What question did this study set out to answer?

The aim is to reconstruct physical fields from sparse measurements without prior knowledge of governing equations.

June 22, 2026Open Access

Equation-Free Field Reconstruction via Structural Discovery

Key Points

The aim is to reconstruct physical fields from sparse measurements without prior knowledge of governing equations.
Developed a structural discovery method for field reconstruction.
Used automated residual cascade to adapt structural complexity.
Applied the method to six standard 2D PDEs and real data.
Achieved R² = 1.000 on six standard 2D PDEs with only 20 measurement points.
Utilized 2-4× fewer sensors than KNN interpolation for equivalent accuracy on real data.
Demonstrated superior performance to KNN on heat conduction (0.998 vs 0.988) and Navier-Stokes velocity (0.991 vs 0.977).

Abstract

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

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