Physics-informed Diffusion Generation for Geomagnetic Map Interpolation

Geomagnetic map interpolation aims to infer unobserved geomagnetic data at spatial points, yielding critical applications in navigation and resource exploration. However, existing methods for scattered data interpolation are not specifically designed for geomagnetic maps, which inevitably leads to suboptimal performance due to detection noise and the laws of physics. Therefore, we propose a Physics-informed Diffusion Generation framework~(PDG) to interpolate incomplete geomagnetic maps. First, we design a physics-informed mask strategy to guide the diffusion generation process based on a local receptive field, effectively eliminating noise interference. Second, we impose a physics-informed constraint on the diffusion generation results following the kriging principle of geomagnetic maps, ensuring strict adherence to the laws of physics. Extensive experiments and in-depth analyses on four real-world datasets demonstrate the superiority and effectiveness of each component of PDG.

💡 Research Summary

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The paper introduces a novel framework called Physics‑informed Diffusion Generation (PDG) for the task of geomagnetic map interpolation, a problem that requires estimating magnetic field intensities at unobserved spatial locations based on scattered measurements. Traditional scattered‑data interpolation methods (e.g., Kriging, inverse‑distance weighting) assume clean data and struggle with large‑scale noisy datasets, while recent deep learning approaches such as Neural Processes, attention‑based models, and Transformers excel at capturing complex correlations but often violate the smoothness and continuity inherent to geomagnetic fields due to their strong non‑linearity.

PDG addresses these two shortcomings by integrating a conditional diffusion model with two physics‑aware components: a physics‑informed mask (PIM) and a Kriging‑guided physics‑informed loss (L_Kriging).

1. Conditional Diffusion Model – The authors adopt a Denoising Diffusion Probabilistic Model (DDPM) where the forward process gradually adds Gaussian noise to the true target magnetic values, and the reverse process learns to denoise. Observed data (coordinates and magnetic intensities) are encoded as conditional inputs, while the noisy target values serve as the generation target at each diffusion step.

2. Physics‑informed Mask (PIM) – At each diffusion step t, the receptive field size K(t) is dynamically adjusted. The mask selects the K(t) geographically nearest observed points for each target point, limiting the attention mechanism to these neighbors. Early diffusion steps use a large K(t) to capture coarse global structure; later steps shrink K(t) to focus on fine‑grained local patterns. This strategy suppresses the influence of distant noisy measurements and encourages the generation of spatially smooth fields.

3. Kriging‑guided Loss (L_Kriging) – Inspired by classical geostatistics, the loss enforces that the empirical variogram of the predicted field matches that of the ground truth. For each target location, the method computes the variogram difference with its t nearest neighbors and minimizes the squared error across all targets. This term directly penalizes violations of the spatial autocorrelation expected in geomagnetic data, complementing the standard mean‑squared error between predicted and true noise (L_ε). The total training objective is L = L_ε + λ·L_Kriging, where λ balances data fidelity and physical consistency.

Experimental Setup – Four real‑world datasets were collected from UAV flight paths over two cities (datasets A‑InX, A‑InZ, A‑OutZ, and B‑InT, representing different magnetic components). Each dataset was split 8:1:1 for training, validation, and testing and evaluated on an NVIDIA A800 GPU. Baselines included Conditional Neural Processes (CNP), Attentive Neural Processes (ANP), Bootstrapping ANP (BANP), NIER‑T, TFR‑Transformer, and HINT.

Results – PDG achieved an average error reduction of about 80% across all metrics (RMSE, MAE, MAPE, MSE). Notably, on the large, noisy B‑InT dataset PDG attained RMSE = 1.092 while all other methods either produced errors >23 or ran out of memory. Qualitative visualizations showed smoother, more physically plausible contour lines; competing methods exhibited spikes and abrupt value changes, especially in high‑gradient regions.

Ablation Studies – Removing the physics‑informed mask increased RMSE (e.g., from 0.552 to 0.609 on A‑InX) and degraded MAPE, confirming the mask’s role in noise mitigation. Excluding the Kriging loss similarly worsened performance, demonstrating the necessity of explicit spatial‑correlation regularization.

Parameter Sensitivity – Sampling steps between 5 and 20 offered the best trade‑off between accuracy and computational cost; beyond 30 steps, gains plateaued while runtime grew. The maximum neighborhood size K_max heavily influenced early‑stage interpolation quality, whereas the minimum size K_min had a modest effect on final refinement.

Conclusion – PDG successfully merges the generative power of diffusion models with domain‑specific physical constraints, delivering robust, noise‑resilient, and physically consistent geomagnetic map interpolation. The approach is readily extensible to other scattered physical fields (e.g., atmospheric pollutants, ocean temperature) where smooth spatial variation and measurement noise coexist. Future work may explore learned mask policies, lightweight variants for real‑time UAV streams, and broader geophysical applications.