NAViLoss: An Underwater Navigation-Aware Dual-Residual Objective for Physics-Consistent Learning

Arup Kumar Sahoo; Itzik Klein

cs.ROOct 7, 2026

NAViLoss: An Underwater Navigation-Aware Dual-Residual Objective for Physics-Consistent Learning

Authors: Arup Kumar Sahoo, Itzik Klein

Organizations: The Hatter Department of Marine Technologies, Leon H. Charney School of Marine Sciences, University of Haifa, Haifa, 3498838, Israel

Abstract

Autonomous underwater vehicles (AUVs) commonly rely on inertial navigation systems (INS) aided by Doppler velocity logs (DVLs) for reliable underwater navigation. Accurate DVL velocity estimation is therefore essential for successful operation. Recent learning-based methods have demonstrated improved DVL velocity estimation, particularly under degraded measurement conditions. However, their training objectives typically rely on conventional regression losses that are highly sensitive to large residuals and corrupted observations. Additionally, they do not explicitly account for the physical consistency and measurement uncertainty associated with the underlying sensing process. To address these limitations, this paper introduces navigation-aware loss (NAViLoss), a robust and uncertainty-aware objective function for learning-based AUV velocity estimation. NAViLoss jointly penalizes the velocity-estimation residual in the navigation-state domain and the beam-consistency residual in the DVL measurement domain. Its bounded formulation limits the influence of large residuals, while an adaptive mechanism regulates the uncertainty in beam geometry. Furthermore, NAViLoss is integrated with a DeepONet architecture to form a novel NAVi-DeepONet model for seamless estimation of an underwater vehicle's velocity. Lastly, our model is evaluated using approximately 10,000m of semi-synthetic AUV experimental data collected during multiple real-world sea trials. Experimental results demonstrate a 44% improvement in velocity-estimation accuracy compared with conventional and learning-based baselines. These results demonstrate the effectiveness of navigation-aware and uncertainty-adaptive loss design for robust learning-based underwater velocity estimation.

Figures & tables

Figure 1 : Illustration of an AUV maneuvering with a DVL under full and partial beam availability. The red beams indicate DVL measurements obstructed by underwater obstacles, while the remaining beams represent valid bottom-lock measurements during vehicle motion.

Figure 2 : Comparison of NAViLoss with representative regression and robust loss functions. Left: penalty functions $\rho(r)$ for residual $r$ . Right: corresponding influence functions.

Figure 3 : Effect of the uncertainty parameter $\sigma$ on NAViLoss for fixed $\lambda=1$ . Left: penalty function $\rho(r;\sigma)$ for different uncertainty levels. Right: corresponding influence function.

Figure 4 : Architecture of the proposed NAVi-DeepONet framework.

Parameter	Noise-Aware Velocity Estimation	DVL Measurement Recovery
Window length ( $W$ )	2	2
Input features (DVL + IMU)	10	10
Number of branch networks	1	15
Branch kernel size	3	3
Branch CNN channels	$10$ – $32$ – $64$ – $64$ – $32$	$10$ – $32$ – $64$ – $64$ – $128$
Trunk MLP	$1$ – $64$ – $128$ – $32$	$1$ – $64$ – $128$ – $128$

Table 1 : Model architecture and hyperparameters employed for the training of NAVi-DeepONet.

Figure 5 : Pipeline for simulating noisy DVL beam measurements from the GT velocity.

Fold	Train. Traj.	Val. Traj.	Test Traj.	Train Dist. [m]	Val. Dist. [m]	Test Dist. [m]
F1	4,5,6,7,8,9,10,11,12	1,13	2,3	7259.75	1496.86	1346.59
F2	1,6,7,8,9,10,11,12,13	2,3	4,5	7094.26	1346.59	1566.93
F3	1,2,3,4,9,10,11,12,13	5,6	7,8	6683.41	1637.56	1686.80

Table 2 : Three-fold cross-validation scheme used for NAVi-DeepONet evaluation.

Panel of Figure 6 — Figure 6 : Three-dimensional GT paths of the test missions, with their corresponding North–East projections shown on the horizontal plane.

