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

2024-12-27

AI Summary

To address substantial parameter estimation bias and challenges in distributional modeling for ordinary differential equation (ODE) systems under repeated cross-sectional (RCS) data, this paper proposes the first end-to-end deep generative framework integrating a physics-informed neural network (PINN) surrogate with a Wasserstein generative adversarial network (WGAN). The method directly learns the posterior distribution of ODE parameters under hard ODE constraints, enabling unbiased, high-fidelity probabilistic inference. Evaluated on exponential growth, logistic population, and Lorenz systems, it significantly outperforms conventional parameter estimation approaches. Moreover, it successfully recovers multimodal parameter distributions observed in Aβ40/42 experimental data—despite sparse and heterogeneous RCS sampling—demonstrating strong generalization. This work establishes a novel, interpretable modeling paradigm for complex dynamic systems in political science, economics, and computational biology.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

Differential equations (DEs) are crucial for modeling the evolution of natural or engineered systems. Traditionally, the parameters in DEs are adjusted to fit data from system observations. However, in fields such as politics, economics, and biology, available data are often independently collected at distinct time points from different subjects (i.e., repeated cross-sectional (RCS) data). 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 called 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 evaluated EIDGM on exponential growth, logistic population models, and the Lorenz system, demonstrating its superior ability to accurately capture parameter distributions. Additionally, we applied EIDGM to an experimental dataset of Amyloid beta 40 and beta 42, successfully capturing diverse parameter distribution shapes. This shows that EIDGM can be applied to model a wide range of systems and extended to uncover the operating principles of systems based on limited data.

Problem

• Differential Equations
• Parameter Estimation
• Data Utilization

Innovation

• EIDGM
• Differential Equations Parameter Estimation
• Data Diversity Handling