Learning continuous network emerging dynamics from scarce observations via data-adaptive stochastic processes

Learning network dynamics from the empirical structure and spatio-temporal observation data is crucial to revealing the interaction mechanisms of complex networks in a wide range of domains. However, most existing methods only aim at learning network dynamic behaviors generated by a specific ordinary differential equation instance, resulting in ineffectiveness for new ones, and generally require dense observations. The observed data, especially from network emerging dynamics, are usually difficult to obtain, which brings trouble to model learning. Therefore, learning accurate network dynamics with sparse, irregularly-sampled, partial, and noisy observations remains a fundamental challenge. We introduce a new concept of the stochastic skeleton and its neural implementation, i.e., neural ODE processes for network dynamics (NDP4ND), a new class of stochastic processes governed by stochastic data-adaptive network dynamics, to overcome the challenge and learn continuous network dynamics from scarce observations. Intensive experiments conducted on various network dynamics in ecological population evolution, phototaxis movement, brain activity, epidemic spreading, and real-world empirical systems, demonstrate that the proposed method has excellent data adaptability and computational efficiency, and can adapt to unseen network emerging dynamics, producing accurate interpolation and extrapolation with reducing the ratio of required observation data to only about 6% and improving the learning speed for new dynamics by three orders of magnitude.