents the experimental results and analysis. Finally, Section 5 concludes the paper.
and measures the Doppler frequency shift of the reflected signal [29]. Let 𝐛𝑖 , 𝑖 = 1, 2, 3, 4, denote the beam direction vectors defined in the DVL frame. Thus, the DVL projection matrix is obtained as ⎡𝐛1 ⎤ ⎢𝐛 ⎥ 𝐇 = ⎢ 2 ⎥ ∈ ℝ4×3 . 𝐛 ⎢ 3⎥ ⎣𝐛4 ⎦ The beam measurement model is [30] 𝐲 = 𝐇𝐯𝑑𝑏 + 𝐧,
This section formulates the DVL velocity estimation problem and highlights the challenges associated with limited and degraded beam measurements.
2.1. Least-Squares Velocity Reconstruction A standard DVL employs four acoustic transducers commonly arranged in a Janus (X) configuration. Each transducer emits an acoustic beam toward the seafloor as depicted in Fig. 1 Page 3 of 15
(2)
where 𝐧 denotes measurement noise, 𝐲 is the measured beam velocity, and 𝐯𝑑𝑏 is the AUV velocity vector expressed in the DVL coordinate frame. The objective of the DVL processing stage is to reconstruct the velocity vector from the beam measurements. The standard DVL solution employs a LS estimator to solve (2). The estimation problem can be formulated as [6]: ‖2 ‖ 𝐯̂ 𝑑𝑏 = arg min ‖𝐲̃ − 𝐇𝐯𝑑𝑏 ‖ . ‖2 𝐯 ‖
(3)
The solution to (3) is 𝐯̂ 𝑑𝑏 = (𝐇𝑇 𝐇)−1 𝐇𝑇 𝐲̃ .
(4)
Although computationally efficient, the LS solution is sensitive to noisy beam measurements, outliers, and limited beam availability. In practical underwater environments, these issues may significantly degrade the algorithm performance.
2.2. Navigation Under Limited DVL Measurements In a DVL-missing-beams, a beam availability vector at time 𝑘 is defined as 𝐦𝑘 = [𝑚1,𝑘 , 𝑚2,𝑘 , 𝑚3,𝑘 , 𝑚4,𝑘 ]𝑇 ,
2. Problem Formulation
(1)
𝑚𝑖,𝑘 ∈ {0, 1}, (5)
The corresponding masking matrix is 𝐌𝑘 = diag(𝐦𝑘 ) ∈ ℝ4×4 .
(6)
The partially observed beam measurements are 𝐲𝑘𝑝 = 𝐌𝑘 𝐲𝑘 .
(7)
Substituting the beam measurement model (2) into the (7) yields
DVL-DeepONet
Figure 1: Geometry of the four-beam Janus DVL mounted on the AUV and the associated body-fixed coordinate system.
(
)
𝐲𝑘𝑝 = 𝐌𝑘 𝐇𝐯𝑑𝑏,𝑘 + 𝐧𝑘 .
(8)
The number of available beams is
𝑁𝑏,𝑘 =
4 ∑
𝑚𝑖,𝑘 .
(9)
𝑖=1
When 𝑁𝑏,𝑘 ≥ 3 a reduced LS solution may still be computed. However, when 𝑁𝑏,𝑘 < 3, the velocity estimation problem becomes underdetermined, as rank(𝐇𝑝𝑘 ) < 3.
(10)
Here 𝐇𝑝𝑘 ∈ ℝ𝑁𝑏,𝑘 ×3 denotes the reduced beam projection matrix obtained by removing the rows of 𝐇 corresponding to unavailable beams at time step 𝑘. Consequently, a unique velocity solution no longer exists, and conventional LS estimation becomes unreliable or infeasible. These challenges motivate the development of robust approaches capable of exploiting both inertial measurements and partial beam observations to estimate the AUV velocity under degraded sensing conditions.
3. Proposed DVL-DeepONet Framework This section discusses a family of physicsguided operator learning models, termed DVLDeepONet, for resilient underwater velocity estimation under varying sensor availability conditions. DeepONet was selected as the backbone architecture because the underwater navigation problem considered in this work can be naturally formulated as an operator-learning task. DeepONet is a neural operator framework that is even capable of learning mappings between infinitedimensional function spaces. Its formulation is mathematically grounded in the universal approximation theorem for operators [26]. Rather than estimating velocity from a single sensor snapshot, the objective is to learn a mapping from a temporal history of beam measurements to the corresponding AUV velocity vector. To further improve physical consistency, DVLDeepONet incorporates an LS beam consistency constraint during training. It ensures that the predicted velocity remains compatible with the underlying DVL measurement geometry. We begin by presenting DVL-DeepONet and then address three different real-world scenarios: (i) noiseresilient estimation, (ii) DVL-only learning, and (iii) beam measurement recovery.
