从聚合数据中学习随机行为
Learning Stochastic Behaviour from Aggregate Data
摘要：
从聚集数据中学习非线性动力学是一个具有挑战性的问题，因为每个个体的完整轨迹是不可用的，即在一个时间点观察到的个体可能在下一个时间点观察不到，或者个体的身份不可用。这与大多数现有方法所基于的具有完整轨迹数据的学习动态形成鲜明对比。我们提出了一种新的方法，使用Fokker Planck方程（FPE）的弱形式-一个偏微分方程-来描述采样形式的数据的密度演化，然后在训练过程中将其与Wasserstein生成对抗网络（WGAN）相结合。在这样一个基于样本的框架中，我们能够从聚合数据中学习非线性动力学，而无需显式求解偏微分方程（PDE）FPE。我们展示了我们的方法的背景下，一系列的合成和现实世界的数据集。
英文摘要：
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.
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