发表机构
University of Trieste; National Institute of Oceanography and Applied Geophysics - OGS; National Biodiversity Future Center（的里雅斯特大学; 国家海洋学与应用地球物理研究所 - OGS; 国家生物多样性未来中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究从数据中进行符号化控制方程发现的可识别性极限，表明吸引子几何中的$\lambda_{\min}(M)$为SINDy和PySR设定上限，混沌对二者影响不同，还引入Soft F1，指出发现首要问题是吸引子允许什么，而非算法。
AI 中文摘要
从数据中进行符号化的控制方程发现不仅受算法设计和数据量限制，还受吸引子几何的限制，即长期动力学允许恢复的程度。通过对Lorenz - 84系统进行系统内设计，一个强迫参数驱动定点、极限环和混沌状态，而控制方程和库保持不变。研究表明，不变测度矩矩阵的最小特征值$\lambda_{\min}(M)$为稀疏回归（SINDy）和进化符号回归（PySR）设定了可识别性上限。它源于伯克霍夫遍历定理，在任何运行前从短参考轨迹获得，衡量吸引子覆盖函数空间的程度。混沌通过扩展吸引子提高$\lambda_{\min}(M)$，但也扩大吸引子并放大噪声，这会使两种方法朝相反方向发展。该框架的无参数机制分数无需重新拟合即可转移到保留的Lorenz - 96系统，证实是机制而非曲线拟合。还引入了Soft F1，一种系数加权结构度量。发现的首要问题不是哪种算法，而是吸引子允许什么。
英文摘要
How the dynamical regime of the observed system affects equation discovery has mainly been investigated through comparisons across systems. However, such comparisons vary both the equations and the dynamics, confounding the effect of the regime with the difficulty of recovering the equations symbolically. We separate the two by varying the forcing of Lorenz-84, moving its fully observed post-transient trajectories through fixed-point, periodic, and chaotic regimes while preserving the equations' functional form. Within each regime, we separately vary the amount of data, the noise, and prior knowledge of which terms the equations contain. We then measure how well two complementary approaches, sparse regression over a fixed library of candidate functions (SINDy) and an evolutionary search over symbolic expression trees (PySR), recover the true equations' terms and coefficients. We find that recovery depends on whether the sampled states distinguish combinations of candidate functions: equations remain poorly recovered from fixed-point data even when the candidate set contains only the true terms. We link the effect of the dynamics on both algorithms to one object: the moment matrix of the candidate functions under the invariant measure of the regime. Small eigenvalues mark weakly distinguishable combinations of candidate functions: we show that more data, less noise, and more prior knowledge can mitigate the resulting recovery difficulties, while a zero eigenvalue makes distinct equations indistinguishable on the visited states. Hence, a more precise prior needs less informative data, with consequences for data collection, method design, and evaluation.
Comments35 pages, 6 figures, 3 tables (main text and appendices)
