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Classical methods of solving spatiotemporal dynamical systems includestatistical approaches such as autoregressive integrated moving average, whichassume linear and stationary relationships between systems' previous outputs.Development and implementation of linear methods are relatively simple, butthey often do not capture non-linear relationships in the data. Thus,artificial neural networks (ANNs) are receiving attention from researchers inanalyzing and forecasting dynamical systems. Recurrent neural networks (RNN),derived from feed-forward ANNs, use internal memory to process variable-lengthsequences of inputs. This allows RNNs to applicable for finding solutions for avast variety of problems in spatiotemporal dynamical systems. Thus, in thispaper, we utilize RNNs to treat some specific issues associated with dynamicalsystems. Specifically, we analyze the performance of RNNs applied to threetasks: reconstruction of correct Lorenz solutions for a system with aformulation error, reconstruction of corrupted collective motion trajectories,and forecasting of streamflow time series possessing spikes, representing threefields, namely, ordinary differential equations, collective motion, andhydrological modeling, respectively. We train and test RNNs uniquely in eachtask to demonstrate the broad applicability of RNNs in reconstruction andforecasting the dynamics of dynamical systems.

Recurrent Neural Networks for Dynamical Systems: Applications to Ordinary Differential Equations, Collective Motion, and Hydrological Modeling