Department of Mathematics and Statistics, Texas A&M University-Corpus Christi, Corpus Christi, Texas 78412, USA.
Department of Physical and Environmental Sciences, Texas A&M University-Corpus Christi, Corpus Christi, Texas 78412, USA.
Chaos. 2023 Jan;33(1):013109. doi: 10.1063/5.0088748.
Classical methods of solving spatiotemporal dynamical systems include statistical approaches such as autoregressive integrated moving average, which assume linear and stationary relationships between systems' previous outputs. Development and implementation of linear methods are relatively simple, but they often do not capture non-linear relationships in the data. Thus, artificial neural networks (ANNs) are receiving attention from researchers in analyzing and forecasting dynamical systems. Recurrent neural networks (RNNs), derived from feed-forward ANNs, use internal memory to process variable-length sequences of inputs. This allows RNNs to be applicable for finding solutions for a vast variety of problems in spatiotemporal dynamical systems. Thus, in this paper, we utilize RNNs to treat some specific issues associated with dynamical systems. Specifically, we analyze the performance of RNNs applied to three tasks: reconstruction of correct Lorenz solutions for a system with a formulation error, reconstruction of corrupted collective motion trajectories, and forecasting of streamflow time series possessing spikes, representing three fields, namely, ordinary differential equations, collective motion, and hydrological modeling, respectively. We train and test RNNs uniquely in each task to demonstrate the broad applicability of RNNs in the reconstruction and forecasting the dynamics of dynamical systems.
经典的时空动力系统求解方法包括自回归综合移动平均等统计方法，这些方法假设系统先前输出之间存在线性和静态关系。线性方法的开发和实施相对简单，但它们通常无法捕捉数据中的非线性关系。因此，人工神经网络 (ANNs) 在分析和预测动力系统方面引起了研究人员的关注。递归神经网络 (RNNs) 源自前馈神经网络，使用内部存储器来处理输入的可变长度序列。这使得 RNN 能够适用于解决时空动力系统中各种各样的问题。因此，在本文中，我们利用 RNN 来处理与动力系统相关的一些具体问题。具体来说，我们分析了 RNN 应用于三个任务的性能：对具有公式错误的系统的正确 Lorenz 解进行重建、对受污染的集体运动轨迹进行重建，以及对具有尖峰的流量时间序列进行预测，分别代表常微分方程、集体运动和水文建模三个领域。我们在每个任务中单独训练和测试 RNN，以展示 RNN 在重建和预测动力系统动态方面的广泛适用性。
