Learning Stochastic Behaviour from Aggregate Data

TL;DR

This paper introduces a novel method to learn nonlinear stochastic dynamics from aggregate data by combining the weak form of the Fokker-Planck Equation with Wasserstein GANs, enabling learning without explicit PDE solutions.

Contribution

It proposes a new sample-based framework that leverages the weak form of FPE and WGANs to learn dynamics from aggregate data, bypassing the need for full trajectory data.

Findings

Successfully applied to synthetic data sets

Effective on real-world aggregate data

Outperforms existing methods in similar settings

Abstract

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.