Recurrent Neural Networks for Dynamical Systems: Applications to Ordinary Differential Equations, Collective Motion, and Hydrological Modeling
This work addresses the challenge of capturing non-linear relationships in dynamical systems for researchers in fields like ordinary differential equations, collective motion, and hydrology, but it is incremental as it applies an existing method (RNNs) to new data without introducing novel methodological advancements.
The paper tackles the problem of analyzing and forecasting spatiotemporal dynamical systems by applying recurrent neural networks (RNNs) to three specific tasks: reconstructing correct Lorenz solutions with formulation errors, reconstructing corrupted collective motion trajectories, and forecasting streamflow time series with spikes, demonstrating RNNs' broad applicability in these domains.
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 (RNN), derived from feed-forward ANNs, use internal memory to process variable-length sequences of inputs. This allows RNNs to 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 reconstruction and forecasting the dynamics of dynamical systems.