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...

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Veröffentlicht in:Journal of computational physics 2023-02, Vol.475, p.111855, Article 111855
Hauptverfasser: Wang, Sifan, Perdikaris, Paris
Format: Artikel
Sprache:eng
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Zusammenfassung: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. •We demonstrate that physics-informed neural networks fail to solve differential equations in long temporal domains.•We develop an effective framework for learning the flow map of arbitrary ordinary and partial differential equations.•The proposed deep learning models can be trained in an entirely self-supervised manner, i.e. without paired input-output data.•We demonstrate how the proposed models can be sequentially evaluated to perform long-time integration of evolution equations.•We consider diverse benchmarks including wave propagation, reaction-diffusion dynamics and stiff chemical kinetics.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2022.111855