Full-Field Reconstruction from Sparse Observations Using Multiscale Gaussian Kernel and Factorized Fourier Neural Operators

Summary

Reconstructing high-dimensional physical fields from sparse observations is a central challenge in subsurface monitoring, where dense measurements are often impractical or unavailable. Existing approaches typically require dense full-state supervision, limiting their applicability when only sparse measurements are available. We propose a reconstruction framework that is trained without exposure to dense grids: It learns directly from sparse observations by transforming them into continuous multiscale representations using a multiscale Gaussian kernel (MSGK) and mapping them to full fields using separable spectral layers via a factorized Fourier neural operator (FFNO). Training is performed solely by enforcing consistency at observed locations, eliminating the need for fully observed ground-truth fields. We evaluate the framework on subsurface waterflooding and carbon dioxide (CO2) injection, together with near-surface air-temperature fields. Across all cases, the method consistently recovers the dominant spatial patterns and remains robust when tested at masking ratios outside the training regime. Compared with baseline reconstruction methods, the proposed approach shows greater resilience under highly noisy observations. These findings demonstrate the promising potential of the proposed framework within controlled, synthetic data settings, where the fields and noise levels are artificially constructed. Although the present study is limited to simplified 2D domains and conceptual masking patterns that may not fully reflect well-placement configurations in many oilfields, the proposed framework provides opportunities for further improvement through artificial intelligence model architectural refinements, enhanced loss formulations, and adaptive selection of MSGK parameters for realistic well layouts.