把方程选择跟神经优化拆开：先冻结场重建，再用SVWS筛项。六个MDBench稀疏场景恢复率最高，KS方程提升最大。
该论文分析耦合神经PDE发现中的优化路径，发现三种行为：正确支撑集可以持续到训练结束、仅短暂出现或始终未出现。作者提出冻结-再选择方法，用结构化场适配器将场分解为空间特征与三次样条时间系数，在不依赖PDE残差的情况下训练。之后用稳定性验证弱选择（SVWS）在独立弱形式系统中识别重复项、重拟合候选支撑并选出最终方程。在MDBench全部六个稀疏场景中，该方法取得了最高的精确支撑恢复率，在Kuramoto-Sivashinsky动力学上比经典和神经基线提升最明显。对固定库之外，方法还能用于遗传编程生成的表达式，从稀疏含噪观测中恢复未知非线性扩散函数的幂律形式。
Freeze, Then Select: Structured Field Adapters and Stability-Validated Weak Selection for PDE Discovery from Sparse Observations
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.