Learning Stochastic Behaviour from Aggregate Data
This addresses a challenge in fields like biology or social sciences where only aggregate data is available, offering a novel approach but with incremental improvements over existing methods.
The paper tackles the problem of learning nonlinear dynamics from aggregate data, where individual trajectories are incomplete or unidentifiable, by proposing a method that combines the weak form of the Fokker-Planck Equation with Wasserstein generative adversarial networks, achieving results on synthetic and real-world datasets without explicitly solving PDEs.
Learning nonlinear dynamics from aggregate data is a challenging problem because the full trajectory of each individual is not available, namely, the individual observed at one time may not be observed at the next time point, or the identity of individual is unavailable. This is in sharp contrast to learning dynamics with full 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) -- a partial differential equation -- to describe the density evolution of data in a sampled form, which is then combined with Wasserstein generative adversarial network (WGAN) in the training process. In such a sample-based framework we are able to learn the nonlinear dynamics from aggregate data without explicitly solving the partial differential equation (PDE) FPE. We demonstrate our approach in the context of a series of synthetic and real-world data sets.