3.1. DVL-DeepONet Framework The proposed framework integrates DeepONet with DVL beam geometry and LS physical Page 4 of 15
DVL-DeepONet
constraints to estimate the velocity vector under noisy, degraded, and partially observable sensing conditions. Let 𝐲𝑘 denote the DVL beam measurements at time step 𝑘, 𝐲𝑘 = [𝑦1,𝑘 , 𝑦2,𝑘 , 𝑦3,𝑘 , 𝑦4,𝑘 ]𝑇 ∈ ℝ4 ,
(11)
where 𝑦𝑖,𝑘 is the velocity measured by the 𝑖-th DVL beam. Similarly, let 𝐮𝑘 denote the synchronized IMU measurement vector,
𝜏𝑘 =
𝑡𝑘 − 𝜇𝑡 . 𝜎𝑡
The branch network 𝐁𝑘 encodes the synchronized IMU/DVL history into a latent representation. 𝐁𝑘 is implemented as a one-dimensional (1-D) convolutional encoder. Since the input to the convolutional layers is arranged as channels by time, the branch input is internally converted to ̃ 𝑘 = 𝐁⊤ ∈ ℝ𝐷×𝑊 . 𝐁 𝑘
𝑇
6
𝐮𝑘 = [𝑓𝑥,𝑘 , 𝑓𝑦,𝑘 , 𝑓𝑧,𝑘 , 𝜔𝑥,𝑘 , 𝜔𝑦,𝑘 , 𝜔𝑧,𝑘 ] ∈ ℝ ,
(12)
where 𝑓𝑥,𝑘 , 𝑓𝑦,𝑘 , and 𝑓𝑧,𝑘 are the averaged specificforce components, and 𝜔𝑥,𝑘 , 𝜔𝑦,𝑘 , and 𝜔𝑧,𝑘 are the averaged angular-rate components associated with the 𝑘-th DVL interval. To align with the DeepONet structure, the inertial readings are averaged between two successive DVL measurements. To capture temporal dependencies, a sliding window of length 𝑊 is employed. The corresponding DVL and IMU histories are 𝐘𝑘 = [𝐲𝑘−𝑊 +1 , … , 𝐲𝑘 ] ∈ ℝ𝑊 ×4 ,
(13)
𝐔𝑘 = [𝐮𝑘−𝑊 +1 , … , 𝐮𝑘 ] ∈ ℝ𝑊 ×6 ,
(14)
where 𝑘 denotes the current discrete-time index. DVL-DeepONet architecture consists of branch and trunk networks. The branch input 𝐁𝑘 is constructed by channel-wise concatenation, 𝐁𝑘 = [𝐘𝑘 || 𝐔𝑘 ] ∈ ℝ𝑊 ×𝐷 ,
(15)
where || denotes concatenation and 𝐷 denotes the number of synchronized sensor channels. For the considered DVL–IMU configuration, 𝐷 = 4 + 6 = 10 channels. The synchronized sensor channels are jointly processed by the branch encoder to learn a latent representation of the recent vehicle dynamics. The objective is to learn a nonlinear operator 𝜃 ∶ (𝐁𝑘 , 𝜏𝑘 ) → 𝐯𝑘 ,
(16)
where 𝐯𝑘 denotes the velocity vector: 𝐯𝑘 = [𝑣𝑥 , 𝑣𝑦 , 𝑣𝑧 ]𝑇 ∈ ℝ3 ,
(17)
and 𝜏𝑘 is the normalized temporal coordinate defined by: Page 5 of 15
(18)
(19)
Then, the convolutional encoder is defined as ( ) (𝑙) (𝑙−1) (𝑙) 𝐡(𝑙) = 𝜎 𝐖 ∗ 𝐡 + 𝐛 , 𝑏 𝑏 𝑏 𝑏
𝑙 = 1, 2, 3, (20)
̃ 𝑘 denotes the branch input, 𝐡(𝑙) is where 𝐡(0) = 𝐁 𝑏 𝑏 the feature map of the 𝑙-th convolutional layer, 𝐖(𝑙) and 𝐛(𝑙) represents the trainable convolution 𝑏 𝑏 kernels and bias parameters, respectively, (∗) denotes 1-D convolution along with the temporal axis, and 𝜎(⋅) is the nonlinear activation function. The output of the final convolutional layer is flattened and passed through a fully connected projection layer to obtain the branch latent representation ( ( )) 𝐳𝑏 = 𝜙𝑏 vec 𝐡(3) ∈ ℝ𝑝 , (21) 𝑏 where vec(⋅) denotes the flattening operation, 𝜙𝑏 (⋅) represents the fully connected projection, and 𝑝 denotes the latent embedding dimension of the DeepONet architecture. Let 𝜃𝑏 denote the branch encoder with trainable parameters 𝜃𝑏 . It consists of three 1-D convolutional layers with 32, 64, and 64 output channels, respectively. Equivalently, the branch encoder can be written compactly as 𝐳𝑏 = 𝜃𝑏 (𝐁𝑘 ) ∈ ℝ𝑝 .
(22)
Thus, the branch network explicitly fuses DVL beam measurements and IMU observations through channel-wise concatenation before temporal convolutional feature extraction. Next, the trunk network maps the normalized temporal coordinate 𝜏𝑘 into the latent operator space through a multilayer perceptron defined as ( ) (𝑙) (𝑙−1) (𝑙) 𝐡(𝑙) = 𝜙 𝐖 𝐡 + 𝐛 , 𝑡 𝑡 𝑡 𝑡
𝑙 = 1, … , 𝐿, (23)
(𝑙) (𝑙) where 𝐡(0) 𝑡 = 𝜏𝑘 , 𝐖𝑡 and 𝐛𝑡 denote the trainable weight matrices and bias vectors, respectively,
DVL-DeepONet
and 𝜙(⋅) is the nonlinear activation function. The output of the final hidden layer is projected to the trunk latent representation 𝐳𝑡 = 𝜃𝑡 (𝜏𝑘 ) ∈ ℝ𝑝 ,
(24)
where 𝜃𝑡 is the trunk encoder MLP parameterized by 𝜃𝑡 . In the implemented architecture, the trunk encoder maps the scalar temporal input to a latent representation of dimension 𝑝 = 128. The trunk encoder consists of three fully connected layers with 64, 128, and 128 neurons, respectively, with tanh activation function. Following the DeepONet principle, the branch and trunk embeddings are fused through element-wise multiplication 𝐳𝑘 = 𝐳𝑏 ⊙ 𝐳𝑡 .
(26)
where 𝜃ℎ (⋅) denotes the prediction head MLP parameterized by 𝜃ℎ , and 𝐯̂ 𝑘 is the predicted vehicle velocity vector. Therefore, the DVL-DeepONet operator approximation results in ( ) 𝐯̂ 𝑘 = 𝜃ℎ 𝜃𝑏 (𝐁𝑘 ) ⊙ 𝜃𝑡 (𝜏𝑘 ) .
