(91g) Forecasting and Reconstructing Chaotic Dynamics from Partial Observable Data

Many applications require the prediction of a time series from only observable data. Experimental measurements of time series are often scalar or poorly spatially resolved, causing challenges in reconstructing and forecasting the system from only the current state. Takens theorem proves that for data lying on a smooth compact manifold the full state can be reconstructed from an embedding of time delayed state observations. These delay coordinates embed the attractor up to a diffeomorphic transformation. While powerful in principle, it is often challenging to learn these functions in practice, particularly for chaotic and highly nonlinear systems. We use artificial neural networks (ANNs) to learn discrete and continuous time forecasting functions as well as reconstruction functions from delay coordinate embeddings of partial observable data. We test our approach on the Lorenz system and the Kuramoto-Sivashinsky equation (KSE). We find the Lorenz attractor can be forecasted and reconstructed from a scalar observable with 2-3 time delays. For the more chaotic KSE, the attractor can be reconstructed from multivariate observations with dimensionality as low as one half the manifold dimension. Our approach provides good short time tracking and longtime statistics with relatively few embedding parameters and ANN hyperparameters.