Generative learning for forecasting the dynamics of high-dimensional complex systems
Han Gao, Sebastian Kaltenbach, Petros Koumoutsakos
Abstract
We introduce generative models for accelerating simulations of high-dimensional systems through learning and evolving their effective dynamics. In the proposed Generative Learning of Effective Dynamics (G-LED), instances of high dimensional data are down sampled to a lower dimensional manifold that is evolved through an auto-regressive attention mechanism. In turn, Bayesian diffusion models, that map this low-dimensional manifold onto its corresponding high-dimensional space, operate on batches of physics correlated, time sequences of data and capture the statistics of the system dynamics. We demonstrate the capabilities and drawbacks of G-LED in simulations of several benchmark systems, including the Kuramoto-Sivashinsky (KS) equation, two-dimensional high Reynolds number flow over a backward-facing step, and simulations of three-dimensional turbulent channel flow. The results demonstrate that generative learning offers new frontiers for the accurate forecasting of the statistical properties of high-dimensional systems at a reduced computational cost. The forecasting of critical phenomena in complex systems governed by partial differential equations remains challenging and computationally expensive. The authors propose a generative learning approach for the forecasting of the statistical properties of high-dimensional systems at a reduced computational cost.