(27)
The proposed framework consists of three complementary operator-learning architectures, which are designed o handle noisy DVL and inertial measurements (DVL-DeepONet-I), DVLonly operation (DVL-DeepONet-II), and partial sensor availability (DVL-DeepONet-III), respectively, as shown in Table 1. The overall framework is illustrated in Fig. 2. In all three DVL-DeepONet variants, the framework exploits temporal sensor histories and nonlinear operator learning to capture the underlying beam-to-velocity dynamics.
3.2. Noise-Resilient Estimation DVL-DeepONet-I is designed for robust velocity estimation under noisy IMU and DVL measurements. For a temporal window of length 𝑊 , the branch input (15) is constructed as Page 6 of 15
Model
IMU
DVL
Objective
DVL-DeepONet-I DVL-DeepONet-II DVL-DeepONet-III
✓ × ✓
✓ ✓ Partial
Noise-Resilient DVL-Only Beam Recovery
𝐁𝑘 = [𝐲𝑘−𝑊 +1 , … , 𝐲𝑘 , 𝐮𝑘−𝑊 +1 , … , 𝐮𝑘 ] ∈ ℝ𝑊 ×10 . (28) The network learns the nonlinear operator (𝐼) ∶ (𝐁𝑘 , 𝜏𝑘 ) → 𝐯𝑘 . 𝜃
(29)
For a mini-batch of size 𝑁, the supervised loss is defined as
(25)
Furthermore, the fused latent representation is subsequently processed by a prediction head MLP consisting of two hidden fully connected layers with 128 and 64 neurons, respectively, followed by an output layer with three neurons corresponding to the velocity components. The prediction head employs the sigmoid linear unit (SiLU) activation function in the hidden layers. It may be compactly written as 𝐯̂ 𝑘 = 𝜃ℎ (𝐳𝑘 ),
Table 1 Summary of DVL-DeepONet variants for estimating the DVL velocity vector in different conditions.
1 ∑ ‖𝐯̂ − 𝐯𝑖 ‖22 . 𝑁 𝑖=1 𝑖 𝑁
𝑣𝑒𝑙 =
(30)
To enforce consistency with the DVL observation model introduced in (2), the predicted velocity is projected back into beam space, 𝐲̂ 𝑖 = 𝐇𝐯̂ 𝑖 .
(31)
where 𝐇 ∈ ℝ4×3 denotes the DVL beam projection matrix (1). Building on the DVL beam projection matrix, the residual is defined as 𝐫𝑖 = 𝐲𝑖 − 𝐇𝐯̂ 𝑖 ,
(32)
Thereby, the physics-informed loss can be written as 1 ∑ 𝑇 𝐿𝑆 = 𝐫 𝐫. 𝑁 𝑖=1 𝑖 𝑖 𝑁
(33)
Combining (30) and (33) leads to the total loss = 𝜆𝑣𝑒𝑙 𝑣𝑒𝑙 + 𝜆𝐿𝑆 𝐿𝑆 .
(34)
where 𝜆𝑣𝑒𝑙 and 𝜆𝐿𝑆 denote positive weighting coefficients. Based on extensive hyperparameter tuning through a series of trial-and-error experiments, the coefficients were fixed at 𝜆𝑣𝑒𝑙 = 1 and 𝜆𝐿𝑆 = 0.1. The LS consistency term acts as a geometryaware physical regularizer that constrains the learned velocity to remain compatible with the DVL beam observation model. The optimized network parameters satisfy Θ∗ = arg min (Θ). Θ
(35)
DVL-DeepONet
Figure 2: Schematic illustration of the proposed DVL-DeepONet architecture.
3.3. DVL-Only Learning DVL-DeepONet-II addresses scenarios where IMU measurements are unavailable or unreliable. In this case, the branch input contains only DVL beam observations. For a temporal window of length 𝑊 , the branch input is constructed as [ ] 𝐁𝑘 = 𝐲𝑘−𝑊 +1 , 𝐲𝑘−𝑊 +2 , … , 𝐲𝑘 ∈ ℝ𝑊 ×4 .
(36)
(39)
The supervised velocity loss associated with partial beams is defined as 1 ∑ ‖𝐯̂ − 𝐯𝑘 ‖22 . 𝑁 𝑘=1 𝑘 𝑁
(37)
DVL-DeepONet-II is trained using the same supervised loss (30), and physics-informed loss (33).
3.4. Beam Measurement Recovery DVL-DeepONet-III is developed for navigation under DVL beam outages. A key distinction between the proposed framework and conventional model-based DVL velocity reconstruction lies in the treatment of missing beam measurements. Unlike the traditional LS estimator formulation in (4), DVL-DeepONet-III does not estimate velocity through matrix inversion. Instead, the DVL observation model is incorporated as a physics-guided constraint, while the velocity is inferred through operator learning. Consequently, the framework is not restricted by the rank requirements of conventional LS reconstruction and can operate even when only one or two DVL beams are available. The available beam measurement provides partial physical information, whereas the missing information is recovered from synchronized IMU observations, temporal context, and previously learned beam-to-velocity mappings. This enables accurate velocity estimation even under severe beam outages. Using the partial DVL measurements in (7) and the synchronized IMU observations in (15), the branch input is evolved as Page 7 of 15
(38)
The corresponding operator is (𝐼𝐼𝐼) ∶ (𝐁𝑘 , 𝜏𝑘 ) → 𝐯𝑘 . 𝜃
The learned operator becomes (𝐼𝐼) ∶ (𝐁𝑘 , 𝜏𝑘 ) → 𝐯𝑘 . 𝜃
𝑝 𝐁𝑘 = [𝐲𝑘−𝑊 , … , 𝐲𝑘𝑝 , 𝐮𝑘−𝑊 +1 , … , 𝐮𝑘 ]. +1
𝑝−𝑣𝑒𝑙 =
(40)
To enforce consistency with the partial DVL beam measurements, a masked physics-informed loss is introduced as 1 ∑∑ 𝑚 (𝑦̂ − 𝑦𝑘,𝑖 )2 , 𝑁 𝑘=1 𝑖=1 𝑖,𝑘 𝑘,𝑖 𝑁
𝑝−𝐿𝑆 =
4
(41)
Consequently, the total loss is = 𝜆𝑝−𝑣𝑒𝑙 𝑝−𝑣𝑒𝑙 + 𝜆𝑝−𝐿𝑆 𝑝−𝐿𝑆 ,
(42)
where 𝜆𝑝−𝑣𝑒𝑙 , 𝜆𝑝−𝐿𝑆 > 0 are weighting coefficients.
