Consistent Long-Term Forecasting of Ergodic Dynamical Systems
This work addresses forecasting challenges in dynamical systems, offering a method to improve consistency and accuracy for applications in fields like climate modeling or finance, though it appears incremental by building on existing operator theory techniques.
The paper tackles the problem of long-term forecasting for ergodic dynamical systems by introducing a learning paradigm that combines eigenvalue deflation and feature centering, resulting in estimators that satisfy uniform learning bounds and conserve mass across forecasted distributions.
We study the evolution of distributions under the action of an ergodic dynamical system, which may be stochastic in nature. By employing tools from Koopman and transfer operator theory one can evolve any initial distribution of the state forward in time, and we investigate how estimators of these operators perform on long-term forecasting. Motivated by the observation that standard estimators may fail at this task, we introduce a learning paradigm that neatly combines classical techniques of eigenvalue deflation from operator theory and feature centering from statistics. This paradigm applies to any operator estimator based on empirical risk minimization, making them satisfy learning bounds which hold uniformly on the entire trajectory of future distributions, and abide to the conservation of mass for each of the forecasted distributions. Numerical experiments illustrates the advantages of our approach in practice.