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PINN-Aided Gravity Anomaly Map Matching for GNSS-Denied Inertial Navigation

Hojung Lim, Souvik Chatterjee, Kenneth Tseng  ·  AME 508 – Spring 2026

PINN Inertial Navigation EKF GNSS-Denied Gravity Anomaly Deep Learning PyTorch

Motivation & Objective

Reliable navigation in GNSS-denied environments is critical for autonomous systems and defense applications. Low-fidelity gravity models fail to capture local anomalies that carry precise positional information. The differentiable nature of PINNs makes them naturally compatible with EKF-based state estimation — eliminating the need for manual linearization.

This project aims to correct INS drift in GNSS-denied environments by training a Physics-Informed Neural Network (PINN) to learn the residual gravity anomaly δg. The PINN-predicted anomaly is compared against onboard gravimeter measurements to generate an innovation term in an Extended Kalman Filter (EKF), where the required Jacobian is computed via automatic differentiation of the PINN.

System Architecture

The proposed pipeline integrates a PINN with an INS and EKF framework. The PINN takes position coordinates (lat, lon, alt) as input and outputs the corresponding gravity anomaly δg through a fully connected network with:

  • 4 hidden layers of 64 neurons each
  • Tanh activation functions with skip connections
  • Jacobian H computed via automatic differentiation

EKF Predict Step

xk|k-1 = F · xk-1|k-1 Pk|k-1 = F · Pk-1|k-1 · Fᵀ + Q

EKF Update Step

vk = δgmeasured − δgPINN(xk|k-1) Kk = Pk|k-1 Hᵀ (H Pk|k-1 Hᵀ + R)⁻¹ xk|k = xk|k-1 + Kk vk Pk|k = (I − Kk H) Pk|k-1 H = ∇ₓ δgPINN (via automatic differentiation)

PINN Training

Training data: 32,768 position–gravity anomaly pairs synthesized from the GGMplus N30W120 tile over the USC campus region (34.02°N, 118.29°W), sampled on a 64×64 uniform grid at 8 altitude levels (0–200 m).

Loss Function

Ltotal = Ldata + λ · Lphysics Ldata — absolute + relative prediction error (prevents low-altitude bias) Lphysics — Laplacian regularization: ||∇²(δgPINN)|| Boundary condition: Δg(r) = ωNN(r)·ΔgNN(r) + ωBC(r)·ΔgBC(r)

Optimizer: Adam (lr = 2⁻⁸, batch size 512, λ = 10⁻³). Data split: 22,937 train / 4,915 val / 4,916 test.

Test 1 vs. Test 2 — Normalization Scope

Two PINN variants were evaluated to study the effect of normalization scope:

  • Test 1: StandardScaler fitted to the local USC subgrid (mean = −75.1 mGal, std = 8.1 mGal)
  • Test 2: StandardScaler fitted to the full GGMplus tile (mean = −105.9 mGal, std = 21.8 mGal) — introduces distribution mismatch

Results

PINN Performance

0.20%
Test 1 Relative Error
0.2434 mGal
Test 1 RMSE
0.19 mGal
USC Origin Prediction Error
0.30%
PINN-GM-III Benchmark
Model RMSE (mGal) MAE (mGal) Relative Error USC δg Prediction USC Error
Test 1 (local norm.) 0.2434 0.1383 0.20% −79.30 mGal 0.19 mGal
Test 2 (tile norm.) 0.5300 0.2821 0.31% −95.39 mGal 16.27 mGal
PINN-GM-III (benchmark) — — 0.30% — —

Test 1 outperforms both Test 2 and the PINN-GM-III paper benchmark. The EKF Jacobian H = [253.3, 3.22, 4.1×10⁻⁵] mGal/deg was successfully obtained via automatic differentiation — confirming suitability for EKF integration at sub-meter position resolution.

EKF Navigation Correction

2,957 m
Raw IMU Dead-Reckoning RMSE
53,181 m
Test 1 + EKF RMSE
26,547 m
Test 2 + Free-Air Correction + EKF RMSE

The EKF diverged in both implementations. The root cause is the USC campus gravity field being nearly flat: δg varies only 0.047 mGal over the 1.5 km path while gravimeter noise is 0.5 mGal (SNR = 0.09). Additionally, the gravity gradient along the East and Up axes is nearly zero (Hx = 2.5×10⁻⁵, Hz = 4.1×10⁻⁵ mGal/m), driving the Kalman gain K ≈ 0 in those directions.

Applying a free-air altitude correction (−0.3086 mGal/m) to the training data and retraining reduced the EKF RMSE from 53,181 m to 26,547 m — demonstrating improved vertical observability through altitude-dependent gravity variation.

Conclusions

This project demonstrated a PINN-based approach to gravity-aided inertial navigation in GNSS-denied environments. Key findings:

  • The trained PINN achieved 0.20% gravity anomaly prediction error, surpassing the PINN-GM-III benchmark of 0.30%.
  • Normalization must be fitted to the local operating region: locally-normalized Test 1 achieved 16× lower prediction error than the tile-normalized variant.
  • EKF-based navigation correction diverged due to insufficient gravity signal variation (SNR = 0.09) and an unobservable system along East and Up axes.
  • Free-air altitude correction reduced EKF RMSE by ~50%, confirming improved vertical observability.

Future Work

  1. Redesign the trajectory over terrain with higher gravity gradients to satisfy EKF observability.
  2. Adopt navigation-grade gravimeter noise (~0.01 mGal) to raise SNR above 1.
  3. Unify the EKF state in local ENU coordinates and calibrate Q to match actual gravity SNR.
  4. Simulate a gradiometer instead of a scalar gravimeter to measure gravity gradients in all three directions.
  5. Aid the PINN with other non-GNSS sensors to bring the observability matrix to full rank.

References

  1. X. Wang, C. Gilliam, A. Kealy, J. Close, and B. Moran, "Probabilistic Map Matching for Robust Inertial Navigation Aiding," NAVIGATION, vol. 70, no. 2, 2023. doi:10.33012/navi.583
  2. J. Martin and H. Schaub, "The Physics-Informed Neural Network Gravity Model: Generation III," Journal of the Astronautical Sciences, 2025. arXiv:2312.10257
  3. C. Tan et al., "Vehicle State Estimation Combining Physics-Informed Neural Network and Unscented Kalman Filtering on Manifolds," Sensors, vol. 23, 2023. doi:10.3390/s23156665

Project Documentation

For more details, view the full report and presentation slides:

📄 Project Report (PDF) 📊 Presentation Slides (PPTX)