Long-time integration of parametric evolution equations with physics-informed DeepONets
This addresses the problem of unreliable long-term PDE simulations in science and engineering, offering a novel method that is incremental in improving temporal stability.
The paper tackles the challenge of stable and accurate long-time predictions for parametric evolution equations by introducing a framework that learns infinite-dimensional operators mapping initial conditions to PDE solutions within short intervals, enabling global predictions through iterative evaluation. This approach achieves accurate simulations for various systems, including wave propagation and stiff chemical kinetics, at a fraction of the computational cost of classical solvers.
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.