Differential equations are pivotal in modeling and understanding the dynamics of various systems, offering insights into their future states through parameter estimation fitted to time series data. In fields such as economy, politics, and biology, the observation data points in the time series are often independently obtained (i.e., Repeated Cross-Sectional (RCS) data). With RCS data, we found that traditional methods for parameter estimation in differential equations, such as using mean values of time trajectories or Gaussian Process-based trajectory generation, have limitations in estimating the shape of parameter distributions, often leading to a significant loss of data information. To address this issue, we introduce a novel method, Estimation of Parameter Distribution (EPD), providing accurate distribution of parameters without loss of data information. EPD operates in three main steps: generating synthetic time trajectories by randomly selecting observed values at each time point, estimating parameters of a differential equation that minimize the discrepancy between these trajectories and the true solution of the equation, and selecting the parameters depending on the scale of discrepancy. We then evaluated the performance of EPD across several models, including exponential growth, logistic population models, and target cell-limited models with delayed virus production, demonstrating its superiority in capturing the shape of parameter distributions. Furthermore, we applied EPD to real-world datasets, capturing various shapes of parameter distributions rather than a normal distribution. These results effectively address the heterogeneity within systems, marking a substantial progression in accurately modeling systems using RCS data.

翻译：微分方程在建模和理解各种系统的动态过程中至关重要，通过拟合时间序列数据来估计参数，从而揭示其未来状态。在经济学、政治学和生物学等领域，时间序列中的观测数据点通常是独立获取的（即重复截面数据）。针对重复截面数据，我们发现传统的微分方程参数估计方法（例如使用时间轨迹的均值或基于高斯过程的轨迹生成）在估计参数分布形态时存在局限性，往往导致数据信息的显著丢失。为解决这一问题，我们提出了一种新方法——参数分布估计法，该方法能够在不损失数据信息的情况下提供准确的参数分布。EPD包含三个主要步骤：通过在每个时间点随机选择观测值生成合成时间轨迹；估计使这些轨迹与方程真实解之间差异最小化的微分方程参数；根据差异程度筛选参数。随后，我们在包括指数增长模型、逻辑斯谛种群模型以及具有延迟病毒产生的靶细胞限制模型等多个模型上评估了EPD的性能，证明了其在捕捉参数分布形态方面的优越性。此外，我们将EPD应用于真实数据集，捕获了非正态分布的各种参数分布形态。这些结果有效解决了系统内的异质性问题，标志着利用重复截面数据精确建模系统的重要进展。