Self-Adapting PDE Field Reconstruction from Sparse Measurements

We present a self-adapting method for reconstructing physical fields from sparse measurements. The method discovers the dominant structural components of a field directly from measurement data, without knowing the governing equation. It automatically determines its own structural complexity — resolution level and number of components — through convergence detection, requiring no domain-specific configuration. We validate on four datasets: synthetic PDEs, simulated heat conduction and fluid dynamics (PDEBench), real thermal camera images (FLIR), and real Particle Image Velocimetry from an optical engine (EngineBench). The method achieves R² = 1.000 on synthetic PDEs from 20 points, needs 2–4× fewer measurements than k-nearest neighbor interpolation on heat and fluid fields, and achieves R² = 0.83 versus KNN's R² = 0.47 on real thermal images. A residual cascade — iterative discovery on successive residuals — surpasses KNN on heat conduction (R² = 0.998 vs 0.988) and Navier-Stokes velocity (R² = 0.991 vs 0.977), though it trails KNN by 0.015 in R² on dense, real PIV reconstruction where all methods have access to the full measurement set. The method uses no training, no neural networks, and no PDE residuals. It adapts automatically from trivial fields (1 component, <1 second) to complex turbulent flows (59 components, 9 minutes). This is a companion paper to "Equation-Free Field Reconstruction via Structural Discovery" (Sitnikov, 2026), extending that work with automatic complexity selection, a residual cascade for full-field reconstruction, and validation on real (non-simulated) experimental data.