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

Arup Kumar Sahoo, Itzik Klein

arXiv:2610.09690v1cs.ROcs.LG

TL;DR

AUV velocity estimation remains difficult when DVL measurements are noisy or beams are missing, while common learning objectives omit physical consistency and measurement uncertainty. NAViLoss combines velocity and beam-consistency residuals with bounded penalties and geometry-aware uncertainty, and is integrated with DeepONet. On semi-synthetic sea-trial data, NAVi-DeepONet reports an average improvement of approximately 44% over baselines.

  • Problem

    Learning-based objectives commonly omit DVL physical consistency and measurement uncertainty, despite variable information from noisy or missing beams.

    Source: §1 · p. 2

  • Method

    NAViLoss combines velocity-state and DVL beam-consistency residuals with bounded penalties and geometry-aware uncertainty, and is integrated with DeepONet.

    Source: §3.1 · p. 5, §3.2 · p. 6, §3.4 · p. 9

  • Results

    Approximately 44% average improvement over baselines is reported for NAVi-DeepONet across full-beam and missing-beam conditions.

    Source: §5 · p. 20, §4.7 · p. 20

  • Takeaways & Limitations

    NAVi-DeepONet provides a learning framework for velocity estimation across complete and degraded DVL measurement conditions.

    Source: §4.7 · p. 20

    The uncertainty formulation reflects beam-geometry informativeness rather than all sensor uncertainties, and generalizability across platforms and environments remains unestablished.

    Source: §5 · p. 20

Abstract

from arXiv · show

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.

1. Introduction

The introduction frames robust AUV velocity estimation under degraded DVL measurements as an open challenge, and proposes NAViLoss to combine navigation-state accuracy with beam consistency, robustness, and geometry-aware uncertainty. The loss is integrated with DeepONet and evaluated on semi-synthetic sea-trial data.

1. Introduction

  • Noisy or incomplete DVL measurements remain a challenge because missing beams alter measurement geometry and reduce information for estimating three-dimensional vehicle velocity.DVL measurements help constrain drift accumulated by inertial navigation systems.Source: §1 · pp. 1–2
  • Existing regression losses either penalize large residuals strongly or operate mainly in prediction space without explicitly exploiting the physical DVL observation model.Under partial beam availability, a learning objective also needs to account for changes in measurement uncertainty.Source: §1 · pp. 2–3
  • NAViLoss jointly penalizes navigation-state and DVL beam-measurement residuals, bounds large-residual influence, and adapts beam-information contribution to available measurement geometry.The paper also establishes analytical properties including small-residual quadratic behavior, uncertainty-aware attenuation, and dual-domain consistency.Source: §1 · p. 3
  • The authors integrate NAViLoss with DeepONet to form two NAVi-DeepONet variants and make the codebase publicly available for reproduction and benchmarking.The paper identifies its analytical loss properties and model integration as contributions.Source: §1 · p. 3
  • 44% average improvement over considered baseline methods is reported for NAVi-DeepONet on semi-synthetic sea-trial data covering approximately 10,000m.The evaluation includes noisy sensing, DVL beam outages, and reduced measurement availability.Source: §1 · p. 3

2. Problem Formulation

The problem formulation connects vehicle velocity to DVL beam measurements through the observation geometry, while identifying shortcomings in velocity-only and conventional physics-guided regression objectives. Partial or corrupted beam data make the fixed weighting and large-residual sensitivity of those objectives problematic.

2. Problem Formulation

  • A four-beam DVL can lose beam measurements when obstacles obstruct acoustic paths, producing partial beam availability during vehicle motion.The remaining beams provide valid bottom-lock measurements.Source: §2 · p. 3, Figure 1 · p. 4
  • Conventional supervised regression minimizes velocity residuals but does not explicitly enforce consistency with the underlying DVL measurements.Its quadratic structure can let residuals from corrupted measurements or sensor outliers dominate optimization.Source: §2 · p. 4
  • The DVL observation model maps predicted vehicle velocity through the beam geometry matrix to predicted velocities for the available beams.The beam-space residual compares predicted and measured DVL beam velocities, with the number of available beams denoted m ≤ 4.Source: §2 · p. 4
  • Conventional physics-guided regression adds measurement consistency but remains sensitive to large residuals and fixes the relative beam-space importance despite changing measurement confidence.Corrupted or partially available beams can change the information carried by the available geometry.Source: §2 · pp. 4–5

3. Proposed Approach

This section develops NAViLoss and pairs it with a physics-guided DeepONet for AUV velocity estimation. It describes the loss’s properties, model architecture, sensing configurations, and training procedure.