3.5. Training Algorithm 1 summarizes the complete training workflow. The proposed DVL-DeepONet framework was trained using the hyperparameters detailed in Table 2.
4. Analysis and Results This section discusses the dataset, evaluation metrics, and results obtained by DVL-DeepONet.
DVL-DeepONet
Algorithm 1: Training procedure of DVLDeepONet (identical for all its three variants). Construct branch input 𝐁𝑘 ; Construct trunk input 𝜏𝑘 ; Initialize network parameters Θ; for each epoch do for each mini-batch do Compute 𝐳𝑏 = 𝜃𝑏 (𝐁𝑘 ); Compute 𝐳𝑡 = 𝜃𝑡 (𝜏𝑘 ); Fuse 𝐳𝑘 = 𝐳𝑏 ⊙ 𝐳𝑡 ; Predict 𝐯̂ 𝑘 = 𝜃ℎ (𝐳𝑘 ); Compute vel using (30), or (40) in the missing beam scenario; Compute LS via (33) or (41) in the missing beam scenario; Compute total loss using (34) or (42) in the missing beam scenario; Update Θ using backpropagation; if < 𝜀 then return Θ return Θ Table 2 Hyperparameters for DVL-DeepONet framework used in all three variants. Parameter
Configuration
Temporal Window (𝑊 ) Branch CNN Trunk MLP Prediction Head MLP Optimizer Learning Rate Batch Size Epochs
2 32-64-64 64-128-128 128-64-3 AdamW 10−3 128 100
4.1. AUV Dataset The AUV dataset was acquired using the University of Haifa’s Snapir AUV during multiple sea trials conducted in the Mediterranean Sea, Israel [31]. Snapir is an [32] ECA Robotics modified A18D mid-size AUV designed for deep-water operations up to 3000 m and capable of missions lasting up to 21 hours. The platform IS mounted with an iXblue Phins Subsea INS [33], and a Teledyne RDI WorkHorse Navigator DVL [34], which provides velocity measurements with a nominal standard deviation of 0.02 m/s. The INS operates at 100 Hz, whereas the DVL provides measurements at 1 Hz. The dataset consists of multiple AUV maneuvers, each spanning approximately 400 s. These missions differ in path geometry, mission lengths, operating depth, and vehicle speed, as presented in Fig. 3. These thirteen Page 8 of 15
Figure 3: Top-view of the 13 AUV trajectories used in this study.
trajectories provide a comprehensive evaluation of the proposed DVL-DeepONet framework under varying operating conditions. For cross-validation purposes, we partition the dataset into three distinct splits. In the first, the training set comprises T1, T6–T13, and the validation set comprises T2 and T3. The remaining two trajectories, T4 and T5, are reserved for testing. The total path lengths of the training and testing datasets are approximately 7095 m and 1566 m, respectively. To maintain clarity and brevity, only the results and graphs obtained from the primary experimental configuration (Split 1) are reported in detail. More information regarding three-fold cross-validation splits are discussed later in Section 4.7.2. To construct the noisy DVL measurements, the DVL velocity measurements recorded during the AUV missions were first projected onto the four DVL beam directions using the beam geometry matrix. The resulting beam measurements are given by 𝐲𝑘 = 𝐇𝐯𝑑𝑏,𝑘 ,
(43)
where 𝐇 ∈ ℝ4×3 denotes the DVL beam geometry matrix and 𝐯𝑑𝑏,𝑘 is the DVL velocity vector expressed in the body frame. To emulate practical underwater environments, scale-factor error, constant bias, and additive Gaussian noise were introduced into the beam measurements 𝐲̃ 𝑘 = (1 + 𝑠𝐷𝑉 𝐿 )𝐇𝐯𝑑𝑏,𝑘 + 𝐛𝐷𝑉 𝐿 + 𝜎𝐷𝑉 𝐿 𝝐 𝑘 ,
(44)
where 𝑠𝐷𝑉 𝐿 = 0.7% is the scale-factor error, 𝐛𝐷𝑉 𝐿 = 0.001 𝟏4 is the bias vector, 𝜎𝐷𝑉 𝐿 = 0.042 is the noise
DVL-DeepONet
standard deviation, and 𝝐 𝑘 ∼ (𝟎, 𝐈4 ). The specific error term values follow those used in [14]. To this end, the corrupted beam measurements 𝐲̃ 𝑘 were subsequently used as inputs in all our evaluations while, the IMU measurements were used directly as is without any addition of error terms.
4.2. Performance Metrics To evaluate the velocity estimation performance, five quantitative metrics are employed. Let the predicted and ground-truth (GT) velocity ̂ 𝑖 ) and 𝐯(𝑡𝑖 ), vectors at time 𝑡𝑖 be denoted by 𝐯(𝑡 respectively. 1. Velocity absolute error (VAE): ̂ 𝑖 ) − 𝐯(𝑡𝑖 )‖ VAE𝑖 = ‖ ‖𝐯(𝑡 ‖2 .
