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.

本文提出了一种用于无配对输入输出观测的深度神经网络参数化的无穷维算子的学习框架，以实现对于参数 ODE/PDE 系统的精确长时间模拟，该方法虽然比传统数值解算法计算成本低，但可靠性更高且能够全局评估。