3.1. Motivation and Overview

  • NAViLoss combines velocity-estimation and DVL beam-consistency residuals, using a bounded penalty and adapting beam-information contribution to DVL geometry.The objective jointly accounts for prediction accuracy, physical consistency, robustness to large residuals, and beam-domain uncertainty.Source: §3.1 · p. 5, §3.2 · p. 6

3.2. NAViLoss Function

  • The objective takes velocity and beam residuals with their respective uncertainty measures, jointly minimizing state-estimation and measurement-consistency errors.A larger uncertainty reduces the contribution of its associated residual during optimization.Source: §3.2 · pp. 5–6

3.3. Theoretical Properties of NAViLoss

  • NAViLoss is non-negative, bounded, and smooth; its penalty stays bounded and influence vanishes for large residuals, while small residuals receive locally quadratic treatment.Increasing uncertainty attenuates both penalty and influence; a zero-loss solution is consistent in both navigation-state and DVL measurement domains.Source: §3.3 · pp. 6–7

3.4. NAVi–DeepONet Architecture

  • NAVi–DeepONet combines branch features from synchronized DVL and IMU windows with a trunk representation of temporal coordinates to estimate three-dimensional AUV velocity.The higher-rate IMU measurements are aligned to DVL epochs; the branch input contains four DVL beam measurements and six IMU channels over a temporal window.Source: §3.4 · pp. 8–9

3.5. DVL Operation mode

  • The framework considers noise-aware velocity estimation with complete DVL and IMU measurements, and DVL measurement recovery with partial beams and complete IMU measurements.The recovery setting tests operation as unavailable beams leave progressively less information about three-dimensional velocity.Source: §3.5 · p. 10, §3.5.2 · p. 10

3.5.1. Noise-Aware Velocity Estimation

  • For noise-aware velocity estimation, DVL and IMU histories are encoded independently, fused into branch features, combined with the trunk representation, and passed to a velocity prediction head.The nominal configuration uses all four DVL beams and complete IMU measurements.Source: §3.5.1 · p. 10

3.5.2. DVL Measurement Recovery

  • For DVL recovery, the model encodes partial DVL history alongside the full IMU history, fuses the representations, and predicts AUV velocity.The setup uses a beam-availability mask and tests progressively reduced DVL geometry.Source: §3.5.2 · pp. 10–11

3.6. Training

  • Training derives beam uncertainty from active-beam geometry: weaker observed directions receive larger uncertainty and consequently less influence in the beam-space loss.Fewer than three available beams receive a predefined large uncertainty; velocity uncertainty is fixed at 1 in these experiments, while DVL uncertainty varies with geometry.Source: §3.6 · pp. 11–12

4. Analysis and Results

Across noisy sensing and reduced DVL availability, NAVi-DeepONet performs well, with its benefits depending on the available beam geometry. The section evaluates the dataset, model performance, ablations, cross-validation, and computational complexity.

4.1. AUV Dataset

  • The semi-synthetic dataset uses Snapir sea-trial measurements and thirteen trajectories totaling 10,000m, with controlled perturbations applied to generate unit-under-test measurements.The beam-noise simulation uses 0.7% scale-factor error, 0.001m/s beam bias, and 0.042m/s additive-noise standard deviation.Source: §4.1 · pp. 12–14, Figure 5 · p. 14

4.3. Noise-Aware Velocity Estimation

  • Under noisy measurements, NAVi-DeepONetV1 records the lowest VRMSE (0.082m/s) and VMAE (0.071m/s), and the highest mean R2 and VAF.It outperforms LS, BeamsNet, and DVL-DeepONet by 54%, 29%, and 11%, respectively.Source: §4.3 · p. 16, Table 3 · p. 16

4.4. DVL Measurement Recovery

  • NAVi-DeepONetV2 achieves best overall performance, reducing VRMSE and VMAE to 0.175 m/s and 0.142 m/s, respectively, approximately 31% and 37% below next-best DVL-DeepONet.Mean R2 increases from 0.312 to 0.694, while VAF improves from 48% to 72%.Source: §4.4 · p. 16