(45)
2. Velocity mean absolute error (VMAE): 1 ∑ VAE𝑖 , 𝑁 𝑖=1 𝑁
VMAE =
(46)
where 𝑁 denotes the total number of samples. 3. Velocity root mean square error (VRMSE): √ √ 𝑁 √1 ∑ VRMSE = √ (VAE𝑖 )2 . (47) 𝑁 𝑖=1 4. Coefficient of determination (𝑅2 ): ∑𝑁 ( 𝑅2 (𝑥̇ 𝑗 , 𝑥̂̇ 𝑗 ) = 1 −
) ̂ 2 𝑖=1 𝑥̇ 𝑗,𝑖 − 𝑥̇ 𝑗,𝑖 ) , ∑𝑁 ( ̄ 2 𝑖=1 𝑥̇ 𝑗,𝑖 − 𝑥̇ 𝑗
(48)
where 𝑥̇ 𝑗,𝑖 and 𝑥̂̇ 𝑗,𝑖 denote the GT and predicted values of the 𝑗-th velocity component, respectively, and 𝑥̄̇ 𝑗 is the mean of the corresponding GT component. 5. Variance accounted for (VAF): [ ( )] var 𝑥̇ 𝑗 − 𝑥̂̇ 𝑗 VAF(𝑥̇ 𝑗 , 𝑥̂̇ 𝑗 ) = 1 − × 100. (49) ( ) var 𝑥̇ 𝑗
4.3. Implementation The proposed DVL-DeepONet framework was implemented in Python using the PyTorch deep learning library. All experiments were conducted on the hardware platform described in Table 3.
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Table 3 Hardware configuration for DVL-DeepONet. Component
Specification
GPU OS Architecture CUDA Version cuDNN Version RAM Memory Tensor Cores CPU Cores
NVIDIA GeForce RTX 4090 Linux (Debian) x86_64 11.8 9.1 67.26 GB 25.28 GB 512 24
4.4. Noise-Resilient Estimation Table 4 summarizes the performance of the proposed DVL-DeepONet-I framework under noisy IMU/DVL measurements and compares it against the model-based LS solution (4) and learning-based BeamsNetV1 [14] model. BeamsNetV1 is a CNN-based velocity reconstruction framework that jointly exploits DVL beam measurements and IMU observations to estimate the AUV velocity vector. Owing to its sensor-fusion capability and demonstrated performance under noisy conditions, it serves as a strong learningbased baseline for comparison. The standard LS estimator yields a VRMSE of 0.129 m/s and a VMAE of 0.115 m/s. Although LS estimator provides a direct analytical reconstruction of the vehicle velocity from the DVL beam measurements, its performance deteriorates under noisy conditions. This is because measurement errors are directly propagated through the inversion process. However, learning-based algorithms such as Vanilla-CNN and BeamsNetV1 improve estimation accuracy marginally over the model-based algorithm due to their inherent noise-reduction ability. The proposed LS-DeepONet-I significantly outperforms all baseline methods. It achieves the lowest velocity reconstruction error, with a VRMSE of 0.105 m/s and a VMAE of 0.093 m/s. Compared with the LS estimator, Vanilla-CNN, and BeamsNetV1 approaches, the proposed model yields performance improvements of 18%, 10%, and 15%, respectively, corresponding to an average improvement of approximately 15%. Beyond average error metrics, DVL-DeepONetI also demonstrates superior agreement with the GT velocity dynamics, achieving the highest mean 𝑅2 score of 0.907 and VAF of 91%. These results indicate that the proposed operatorlearning framework more accurately captures the underlying nonlinear relationship between noisy DVL/IMU measurements and vehicle velocity. Figures 4a and 4b illustrate the velocity reconstruction error profiles for the two unseen test trajectories in split 1. For Trajectory 4, the
DVL-DeepONet Table 4 Performance comparison for noisy IMU/DVL measurements. Model Classical LS Vanilla-CNN BeamsNetV1 DVL-DeepONet-I
VRMSE ↓
VMAE ↓
Mean 𝑅2 ↑
VAF (%) ↑
VRMSE Gain (%) ↑
0.129 0.116 0.123 0.105
0.115 0.097 0.110 0.093
0.846 0.880 0.867 0.907
84 88 86 91
18 10 15 –
(a) Test Trajectory 4
(b) Test Trajectory 5
Figure 4: Velocity reconstruction error profiles of Trajectories 4 and 5 using DVL-DeepONet-I.
mean velocity error remains relatively consistent throughout the mission, oscillating around 0.08– 0.12 m/s. Trajectory 5 exhibits slightly larger error fluctuations, as the mean error increases to approximately 0.15–0.17 m/s. However, none of the trajectories exhibits a sustained growth in error over mission progress. This behavior suggests that DVL-DeepONet does not accumulate drift over time and maintains stable performance over long-duration missions.
4.5. DVL-Only Learning To evaluate the DVL-DeepONet-II framework under reduced sensor availability, a second experiment was conducted using only DVL beam measurements, without any IMU information. This scenario is particularly relevant for lowcost underwater platforms and situations where inertial sensors become unavailable or unreliable. In such cases, velocity estimation must rely solely on the information contained in the four DVL beams. For the baseline comparison, we have used model-based LS estimator alongside learning-based vanilla-CNN and BeamsNetV2 [14] models. BeamsNetV2 is a DVL-only neural baseline that maps fourbeam DVL measurements to the velocity vector. The model combines the current DVL beam vector with temporal features derived from a short history of previous beam measurements to predict the AUV velocity components. Table 5 gives the performance of different baseline models when only DVL measurements are available. The proposed DVL-DeepONet-II Page 10 of 15
achieves substantially the best overall performance with a VRMSE of 0.095 m/s and a VMAE of 0.085 m/s. Compared with the LS estimator, DVL-DeepONet-II reduces both VRMSE and VMAE by approximately 27%. Furthermore, the proposed method outperforms VanillaCNN and BeamsNetV2 baselines, achieving VRMSE reductions of 19% and 68%, respectively. Overall, these results correspond to an average improvement of approximately 38% over the considered baseline methods. In addition, DVL-DeepONet-II attains the highest mean 𝑅2 value of 0.905 and the highest VAF of 91%, indicating a stronger agreement with the GT velocity dynamics. Although both DNN and Vanilla-CNN improve upon the classical LS solution, their performance remains inferior to that of DVL-DeepONet-II in all reported metrics. The results suggest that purely data-driven architectures are unable to fully exploit the underlying physical relationship between DVL beam measurements and vehicle velocity.
