  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.
