AI 中文总结
该研究针对部分观测的活跃-静息系统，提出可观测性约简引导的稀疏回归方法，构建适配观测过程的回归库，可实现可解释的方程学习，优于标准多项式 SINDy。
AI 中文摘要
生物种群中存在活跃-静息切换现象：细胞生长被限制在增殖活跃状态，同时可可逆进入非增殖静息状态。实验通常仅观测该过程的一部分，观测方式包括活跃状态标记物、种群总量测量，或结合稀疏活跃状态观测的总量数据。当标记物 panel 有限、活跃状态检测需固定或终点采样，或仅报告总细胞数、光密度、肿瘤负荷或总荧光时，就会产生此类测量结果。这些观测选择会使稀疏回归方法（如非线性动力学稀疏识别 SINDy）变得复杂，因为测量变量满足的方程可能与潜在的活跃-静息系统不同。因此，为错误可观测性选择的库可能拟合轨迹，却无法保留机理解释性或可迁移性。我们采用双室常微分方程模型研究该问题，将可观测性约简定义为消除隐藏状态以得到测量变量满足的微分方程。针对多种生物学相关生长规律，我们推导了特定观测的约简结果，并用于构建稀疏回归库。通过合成数据，我们将这些结构化库与标准多项式 SINDy 进行比较：多项式库可拟合训练轨迹，但无法通过系数关系和迁移测试；而当观测区域具有信息性时，约简引导库可恢复可解释的系数映射。这些结果表明，在隐藏室系统中进行可解释方程学习，需要使回归目标和候选库均与观测过程相匹配。
英文摘要
Active-quiescent switching occurs in biological populations in which growth is confined to a proliferative active state, while cells may reversibly enter a nonproliferative quiescent state. Experiments often observe only part of this process, through active-state markers, aggregate population measurements, or aggregate data supplemented by sparse active-state observations. Such measurements arise when marker panels are limited, active-state assays require fixation or endpoint sampling, or only total cell number, optical density, tumor burden, or aggregate fluorescence is reported. These observation choices complicate sparse regression methods such as sparse identification of nonlinear dynamics (SINDy), because the measured variable may satisfy a different equation from the underlying active-quiescent system. A library chosen for the wrong observable may therefore fit a trajectory without preserving mechanistic interpretation or transferability. We study this issue using a two-compartment ordinary differential equation model. We define observable reduction as the elimination of hidden states to obtain the differential equation satisfied by the measured variable. For several biologically relevant growth laws, we derive observation-specific reductions and use them to construct sparse-regression libraries. Using synthetic data, we compare these structured libraries with standard polynomial SINDy. Polynomial libraries can match training trajectories while failing coefficient-relation and transfer tests, whereas reduction-guided libraries recover interpretable coefficient maps when the observed regime is informative. These results show that interpretable equation learning in hidden-compartment systems requires matching both the regression target and candidate library to the observation process.
