Inverse problems involving partial differential equations (PDEs) can be seen as discovering a mapping from measurement data to unknown quantities, often framed within an operator learning approach. However, existing methods typically rely on large amounts of labeled training data, which is impractical for most real-world applications. Moreover, these supervised models may fail to capture the underlying physical principles accurately. To address these limitations, we propose a novel architecture called Physics-Informed Deep Inverse Operator Networks (PI-DIONs), which can learn the solution operator of PDE-based inverse problems without any labeled training data. We extend the stability estimates established in the inverse problem literature to the operator learning framework, thereby providing a robust theoretical foundation for our method. These estimates guarantee that the proposed model, trained on a finite sample and grid, generalizes effectively across the entire domain and function space. Extensive experiments are conducted to demonstrate that PI-DIONs can effectively and accurately learn the solution operators of the inverse problems without the need for labeled data.

Differential equations are pivotal in modeling and understanding the dynamics of various systems, as they offer 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). RCS data showed that traditional methods for parameter estimation in differential equations, such as using mean values of RCS data over time, Gaussian Process-based trajectory generation, and Bayesian-based methods, have limitations in estimating the shape of parameter distributions, leading to a significant loss of data information. To address this issue, this study proposes a novel method called Estimation of Parameter Distribution (EPD) that provides 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 minimizes 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, thereby demonstrating the ability of the proposed method in capturing the shape of parameter distributions. Furthermore, we applied EPD to real-world datasets, capturing various shapes of parameter distributions over a normal distribution. These results address the heterogeneity within systems, marking a substantial progression in accurately modeling systems using RCS data. Therefore, EPD marks a significant advancement in accurately modeling systems with RCS data, realizing a deeper understanding of system dynamics and parameter variability.