Learning Stochastic Behaviour from Aggregate Data
Authors: Shaojun Ma, Shu Liu, Hongyuan Zha, Hao-Min Zhou
Keywords: Fokker Planck Equation, weak form, Wasserstein GAN
Abstract: Learning nonlinear dynamics from aggregate data is a challenging problem since the full trajectory of each individual is not available, namely, the individual observed at one time point may not be observed at next time point, or the identity of individual is unavailable. This is in sharp contrast to learning dynamics with trajectory data, on which the majority of existing methods are based. We propose a novel method using the weak form of Fokker Planck Equation (FPE) to describe density evolution of data in a sampling form, which is then combined with Wasserstein generative adversarial network (WGAN) in training process. In such a sample-based framework we are able to study nonlinear dynamics from aggregate data without solving the partial differential equation (PDE). The model can also handle high dimensional cases with the help of deep neural networks. We demonstrate our approach in the context of a series of synthetic and real-world data sets.