Freeze, Then Select: Structured Field Adapters and Stability-Validated Weak Selection for PDE Discovery from Sparse Observations

2026-07-31

AI Summary

This work addresses the challenge of discovering governing partial differential equations (PDEs) from sparse and noisy observations, which requires simultaneously reconstructing continuous fields and identifying the correct differential terms—a task where conventional methods often struggle to balance stability and accuracy. The authors propose a “freeze-and-reselect” strategy: first, a structured neural field adapter—integrating spatial features with cubic spline time coefficients—reconstructs the continuous field without relying on residual-based training; then, this reconstructed field is frozen, and a cross-system term selection mechanism, combining weak-form residuals with stability validation, robustly identifies the PDE structure within a space of candidate expressions generated by genetic programming. Evaluated on all six sparse scenarios in MDBench, the method achieves the highest support recovery accuracy, significantly outperforming existing baselines, particularly on challenging dynamics such as the Kuramoto–Sivashinsky equation.

Abstract

PDE discovery from sparse observations requires reconstructing a continuous field and selecting the correct differential terms. Our analysis of optimization paths in coupled neural PDE discovery reveals three behaviors: the exact support can persist to the end of training, appear only transiently, or fail to emerge. To decouple equation selection from neural optimization, we develop a freeze-then-select method combining a structured field adapter with Stability-Validated Weak Selection (SVWS). Trained from observations without a PDE residual, the adapter factorizes the field into learned spatial features and temporal coefficients represented by cubic splines. After freezing the field, SVWS identifies recurrent terms across independent weak-form systems, refits candidate supports, and selects the final equation on held-out weak-form systems. Beyond fixed libraries, we apply the same principle to expressions generated by genetic programming and recover the power-law form of an unknown nonlinear diffusion function from sparse, noisy observations. Across all six sparse MDBench regimes, our method attains the highest exact support recovery rate, with its clearest gains over classical and neural baselines on challenging Kuramoto-Sivashinsky dynamics.

Problem

• PDE discovery
• sparse observations
• equation selection
• weak-form systems
• nonlinear diffusion

Innovation

• freeze-then-select
• structured field adapter
• Stability-Validated Weak Selection
• PDE discovery
• sparse observations