Estimation of System Parameters Including Repeated Cross-Sectional Data through Emulator-Informed Deep Generative Model

Abstract. Differential equations (DEs) are widely used to model the evolution of natural or engineered systems. Traditionally, the parameters in DEs are adjusted to fit data obtained through system observations. However, in fields such as politics, economics, and biology, the observed data points are often collected independently (i.e., repeated cross-sectional (RCS) data). In this paper, we first demonstrate that conventional optimization techniques struggle to accurately estimate DE parameters when RCS data exhibit various heterogeneities, leading to a significant loss of information. To address this issue, we propose a new estimation method, the emulator-informed deep-generative model (EIDGM), designed to handle RCS data. Specifically, EIDGM integrates a physics-informed neural network–based emulator that immediately generates DE solutions and a Wasserstein-generative adversarial network–based parameter generator that can effectively mimic the RCS data. We then evaluated the effectiveness of EIDGM across various models, including exponential and logistic growth models, and the Lorenz system, demonstrating its superior ability to accurately capture parameter distributions. In addition, we applied EIDGM to real-world datasets, successfully capturing diverse shapes of parameter distributions. This result highlights that EIDGM can be applied to model a wide range of systems with limited data availability, and our approach can be significantly extended to uncover the operating principles of systems based on limited data. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/CHWmath/EIDGM/tree/main and in the supplementary materials ( EIDGM-main.zip [19.5MB]). [Formula: see text]