Abstract: We present a numerical approach for modeling unknown dynamical systems using partially observed data, with a focus on biological systems with (relatively) complex dynamical behavior. As an extension of the recently developed deep neural network (DNN) learning methods, our approach is particularly suitable for practical situations when (i) measurement data are available for only a subset of the state variables, and (ii) the system parameters cannot be observed or measured at all. We demonstrate that, with a properly designed DNN structure with memory terms, effective DNN models can be learned from such partially observed data containing hidden parameters. The learned DNN model serves as an accurate predictive tool for system analysis. Through a few representative biological problems, we demonstrate that such DNN models can capture qualitative dynamical behavior changes in the system, such as bifurcations, even when the parameters controlling such behavior changes are completely unknown throughout not only the model learning process but also the system prediction process. The learned DNN model effectively creates a “closed” model involving only the observables when such a closed-form model does not exist mathematically.

Key words: Deep neural network (DNN), Governing equation discovery, Biological system, Partial observation