从稀疏观测中精准发现偏微分方程，关键在于先冻结再筛选。
Freeze, Then Select: Structured Field Adapters and Stability-Validated Weak Selection for PDE Discovery from Sparse Observations
- 先训练神经场适配器重建连续场，再冻结后通过弱形式系统筛选方程项
- 在六个稀疏基准测试中实现最高精确项支持恢复率，尤其在复杂动力学上优势显著
- 适用于非线性扩散等未知函数形式，适合科学计算与物理建模研究者
从稀疏观测中发现偏微分方程需同时重建连续场并选择正确微分项。我们对耦合神经PDE发现优化路径的分析揭示三种行为：精确项支持可持久存在、仅短暂出现或完全无法生成。为解耦方程选择与神经优化，提出‘冻结-再筛选’方法，结合结构化场适配器与稳定性验证弱选择（SVWS）。适配器从无残差的观测数据训练，将场分解为学习到的空间特征与由三次样条表示的时间系数。冻结场后，SVWS在独立弱形式系统中识别重复项，重新拟合候选支持，并在保留弱形式系统上选择最终方程。该方法不仅适用于固定项库，还可应用于遗传编程生成的表达式，从稀疏噪声观测中恢复未知非线性扩散函数的幂律形式。在所有六个稀疏MDBench场景中，本方法达到最高精确项支持恢复率，尤其在挑战性的Kuramoto-Sivashinsky动力学上显著优于经典与神经基线。
原文摘要 · Abstract (English)
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.
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