Long-time integration of parametric evolution equations with physics-informed DeepONets
Long-time integration of parametric evolution equations with physics-informed DeepONets
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DOI：
10.1016/j.jcp.2022.111855
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发表时间：
2021-06
期刊：
影响因子：
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通讯作者：
Sifan Wang;P. Perdikaris
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文献类型：
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作者：
Sifan Wang;P. Perdikaris
常微分方程和偏微分方程（ODE/PDE）在分析和模拟科学和工程各个领域的复杂动态过程中发挥着至关重要的作用。近年来，机器学习工具正致力于引入新的有效方法来模拟这些方程，然而现有的方法无法在长时间范围内可靠地返回稳定和准确的预测。我们的目标是通过引入一个有效的框架来学习进化算子，将随机初始条件映射到短时间间隔内的相关ODE/PDE解决方案，以应对这一挑战。这样的算子可以通过深度神经网络进行参数化，这些神经网络以完全自我监督的方式进行训练，而不需要生成任何成对的输入输出观察。然后，可以通过使用每个预测作为下一个评估步骤的初始条件来迭代地评估训练模型，从而获得跨一系列初始条件的全局长期预测。这介绍了一种新的时域分解方法，该方法在对广泛的参数ODE和PDE系统进行精确的长时间模拟方面是有效的，从波传播到反应扩散动力学和刚性化学动力学，介绍了一种快速模拟科学和工程中非平衡过程的新方法。
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.