Algebraically Observable Physics-Informed Neural Network and its Application to Epidemiological Modelling

Physics-Informed Neural Network (PINN) is a deep learning framework that integrates the governing equations underlying data into a loss function. In this study, we consider the problem of estimating state variables and parameters in epidemiological models governed by ordinary differential equations using PINNs. In practice, not all trajectory data corresponding to the population described by models can be measured. Learning PINNs to estimate the unmeasured state variables and epidemiological parameters using partial measurements is challenging. Accordingly, we introduce the concept of algebraic observability of the state variables. Specifically, we propose augmenting the unmeasured data based on algebraic observability analysis. The validity of the proposed method is demonstrated through numerical experiments under three scenarios in the context of epidemiological modelling. Specifically, given noisy and partial measurements, the accuracy of unmeasured states and parameter estimation of the proposed method is shown to be higher than that of the conventional methods. The proposed method is also shown to be effective in practical scenarios, such as when the data corresponding to certain variables cannot be reconstructed from the measurements.