Bridging Physics-based and Data-driven modeling for Learning Dynamical Systems

How can we learn a dynamical system to make forecasts, when some variables are unobserved? For instance, in COVID-19, we want to forecast the number of infected patients and death cases but we do not know the count of susceptible and exposed people. How to proceed? While mechanics compartment models are widely-used in epidemic modeling, data-driven models are emerging for disease forecasting. As a case study, we compare these two types of models for COVID-19 forecasting and notice that physics-based models significantly outperform deep learning models. We present a hybrid approach, AutoODE-COVID, which combines a novel compartmental model with automatic differentiation. Our method obtains a 57.4% reduction in mean absolute errors of the 7-day ahead COVID-19 trajectories prediction compared with the best deep learning competitor. To understand the inferior performance of deep learning, we investigate the generalization problem in forecasting. Through systematic experiments, we found that deep learning models fail to forecast under shifted distributions either in the data and parameter domains of dynamical systems. This calls attention to rethink generalization especially for learning dynamical systems.