4.5.1. Effect of Temporal Window Length

  • A two-step temporal window gives the best missing-beam result, with VRMSE of 0.175m/s; larger windows yield 0.188–0.207m/s.The selected W = 2 corresponds to a two-second window in this dataset.Source: §4.5.1 · p. 17, Table 5 · p. 17

4.5.2. Effect of Inertial Data

  • Adding inertial data improves performance by 9% in noisy velocity estimation and 38% in DVL measurement recovery, with greater benefit under degraded beam geometry.The reported improvements compare the model with DVL-only learning.Source: §4.5.2 · p. 17, Table 6 · p. 17, Table 7 · p. 18

4.6. Model Complexity

  • NAVi-DeepONetV2 uses approximately 75% more parameters than V1, but activates only the branch matching the observed beam pattern, keeping MACs and FLOPs comparable.The authors report real-time computational feasibility for the proposed framework.Source: §4.6 · p. 18, Table 9 · p. 18

4.5.3. Three-Fold Cross Validation

  • Across six evaluations per method, NAVi-DeepONetV2 maintains its advantage across folds and random initializations, with lower mean VRMSE as beam outages increase.The six evaluations use three folds and two random seeds; Figure 7 compares performance with one, two, and three missing beams.Source: §4.5.3 · p. 18, Figure 7 · p. 19

4.7. Summary

  • With full-beam inputs, NAVi-DeepONetV1 improves VRMSE by at least 11%; in measurement recovery, NAVi-DeepONetV2 improves VRMSE by at least 31% and VMAE by at least 37% versus baselines.The reported gains under reduced beam availability are more pronounced than with complete DVL geometry; the authors attribute this to NAViLoss adapting beam-domain information to available geometry.Source: §4.7 · pp. 19–20

5. Conclusion

NAViLoss combines bounded navigation-state and DVL beam residuals with geometry-adaptive uncertainty handling. NAVi-DeepONet improves velocity-estimation accuracy by approximately 44%, while generalizability across platforms and environments remains untested.

5. Conclusion

  • NAViLoss combines bounded navigation-state and DVL beam residuals with uncertainty weighting that adapts to available beam geometry.The objective is designed for nominal and degraded DVL measurement conditions.Source: §5 · p. 20
  • Approximately 44% average improvement over baselines was achieved by the proposed NAVi-DeepONet model.Under full-beam measurements, NAVi-DeepONetV1 improves accuracy by an average of 31%; under missing-beam conditions, NAVi-DeepONetV2 improves by 56%.Source: §5 · p. 20
  • The uncertainty formulation reflects geometric measurement informativeness, parameters are selected empirically, and cross-platform and cross-environment generalizability remains to be investigated.The study identifies different AUV platforms, DVL geometries, seabed conditions, and acoustic environments as validation targets.Source: §5 · p. 20

Funding Declaration

The authors declare that they received no funding for this work.

Funding Declaration

A. Proof of Proposition 1

The proof establishes that the NAViLoss residual terms and their weighted sum are smooth and nonsingular.

A. Proof of Proposition 1

  • Strictly positive uncertainty denominators make the residual terms nonsingular and infinitely differentiable, and their weighted sum is smooth.Source: A. Proof of Proposition 1 · p. 21

A.1. Proof of Lemma 1

The proof characterizes the penalty's bounded and redescending behavior for large residuals.

A.1. Proof of Lemma 1

  • For large |r|, the penalty approaches the finite upper bound 1/λ while the residual's gradient contribution vanishes.Source: §A.1 · p. 22

A.2. Proof of Lemma 2

The passages state that the penalty has locally quadratic behavior and that increasing uncertainty attenuates the penalty for a fixed nonzero residual.

A.2. Proof of Lemma 2

  • The penalty exhibits the locally quadratic behavior stated in Lemma 2.Source: §A.2 · p. 22
  • For any fixed nonzero residual, the penalty strictly decreases as the uncertainty parameter increases.The passage also states that this attenuation proves the result.Source: §A.2 · p. 23

A.4. Proof of Proposition 2

The proof establishes that the NAViLoss component losses vanish exactly when their corresponding residuals are zero.

A.4. Proof of Proposition 2

  • Each component loss vanishes if and only if its corresponding residual is zero.Source: §A.4 · p. 23
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