We propose two novel learning frameworks using neural mean-field (NMF) dynamics for inference and estimation problems on heterogeneous diffusion networks in discrete-time and continuous-time setting, respectively. The frameworks leverages the Mori-Zwanzig formalism to obtain an exact evolution equation of the individual node infection probabilities, which renders a delay differential equation with memory integral approximated by learnable time convolution operators.

Directly using information diffusion cascade data, our frameworks can simultaneously learn the structure of the diffusion network and the evolution of node infection probabilities. Connections between parameter learning and optimal control are also established, leading to a rigorous and implementable algorithm for training NMF. Moreover, we show that the projected gradient descent method can be employed to solve the challenging influence maximization problem, where the gradient is computed extremely fast by integrating NMF forward in time just once in each iteration. Extensive empirical studies show that our approach is versatile and robust to variations of the underlying diffusion network models, and significantly outperform existing approaches in accuracy and efficiency on both synthetic and real-world data.