Enhancing computational efficiency in multiscale systems using deep learning of coordinates and flow maps

Abstract

Multiscale systems are expensive to simulate because fast dynamics require small time-steps, while slow dynamics require long prediction horizons. We propose latent hierarchical time-stepping (L-HiTS), which combines nonlinear coordinate discovery with multiscale flow-map learning. A deep autoencoder first compresses the high-dimensional PDE state into a validated low-dimensional latent space. Residual neural network time-steppers are then trained and coupled directly in this reduced space using validation-based hierarchy selection and vectorized prediction. Unlike multiscale HiTS, L-HiTS performs recursive forecasting in latent coordinates and reconstructs the full state only after prediction. The method is validated on the FitzHugh–Nagumo model, the chaotic Kuramoto–Sivashinsky equation, and a two-dimensional Burgers’ system. L-HiTS achieves comparable prediction accuracy to multiscale HiTS while substantially reducing training and prediction costs, with near order-of-magnitude prediction-time savings in the reported cases.