4.6. Beam Measurement Recovery To evaluate the robustness of DVL-DeepONetIII under degraded sensing conditions, artificial beam outages were introduced in both the training and testing datasets. Specifically, one or two DVL beam measurements were randomly removed every two seconds, corresponding to every second DVL epoch. The corresponding unavailable beam measurements were masked using the beam-availability matrix defined in (5) and (6), while the synchronized IMU
DVL-DeepONet Table 5 Performance comparison under DVL-only measurements. Model Classical LS Vanilla-CNN BeamsNetV2 DVL-DeepONet-II
VRMSE ↓
VMAE ↓
Mean 𝑅2 ↑
VAF (%) ↑
VRMSE Gain (%) ↑
0.129 0.117 0.300 0.096
0.115 0.100 0.258 0.085
0.846 0.866 -0.059 0.905
84 87 78 91
27 19 68 –
Table 6 Performance comparison under partial DVL beam availability. Model ELC MissBeamNet DVL-DeepONet-III
VRMSE ↓
VMAE ↓
Mean 𝑅2 ↑
VAF (%) ↑
VRMSE Gain (%) ↑
1.798 0.183 0.108
1.496 0.153 0.095
-38.428 0.601 0.886
-929 62 89
92 37 –
measurements remained available throughout the experiment. Then, our proposed algorithm is compared with baseline model-based extended loosely coupled (ELC) approach [25], and with learningbased MissBeamNet [22]. In the ELC approach, the missing beams are generated by virtual beams using the assumption of zero sway velocity or taking the last estimated velocity vector. Both approaches outperformed a simple average on the missing beams history. For the learning-based baselines, MissBeamNet employs an LSTM network to estimate missing DVL beam measurements from the available beams and their temporal evolution. The reconstructed beam vector is subsequently processed using the LS estimator to obtain the velocity vector, thereby providing a strong learning-based baseline for missing-beam scenarios. Table 6 presents the performance of proposed algorithm under partial DVL beam availability. Under this challenging scenario, ELC approach exhibits substantial performance degradation, yielding VRMSE values close to 2 m/s. The corresponding negative 𝑅2 and VAF values indicate that the traditional model-based techniques are less effective. MissBeamNet improves the reconstruction accuracy by LSTM algorithm. Consequently, the VRMSE decreases from approximately 1.8 m/s to 0.182 m/s, while the mean 𝑅2 and VAF increase to 0.624 and 64%, respectively. The proposed DVL-DeepONet-III achieves the best performance across all evaluation metrics, obtaining a VRMSE of 0.114 m/s and a VMAE of 0.099 m/s. Compared with ELC, the proposed method reduces the VRMSE by approximately 92%, respectively. Furthermore, DVL-DeepONetIII outperforms MissBeamNet by reducing the Page 11 of 15
VRMSE by approximately 37%. The proposed framework also achieves the highest mean 𝑅2 score of 0.873 and VAF of 88%, indicating a significantly stronger agreement with the GT velocity dynamics. Overall, DVL-DeepONet-III achieves 65% improvement over baselines.
4.7. Ablation Study To better understand the contribution of the individual components of the proposed framework, an ablation study was conducted focusing on two aspects of DVL-DeepONet-I (i) temporal observation window length and (ii) crossvalidation.
4.7.1. Effect of Temporal Window Length The first ablation study investigated the influence of the temporal observation window size (𝑊 ) used by the branch network. To this end, the model was trained and evaluated using window sizes ranging from 𝑊 = 2 to 𝑊 = 20, while keeping all other hyperparameters unchanged. The VRMSE results as a function of the window size are presented in Fig. 5. As shown in Fig. 5, the estimation accuracy initially improves as the window size increases. The best performance is achieved at 𝑊 = 10, yielding a VRMSE of 0.078 m/s. Compared with 𝑊 = 2, the VRMSE is reduced by approximately 17%, demonstrating the benefit of incorporate additional temporal information. However, further increasing the window size beyond 𝑊 = 10 leads to a gradual increase in VRMSE. This behavior suggests that excessively long windows may introduce redundant or less informative historical measurements. 4.7.2. Cross-Validation Study Table 7 summarizes the trajectory distribution used in the three-fold cross-validation study. In
DVL-DeepONet Table 7 Cross-validation splits used for the evaluations. Split
Train. Traj.
S1 1,6,7,8,9,10,11,12,13 S2 4,5,6,7,8,9,10,11,12 S3 1,2,3,4,5,6,11,12,13
Val. Traj. Test Traj. Train Dist. (m) Val. Dist. (m) Test Dist. (m) 2,3 1,13 7,8
4,5 2,3 9,10
7094.26 7259.75 6712.22
1346.59 1496.86 1686.80
1566.93 1346.59 1608.74
Table 8 Cross-validation results for the noise-resilient estimation scenario. Split
Method
VRMSE𝑥
VRMSE𝑦
VRMSE𝑧
Improvement over LS (%)
S1
Classical LS BeamsNetV1 DVL-DeepONet-I
0.088 0.082 0.070
0.090 0.088 0.077
0.023 0.021 0.017
N/A 05 18
S2
Classical LS BeamsNetV1 DVL-DeepONet-I
0.088 0.084 0.060
0.089 0.084 0.062
0.023 0.021 0.012
N/A 04 31
S3
Classical LS BeamsNetV1 DVL-DeepONet-I
0.086 0.081 0.071
0.087 0.082 0.088
0.024 0.021 0.011
N/A 06 14
Avg. Improvement
21
Table 9 Summary of the three DVL-DeepONet configurations with the best VRMSE improvement achieved in each scenario. Scenario Noise-Resilient Estimation DVL-Only Learning Beams Measurement Recovery
VRMSE ↓
Best Gain (%) ↑
0.105 0.096 0.114
18 68 92
gains across all splits demonstrate that the proposed model is a reliable solution for practical underwater navigation scenarios.
