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, 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 generated by projecting the hindcast portion of these modes onto new data, 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. The only tunable parameters are the truncation rank and the hindcast length; no additional hyperparameters are required. Hindcast accuracy serves as a reliable indicator for short-term forecast accuracy. The method’s efficacy 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 dynamic mode decomposition (DMD) and long short-term memory (LSTM) networks (acknowledging that alternative architectures or training strategies may yield different outcomes) STP achieved the lowest errors at short and long lead times and was comparable at intermediate horizons. Considering its simplicity and robust performance, STP offers an interpretable and competitive baseline for forecasting high-dimensional transient chaotic processes, relying purely on spatio-temporal correlation.