Title: PDE-Net: Learning PDEs from Data,
Abstract: In this paper, we present an initial attempt to learn evolution PDEs from
data. Inspired by the latest development of neural network designs in deep
learning, we propose a new feed-forward deep network, called PDE-Net, to
fulfill two objectives at the same time: to accurately predict dynamics of
complex systems and to uncover the underlying hidden PDE models. The basic idea
of the proposed PDE-Net is to learn differential operators by learning
convolution kernels (filters), and apply neural networks or other machine
learning methods to approximate the unknown nonlinear responses. Comparing with
existing approaches, which either assume the form of the nonlinear response is
known or fix certain finite difference approximations of differential
operators, our approach has the most flexibility by learning both differential
operators and the nonlinear responses. A special feature of the proposed
PDE-Net is that all filters are properly constrained, which enables us to
easily identify the governing PDE models while still maintaining the expressive
and predictive power of the network. These constrains are carefully designed by
fully exploiting the relation between the orders of differential operators and
the orders of sum rules of filters (an important concept originated from
wavelet theory). We also discuss relations of the PDE-Net with some existing
networks in computer vision such as Network-In-Network (NIN) and Residual
Neural Network (ResNet). Numerical experiments show that the PDE-Net 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.