4.8. Summary Figure 5: Ablation study of the influence of temporal window size 𝑊 on the VRMSE of DVL-DeepONet in the noise-resilient estimation scenario.
each fold, nine trajectories are used for training, two trajectories for validation, and two for testing, corresponding to an approximate 70%-15%-15% train-validation-test split. Fig. 6 shows the GT trajectories used as unseen test missions in the three-fold cross-validation study. Table 8 incorporates cross-validation results of the proposed DVL-DeepONet-I framework in the noise-resilient estimation scenario. While BeamsNet provides only marginal gains of 5% on average, over the LS estimator, DVL-DeepONet-I achieves substantially larger improvements of 21%. The consistent
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The proposed DVL-DeepONet framework was evaluated in three challenging underwater navigation scenarios and demonstrated supervisor performance against other model-based and data-driven approaches as shown in Table 9. For the noisy IMU/DVL scenario, DVLDeepONet-I achieved average performance improvements of 15% over the baselines. Similarly, in the DVL-only scenario, the average gain by LS-DeepONet-II is 38%. Finally, under the measurement recovery, our model registered 65% improvement. Moreover, across all scenarios, the proposed operator-learning framework consistently outperformed both model-based approaches and conventional deep-learning baselines. Additionally, it maintains around 90% agreement with the GT, as evidenced by mean 𝑅2 values.
DVL-DeepONet
(a) Trajectory 2
(b) Trajectory 3
(c) Trajectory 4
(d) Trajectory 5
(e) Trajectory 9
(f) Trajectory 10
Figure 6: GT trajectories used as unseen test missions in the three-fold cross-validation study.
5. Conclusion Existing DVL-based velocity estimation methods either rely on model-based LS estimators or employ purely data-driven models. Moreover, the model-based algorithms become unreliable under beam outages and rank-deficient measurement configurations, and data-driven models are black-box in nature and do not explicitly enforce the physical relationship between DVL beam measurements and AUV velocity. To address these limitations, this paper proposed DVLDeepONet, a physics-guided operator-learning framework for velocity estimation of AUV. The proposed method learns a nonlinear mapping from inertial/DVL observations to vehicle velocity and incorporates the DVL observation model as a physics-based consistency constraint during training. Unlike conventional LS reconstruction, the framework does not require direct inversion of the beam geometry matrix and therefore remains applicable even in beam outrages scenarios. Furthermore, the integration of physical constraints improves interpretability and preserves consistency between the predicted velocity vector and the measured DVL beams. As a result, DVL-DeepONet provides robust velocity estimation under noisy measurements, partial beam availability, and degraded beam conditions.
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The proposed framework has been evaluated using approximately 10,000 m of real-world AUV experimental data collected during multiple sea trials. Experimental results demonstrated that, DVL-DeepONet-I achieved average performance improvements of approximately 15% under noisy inertial/DVL measurements, DVL-DeepONet-II achieved improvements of about 38% in the DVL-only scenario, and DVL-DeepONet-III gains exceeding 65% under partial measurement availability. Overall, the proposed framework achieves an average improvement of approximately 40% compared with the baseline methods. These results demonstrate that the proposed operator-learning frameworks successfully handles multiple practical operational scenarios using a unified architecture. The study further shows that integrating physics constraints within DeepONet architectures improves accuracy, robustness and generalization. Moreover, the results indicate that physically informed operator learning can serve as a viable alternative to purely model-based and purely data-driven approaches for underwater navigation tasks. Nonetheless, several limitations warrant consideration. First, DVL-DeepONet requires a minimum temporal window size of two samples, whereas the model-based LS estimator can operate using only the current measurement. Second, the training stage involves multiple
DVL-DeepONet
CNN and MLP components, which require GPU acceleration or higher computational power. However, the trained model remains lightweight during inference and is suitable for real-time deployment on AUVs with limited onboard computational resources. DVL-DeepONet offers a white-box solution for resilient underwater navigation, improving mission safety, and operational effectiveness under degraded sensing conditions. Since the imposed physical constraints originate from the DVL measurement process itself and are independent of a particular operating region, the proposed framework is expected to generalize more effectively across diverse underwater environments than purely data-driven approaches. Future work will focus on extending the proposed framework to full navigation-state estimation (position, velocity, and orientation) and integrating uncertaintyaware learning strategies.
Conflict of Interest Statement The authors confirm that they have no conflicts of interest related to this paper.
Funding Declaration The authors confirm that they did not receive any funding to carry out this work.
Data Availability The dataset used in this study is publicly available at: https://github.com/ansfl/A-KIT/tree/main.
Code Availability The source code associated with this study is publicly available at: https://github.com/ansfl/DVL -DeepONet.
CRediT authorship contribution statement Arup Kumar Sahoo: Conceptualization, Methodology, Software, Validation, Investigation, Writing - original draft. Itzik Klein: Supervision, Methodology, Writing - review & editing.
References [1] L. Paull, S. Saeedi, M. Seto, H. Li, AUV navigation and localization: A review, IEEE Journal of Oceanic Engineering 39 (1) (2013) 131–149.
