AI 中文总结
该研究针对稀疏观测的PDE发现问题，提出冻结后选择方法，结合结构化场适配器与SVWS，在6种稀疏MDBench场景中实现最高精确支持集恢复率，尤其在Kuramoto-Sivashinsky动力学上优势显著。
AI 中文摘要
从稀疏观测中发现偏微分方程（PDE）需要重建连续场并选择正确的微分项。我们对耦合神经PDE发现中的优化路径分析揭示了三种行为：精确支持集可持续到训练结束、仅短暂出现或无法出现。为将方程选择与神经优化解耦，我们开发了冻结后选择方法，结合结构化场适配器与稳定性验证弱选择（SVWS）。该适配器在无PDE残差的观测上训练，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识别独立弱形式系统中的重复项，重新拟合候选支持集并在保留的弱形式系统上选择最终方程。除固定库外，我们将相同原理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6种稀疏MDBench场景中，我们的方法达到最高的精确支持集恢复率，在具有挑战性的Kuramoto-Sivashinsky动力学上相比经典与神经基准取得最明显增益。
英文摘要
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.
Comments18 pages, 5 figures, and 17 tables; includes supplementary material
