Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks

We introduce a data-driven forecasting method for high-dimensional chaotic systems using long short-term memory (LSTM) recurrent neural networks. The proposed LSTM neural networks perform inference of high-dimensional dynamical systems in their reduced order space and are shown to be an effective set of nonlinear approximators of their attractor. We demonstrate the forecasting performance of the LSTM and compare it with Gaussian processes (GPs) in time series obtained from the Lorenz 96 system, the Kuramoto-Sivashinsky equation and a prototype climate model. The LSTM networks outperform the GPs in short-term forecasting accuracy in all applications considered. A hybrid architecture, extending the LSTM with a mean stochastic model (MSM-LSTM), is proposed to ensure convergence to the invariant measure. This novel hybrid method is fully data-driven and extends the forecasting capabilities of LSTM networks.Comment: 31 page

RefreshNet: Learning Multiscale Dynamics through Hierarchical Refreshing

Forecasting complex system dynamics, particularly for long-term predictions, is persistently hindered by error accumulation and computational burdens. This study presents RefreshNet, a multiscale framework developed to overcome these challenges, delivering an unprecedented balance between computational efficiency and predictive accuracy. RefreshNet incorporates convolutional autoencoders to identify a reduced order latent space capturing essential features of the dynamics, and strategically employs multiple recurrent neural network (RNN) blocks operating at varying temporal resolutions within the latent space, thus allowing the capture of latent dynamics at multiple temporal scales. The unique "refreshing" mechanism in RefreshNet allows coarser blocks to reset inputs of finer blocks, effectively controlling and alleviating error accumulation. This design demonstrates superiority over existing techniques regarding computational efficiency and predictive accuracy, especially in long-term forecasting. The framework is validated using three benchmark applications: the FitzHugh-Nagumo system, the Reaction-Diffusion equation, and Kuramoto-Sivashinsky dynamics. RefreshNet significantly outperforms state-of-the-art methods in long-term forecasting accuracy and speed, marking a significant advancement in modeling complex systems and opening new avenues in understanding and predicting their behavior

Data-assisted reduced-order modeling of extreme events in complex dynamical systems

Dynamical systems with high intrinsic dimensionality are often characterized by extreme events having the form of rare transitions several standard deviations away from the mean. For such systems, order-reduction methods through projection of the governing equations have limited applicability due to the large intrinsic dimensionality of the underlying attractor but also the complexity of the transient events. An alternative approach is data-driven techniques that aim to quantify the dynamics of specific modes utilizing data-streams. Several of these approaches have improved performance by expanding the state representation using delayed coordinates. However, such strategies are limited in regions of the phase space where there is a small amount of data available, as is the case for extreme events. In this work, we develop a blended framework that integrates an imperfect model, obtained from projecting equations into a subspace that still contains crucial dynamical information, with data-streams through a recurrent neural network (RNN) architecture. In particular, we employ the long-short-term memory (LSTM), to model portions of the dynamics which cannot be accounted by the equations. The RNN is trained by analyzing the mismatch between the imperfect model and the data-streams, projected in the reduced-order space. In this way, the data-driven model improves the imperfect model in regions where data is available, while for locations where data is sparse the imperfect model still provides a baseline for the prediction of the system dynamics. We assess the developed framework on two challenging prototype systems exhibiting extreme events and show that the blended approach has improved performance compared with methods that use either data streams or the imperfect model alone. The improvement is more significant in regions associated with extreme events, where data is sparse.Comment: Submitted to PLOS ONE on March 8, 201

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