标题： PDE-Net 2.0：从数据中学习PDE的数值-符号混合深度网络 显示英文标题

标题： PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network

摘要： 偏微分方程（PDEs）通常基于经验观测来推导。然而，技术的最新进展使我们能够收集和存储大量数据，这为基于数据驱动的PDE发现提供了新的机会。本文提出了一种新的深度神经网络，称为PDE-Net 2.0，用于从观察到的动态数据中发现（时间相关的）PDE，而对潜在机制只需少量先验知识。PDE-Net 2.0的设计基于我们之前的工作\cite{Long2018PDE}，其中提出了PDE-Net的原始版本。PDE-Net 2.0结合了通过卷积进行微分算子的数值逼近和一个用于模型恢复的符号多层神经网络。与现有方法相比，PDE-Net 2.0具有最大的灵活性和表达能力，因为它同时学习微分算子和潜在PDE模型的非线性响应函数。数值实验表明，PDE-Net 2.0有可能揭示观察到的动力学的隐藏PDE，并在相对较长的时间内预测动力学行为，即使在嘈杂的环境中也是如此。 显示英文摘要

摘要： Partial differential equations (PDEs) are commonly derived based on empirical observations. However, recent advances of technology enable us to collect and store massive amount of data, which offers new opportunities for data-driven discovery of PDEs. In this paper, we propose a new deep neural network, called PDE-Net 2.0, to discover (time-dependent) PDEs from observed dynamic data with minor prior knowledge on the underlying mechanism that drives the dynamics. The design of PDE-Net 2.0 is based on our earlier work \cite{Long2018PDE} where the original version of PDE-Net was proposed. PDE-Net 2.0 is a combination of numerical approximation of differential operators by convolutions and a symbolic multi-layer neural network for model recovery. Comparing with existing approaches, PDE-Net 2.0 has the most flexibility and expressive power by learning both differential operators and the nonlinear response function of the underlying PDE model. Numerical experiments show that the PDE-Net 2.0 has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.