Performance metrics used throughout the paper.

Characteristics of individuals in a population, such as age and size, play a key role in determining how populations change over time. In contexts of population dynamics, identifying effective model features, such as fecundity and mortality rates, is generally a complex and computationally intensive process, especially when the dynamics are heterogeneous across the population. In this work, we propose a Weak form Scientific Machine Learning-based method for selecting appropriate model ingredients from a library of scientifically feasible functions used to model structured populations. This paper presents extensions of the Weak form Sparse Identification of Nonlinear Dynamics (WSINDy) method to select the best-fitting ingredients from noisy time-series histogram data. This extension includes learning heterogeneous dynamics and also learning the boundary processes (such as birth) of the model directly from the data. We additionally incorporate a cross-validation method which helps fine tune the recovered boundary process hyperparameters to the data.Several test cases are considered, demonstrating the method’s performance for several standard models from population modeling, including age and size-structured models. Through these examples, we examine both the advantages and limitations of the method, with a particular focus on the distinguishability of terms in the library.种群中个体的特征（如年龄、体型）在决定种群随时间的变化规律中起着关键作用。在种群动力学研究范畴内，识别有效的模型特征（如繁殖率、死亡率）通常是一个复杂且计算量庞大的过程，尤其是当种群内部动力学存在异质性时。本研究提出了一种基于弱形式科学机器学习（Weak form Scientific Machine Learning）的方法，用于从可用于建模结构化种群的科学可行函数库中筛选合适的模型组分。本文将非线性动力学弱形式稀疏识别（Weak form Sparse Identification of Nonlinear Dynamics, WSINDy）方法进行扩展，使其能够从带噪声的时间序列直方图数据中选取最优拟合的模型组分。该扩展版本不仅可学习异质性动力学，还能直接从数据中习得模型的边界过程（如出生过程）。此外，我们引入了一种交叉验证方法，可辅助将恢复得到的边界过程超参数微调至适配给定数据集。本文设置了多个测试案例，验证了该方法在若干种群建模标准模型（包括年龄结构化、体型结构化模型）中的性能表现。通过这些示例，我们剖析了该方法的优势与局限性，尤其聚焦于函数库中各项的可区分性。