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.

本文提出一种新的深度神经网络PDE-Net 2.0，利用先前成果基于微分算子卷积数值逼近及符号多层神经网络模型恢复，从大量动态数据中探索时变偏微分方程(PDEs)的复杂性及本质特点，具有较高的灵活性和表达能力，研究实验证明PDE-Net 2.0对于探测并预测噪声环境下的隐藏式动态过程有很好的潜力。

PDE-Net 2.0: 数字符号混合深度网络学习从数据中推导偏微分方程