Page 14 of 15
[2] Y. Zhang, H. Zhang, J. Liu, S. Zhang, Z. Liu, E. Lyu, W. Chen, Submarine pipeline tracking technology based on AUVs with forward looking sonar, Applied Ocean Research 122 (2022) 103128. [3] P. Groves, Principles of GNSS, Inertial and Multi-Sensor Integrated Navigation Systems, Artech House, UK, 2013. [4] D. Titterton, J. L. Weston, Strapdown inertial navigation technology, Vol. 17, IET, 2004. [5] S. Wadoo, Autonomous underwater vehicles: modeling, control design and simulation, CRC press, 2017. [6] B. Braginsky, A. Baruch, H. Guterman, Correction of DVL error caused by seafloor gradient, IEEE Sensors Journal 20 (19) (2020) 11652–11659. [7] J. L. Farrell, GNSS aided navigation & tracking: inertially augmented or autonomous, American Literary Press Baltimore, Maryland, 2007. [8] A. K. Sahoo, I. Klein, PiDR: Physics-informed inertial dead reckoning for autonomous platforms, arXiv preprint arXiv:2601.03040 (2026). [9] B. Zhang, D. Ji, S. Liu, X. Zhu, W. Xu, Autonomous underwater vehicle navigation: A review, Ocean Engineering 273 (2023) 113861. [10] D. Wang, X. Xu, Y. Yao, T. Zhang, Y. Zhu, A novel SINS/DVL tightly integrated navigation method for complex environment, IEEE Transactions on Instrumentation and Measurement 69 (7) (2019) 5183–5196. [11] D. Engelsman, I. Klein, Information-aided inertial navigation: A review, IEEE Transactions on Instrumentation and Measurement 72 (2023) 1–18. [12] S. Cheng, Y. Wang, Q. Zhao, H. Zhu, X. Qu, A robust INS/USBL/DVL integrated navigation method based on adaptive correlation entropy factor graph optimization, Ocean Engineering 356 (2026) 125234. [13] H. Zhang, C. Li, T. Zhang, G. Wang, D. Wang, Novel algorithm for the calibration of DVL in underwater integrated navigation system, Ocean Engineering 353 (2026) 124676. [14] N. Cohen, I. Klein, BeamsNet: A data-driven approach enhancing Doppler velocity log
DVL-DeepONet
measurements for autonomous underwater vehicle navigation, Engineering Applications of Artificial Intelligence 114 (2022) 105216. [15] G. Damari, I. Klein, ResAlignNet: A datadriven approach for INS/DVL alignment, Ocean Engineering 356 (2026) 125277. [16] M. Batoš, Ð. Nađ, DMIAN: deep learningbased multi-IMU fusion for enhanced marine aided navigation, Control engineering practice 173 (2026) 106991. [17] L. Kang, K. He, J. Zhao, X. Wang, P. Tan, A hybrid-kernel-based adaptive robust Kalman filter for INS/DVL integrated underwater navigation, Ocean Engineering 350 (2026) 124269. [18] H. Mo, H. Yang, Y. Zhang, D. Pan, G. Yang, W. Li, A hybrid physics–data-driven navigation method for AUVs fusing hydrographic information with INS/DVL integration, IEEE Sensors Journal (2026). [19] Z. Yampolsky, I. Klein, DCNet: A datadriven framework for DVL calibration, Applied Ocean Research 158 (2025) 104525. [20] F. Zhang, S. Zhao, L. Li, C. Cao, Underwater DVL optimization network (UDON): A learning-based DVL velocity optimizing method for underwater navigation, Drones 9 (1) (2025) 56. [21] N. Cohen, I. Klein, LiBeamsNet: AUV velocity vector estimation in situations of limited DVL beam measurements, in: OCEANS 2022, Hampton Roads, IEEE, 2022, pp. 1–5. [22] M. Yona, I. Klein, MissBeamNet: Learning missing Doppler velocity log beam measurements, Neural Computing and Applications 36 (9) (2024) 4947–4958. [23] Y. Miao, X. Liu, Y. Sun, X. Liu, C. Shen, C. Wang, J. Tang, J. Liu, Physics-guided adaptive UKF for robust AUV integrated navigation under degraded underwater observations, Ocean Engineering 362 (2026) 126542. [24] X. Mu, B. He, X. Zhang, Y. Song, Y. Shen, C. Feng, End-to-end navigation for autonomous underwater vehicle with hybrid recurrent neural networks, Ocean Engineering 194 (2019) 106602. [25] N. Cohen, I. Klein, Seamless underwater navigation with limited Doppler velocity log measurements, IEEE Transactions on Intelligent Vehicles (2024). Page 15 of 15
[26] L. Lu, P. Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators, Nature Machine Intelligence 3 (2021) 218–229. [27] S. Chakraverty, A. K. Sahoo, D. Mohapatra, Artficial Neural Networks and Type-2 Fuzzy Set: Elements of Soft Computing and Its Applications, Elsevier, 2025. [28] C. Tan, Y. Cai, H. Wang, L. Chen, Y. Lian, Modeling vehicle dynamics with physicsinformed deep operator network, Vehicle System Dynamics (2025) 1–29. [29] N. A. Brokloff, Matrix algorithm for Doppler sonar navigation, in: Proceedings of OCEANS’94, Vol. 3, IEEE, 1994, pp. III–378. [30] P. Liu, B. Wang, Z. Deng, M. Fu, INS/DVL/PS tightly coupled underwater navigation method with limited DVL measurements, IEEE Sensors Journal 18 (7) (2018) 2994–3002. [31] A. Shurin, A. Saraev, M. Yona, Y. Gutnik, S. Faber, A. Etzion, I. Klein, The autonomous platforms inertial dataset, IEEE Access 10 (2022) 10191–10201. [32] ECA Group, A18-D AUV: Autonomous Underwater Vehicle, https://www.ecagroup.com/en/so lutions/a18-d-auv-autonomous-underwater-vehicle, accessed: Dec. 2025 (2023). [33] iXblue, PHINS Subsea, https://www.ixblue.c om/store/phins- subsea/, accessed: Dec. 2025 (2023). [34] Teledyne Marine, Doppler Velocity Logs, http s://www.teledynemarine.com/products/product-lin e/navigation-positioning/{Doppler}-velocity-logs,
accessed: Dec. 2025 (2023).