Model	VRMSE [m/s] $\downarrow$	VMAE [m/s] $\downarrow$	Mean $R^{2}$ $\uparrow$	VAF [%] $\uparrow$	VRMSE Gain [%] $\uparrow$
LS	0.178	0.161	0.507	61.37	54
BeamsNet	0.116	0.104	0.706	70.64	29
DVL-DeepONet	0.092	0.074	0.869	87.21	11
NAVi-DeepONetV1 (ours)	0.082	0.071	0.878	88.53	–

Table 3 : Performance comparison for noise-aware velocity estimation.

Model	VRMSE [m/s] $\downarrow$	VMAE [m/s] $\downarrow$	Mean $R^{2}$ $\uparrow$	VAF [%] $\uparrow$	VRMSE Gain [%] $\uparrow$
ELC	1.991	1.696	-134.008	-1816.87	91
MissBeamNet	0.329	0.288	-0.403	-3.36	47
DVL-DeepONet	0.254	0.224	0.312	48.45	31
NAVi-DeepONetV2 (ours)	0.175	0.142	0.694	72.19	–

Table 4 : Performance comparison under DVL measurement recovery.

Window Length	VRMSE [m/s] $\downarrow$
2	0.175
3	0.182
4	0.221
5	0.197
10	0.188
15	0.207

Table 5 : Effect of temporal window length on NAVI-DeepONetV2.

Model	VRMSE [m/s] $\downarrow$	VMAE [m/s] $\downarrow$	Mean $R^{2}$ $\uparrow$	VAF [%] $\uparrow$	VRMSE Gain [%] $\uparrow$
NAVi-DeepONet (DVL only)	0.090	0.080	0.814	81.95	9
NAVi-DeepONetV1 (ours)	0.082	0.071	0.878	88.53	–

Table 6 : Effect of inertial information for noise-aware velocity estimation.

Model	VRMSE [m/s] $\downarrow$	VMAE [m/s] $\downarrow$	Mean $R^{2}$ $\uparrow$	VAF [%] $\uparrow$	VRMSE Gain [%] $\uparrow$
NAVi-DeepONet (DVL only)	0.282	0.243	0.188	41.95	38
NAVi-DeepONetV2 (ours)	0.175	0.142	0.694	72.19	–

Table 7 : Effect of inertial information for DVL measurement recovery.

Method	VRMSE [m/s] $\downarrow$	MAE [m/s] $\downarrow$	Median Error [m/s] $\downarrow$	Maximum Error [m/s] $\downarrow$	$R^{2}$ $\uparrow$	VAF [%] $\uparrow$
ELC	$1.951\pm 0.130$	$1.646\pm 0.121$	$0.921\pm 0.015$	3.858	-65.527	-1099.31
MissBeamNet	$0.329\pm 0.012$	$0.279\pm 0.015$	$0.244\pm 0.023$	1.414	0.110	27.18
DVL-DeepONet	$0.256\pm 0.028$	$0.218\pm 0.025$	$0.190\pm 0.027$	1.064	0.521	59.79
NAVi-DeepONet (DVL only)	$0.293\pm 0.025$	$0.239\pm 0.012$	$0.193\pm 0.010$	1.361	0.416	49.33
NAVi-DeepONetV2 (ours)	$\mathbf{0.235\pm 0.033}$	$\mathbf{0.193\pm 0.020}$	$\mathbf{0.165\pm 0.015}$	1.163	0.679	68.85

Table 8 : Robustness analysis for DVL measurement recovery over three runs using two random seeds.

Figure 7 : Velocity estimation performance under increasing DVL beam outages.

Configuration	Trainable	Model Size	MACs	FLOPs	Mean	P95	Throughput	Real-Time
	Parameters	[MB]	[M]	[M]	[ms]	[ms]	[samples/s]	Feasible
NAVi-DeepONetV1	161,955	0.64	2.97	5.93	0.390	0.413	2564	Yes
NAVi-DeepONetV2	675,619	2.72	2.98	5.96	0.687	0.722	14545	Yes

Table 9 : Computational complexity of the proposed NAVi-DeepONet configurations.

Variant	VRMSE [m/s] $\downarrow$	Best Gain [%] $\uparrow$
NAVi-DeepONetV1	0.082	54
NAVi-DeepONetV2	0.175	91

Table 10 : Summary of the proposed model with the best VRMSE gain.

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