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

Imagine you're trying to model complex physical systems, like fluid dynamics or heat transfer, using data that’s sparse and noisy. The challenge lies in accurately identifying the underlying partial differential equations (PDEs) that govern these systems. Traditional methods often struggle because they rely heavily on complete datasets or predefined equation libraries, which can lead to missing critical dynamics or misidentifying the governing equations. This is what's called the equation selection problem, where the right terms may not emerge during the modeling process, leading to inaccurate predictions and insights. To address this, the authors propose a method that first reconstructs a continuous field from the sparse observations and then selects the correct differential terms through a two-step process. They use a structured field adapter that learns spatial features and temporal coefficients, which are then frozen to allow for a more focused selection of equations using Stability-Validated Weak Selection (SVWS). This approach not only improves the identification of the correct PDEs but also allows for the exploration of expressions generated through genetic programming, enhancing the flexibility of the method. Compared to previous work, this method achieves the highest recovery rates of exact supports across various challenging scenarios, particularly excelling in complex dynamics like the Kuramoto-Sivashinsky equation. For builders, this means a more reliable way to extract meaningful models from limited data, potentially leading to better predictions and insights in real-world applications.

The method introduces a novel approach to PDE discovery that combines structured field adaptation with a unique selection process.

The results show significant improvements over classical and neural baselines across multiple regimes, indicating solid performance.