Equation-Free Field Reconstruction via Structural Discovery

A structural discovery method for reconstructing physical fields from sparse measurements without knowing the governing PDE. The method identifies dominant structural components directly from data, producing an interpretable decomposition. An automatic residual cascade adapts structural complexity to the data — no configuration or domain knowledge required. Key results: - R² = 1.000 on six standard 2D PDEs from 20 measurement points - 2-4× fewer sensors than KNN interpolation for equivalent accuracy on real data - Residual cascade beats KNN on heat conduction (0.998 vs 0.988) and Navier-Stokes velocity (0.991 vs 0.977) - No training, no neural networks, no PDE residuals