Data-Driven Forecasting of High-Dimensional Transient and Stationary Processes via Space-Time Projection

Abstract: Space-Time Projection (STP) is introduced as a data-driven forecasting approach for high-dimensional and time-resolved data. The method computes extended space-time proper orthogonal modes from training data spanning a prediction horizon comprising both hindcast and forecast intervals. Forecasts are then generated by projecting the hindcast portion of these modes onto new data, simultaneously leveraging their orthogonality and optimal correlation with the forecast extension. Rooted in Proper Orthogonal Decomposition (POD) theory, dimensionality reduction and time-delay embedding are intrinsic to the approach. For a given ensemble and fixed prediction horizon, the only tunable parameter is the truncation rank--no additional hyperparameters are required. The hindcast accuracy serves as a reliable indicator for short-term forecast accuracy and establishes a lower bound on forecast errors. The efficacy of the method is demonstrated using two datasets: transient, highly anisotropic simulations of supernova explosions in a turbulent interstellar medium, and experimental velocity fields of a turbulent high-subsonic engineering flow. In a comparative study with standard Long Short-Term Memory (LSTM) neural networks--acknowledging that alternative architectures or training strategies may yield different outcomes--the method consistently provided more accurate forecasts. Considering its simplicity and robust performance, STP offers an interpretable and competitive benchmark for forecasting high-dimensional transient and chaotic processes, relying purely on spatiotemporal correlation information.