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

TL;DR

This work introduces Space-Time Projection (STP), a data-driven method for forecasting high-dimensional, time-resolved processes using extended space-time POD modes. By constructing STP modes from an ensemble over a prediction horizon and projecting new hindcast data, STP achieves accurate forecasts with only the rank truncation as a tunable parameter, and without needing extra hyperparameters. The approach is demonstrated on a transient supernova dataset and a stationary cavity flow dataset, showing strong predictive performance and interpretability of the modes, and it compares favorably against LSTM-based regression on POD coefficients. STP provides a robust, computationally efficient benchmark for forecasting complex spatiotemporal dynamics and offers potential extensions to parameterized settings and sensor-driven forecasting.

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.

Figures (14)

• Figure 1: Schematic of the STP algorithm. Step 1: Construct the hindcast data matrix and solve the ensemble space-time POD eigenvalue problem, equation (\ref{['eqn:evp']}), to obtain the hindcast expansion coefficients. Step 2: Compute the STP (extended hindcast) modes using equation (\ref{['eqn:phipm']}). Step 3: Approximate the hindcast expansion coefficients via projection, equation (\ref{['eqn:ahind']}), and expand the predicted trajectory using equation (\ref{['eqn:qstarpm']}).
• Figure 2: Overview of the numerical supernova simulation: The top row shows instantaneous isocontours of temperature at 1% of its maximum value, highlighting the expanding supernova shell at six representative time instances. The bottom row presents the corresponding temperature fields in the $y$-$z$ plane at $x=0$. The magenta contour line indicates the isovalue used in the 3D visualization of the shell. The first realization from the training dataset is shown as an example.
• Figure 3: Overview of experimental cavity flow: (a) instantaneous streamwise velocity $u$; (b) instantaneous normal velocity $v$; (c) power spectrum at $(x,y)=(5,0)$; (d) frequency–wavenumber diagram along $(x,y=0)$. The peaks identified in (c) and (d) correspond to the two dominant Rossiter tones. The incoming flow travels over the cavity ($y>0$) at Mach 0.6 and recirculates within the cavity walls ($y<0$) while undergoing violent oscillations.
• Figure 4: STP prediction of the supernova simulation with a hindcast horizon of ${n}=30$ and forecast horizon of ${m}=29$: (a) hindcast mode variance; (b) prediction error, where gray lines represent individual forecasts, the blue line indicates the mean, the solid circle indicates the first time step of the forecastand, and the light blue band shows the standard deviation. The leading $r=100$ retained modes, out of a total of 400, capture roughly 80% of the total variance. In (b), the time step index $i$ is used because the time steps are non-uniform.
• Figure 5: Study of the prediction error in the supernova prediction: (a) mean prediction error as a function of hindcast interval length ${n}$ without basis truncation; (b) mean prediction error for different numbers $r$ of retained modes, with a fixed hindcast interval length of ${n}=30$. In all cases, the entire trajectory is considered, such that ${n}+{m} = 59$. Solid circles indicate the first time step of the forecast.