We introduce a self-consistent deep-learning framework which, for a noisy deterministic time series, provides unsupervised filtering, state-space reconstruction, identification of the underlying differential equations and forecasting. Without a priori information on the signal, we embed the time series in a state space, where deterministic structures, i.e. attractors, are revealed. Under the assumption that the evolution of solution trajectories is described by an unknown dynamical system, we filter out stochastic outliers. The embedding function, the solution trajectories and the dynamical systems are constructed using deep neural networks, respectively. By exploiting the differentiability of the neural solution trajectory, the neural dynamical system is defined locally at each time, mitigating the need for propagating gradients through numerical solvers. On a chaotic time series masked by additive Gaussian noise, we demonstrate the filtering ability and the predictive power of the proposed framework.
翻译:我们引入了自成一体的深层学习框架, 用于噪音的确定时间序列, 提供不受监督的过滤、 州- 空间重建、 定位基本差异方程和预测。 没有信号的先验信息, 我们将时间序列嵌入一个状态空间, 从而显示确定性结构, 即吸引器。 根据解决方案轨迹的演变由未知动态系统描述的假设, 我们过滤了随机外线。 嵌入功能、 解决方案轨迹和动态系统分别使用深神经网络构建。 通过利用神经溶液轨迹的可变性, 神经动态系统每次都在当地定义, 减轻通过数字解析器传播梯度的需要 。 在由添加高素噪音遮掩的混乱时间序列中, 我们展示了拟议框架的过滤能力和预测能力 。