Long-time integration of parametric evolution equations with physics-informed DeepONets

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 such equations, 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 evolution operators that map random initial conditions to associated ODE/PDE solutions within a short time interval. Such operators can be parametrized by deep neural networks that are trained in an entirely self-supervised manner without requiring one to generate 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, introducing a new way of rapidly emulating non-equilibrium processes in science and engineering.(c) 2022 Elsevier Inc. All rights reserved.

Automated summary

Ordinary and partial differential equations (ODEs/PDEs) are crucial in analyzing and simulating complex dynamic processes in various fields of science and engineering. Current machine learning approaches have limitations in accurately predicting long-term behaviors of these equations. This study introduces an effective learning framework for evolution operators that can map random initial conditions to associated ODE/PDE solutions within a short time interval. The framework utilizes deep neural networks and does not require paired input-output observations. The proposed approach of temporal domain decomposition provides accurate long-term simulations for a wide range of parametric ODE and PDE systems, revolutionizing the emulation of non-equilibrium processes in science and engineering.