  Ordinary and partial differential equations (ODEs/PDEs) play a paramount role
in analyzing and simulating complex dynamic processes across all corners of
science and engineering. In recent years machine learning tools are aspiring to
introduce new effective ways of simulating PDEs, however existing approaches
are not able to reliably return stable and accurate predictions across long
temporal horizons. We aim to address this challenge by introducing an effective
framework for learning infinite-dimensional operators that map random initial
conditions to associated PDE solutions within a short time interval. Such
latent operators can be parametrized by deep neural networks that are trained
in an entirely self-supervised manner without requiring any paired input-output
observations. Global long-time predictions across a range of initial conditions
can be then obtained by iteratively evaluating the trained model using each
prediction as the initial condition for the next evaluation step. This
introduces a new approach to temporal domain decomposition that is shown to be
effective in performing accurate long-time simulations for a wide range of
parametric ODE and PDE systems, from wave propagation, to reaction-diffusion
dynamics and stiff chemical kinetics, all at a fraction of the computational
cost needed by classical numerical solvers.
