PDE-Net: Learning PDEs from data

Zichao Long, Yiping Lu, Xianzhong Ma, Bin Dong

Research output: Contribution to conferencePaper

8 Citations (Scopus)

Abstract

Partial differential equations (PDEs) play a prominent role in many disciplines such as applied mathematics, physics, chemistry, material science, computer science, etc. PDEs are commonly derived based on physical laws or empirical observations. However, the governing equations for many complex systems in modern applications are still not fully known. With the rapid development of sensors, computational power, and data storage in the past decade, huge quantities of data can be easily collected and efficiently stored. Such vast quantity of data offers new opportunities for data-driven discovery of hidden physical laws. 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.

Original languageEnglish (US)
StatePublished - Jan 1 2018
Event6th International Conference on Learning Representations, ICLR 2018 - Vancouver, Canada
Duration: Apr 30 2018May 3 2018

Conference

Conference6th International Conference on Learning Representations, ICLR 2018
CountryCanada
CityVancouver
Period4/30/185/3/18

Fingerprint

Partial differential equations
neural network
learning
data storage
Law
learning method
Neural networks
computer science
physics
Large scale systems
flexibility
chemistry
Differential Equations
mathematics
experiment
Materials science
science
Convolution
Computer science
Computer vision

ASJC Scopus subject areas

  • Education
  • Computer Science Applications
  • Linguistics and Language
  • Language and Linguistics

Cite this

Long, Z., Lu, Y., Ma, X., & Dong, B. (2018). PDE-Net: Learning PDEs from data. Paper presented at 6th International Conference on Learning Representations, ICLR 2018, Vancouver, Canada.

PDE-Net : Learning PDEs from data. / Long, Zichao; Lu, Yiping; Ma, Xianzhong; Dong, Bin.

2018. Paper presented at 6th International Conference on Learning Representations, ICLR 2018, Vancouver, Canada.

Research output: Contribution to conferencePaper

Long, Z, Lu, Y, Ma, X & Dong, B 2018, 'PDE-Net: Learning PDEs from data' Paper presented at 6th International Conference on Learning Representations, ICLR 2018, Vancouver, Canada, 4/30/18 - 5/3/18, .
Long Z, Lu Y, Ma X, Dong B. PDE-Net: Learning PDEs from data. 2018. Paper presented at 6th International Conference on Learning Representations, ICLR 2018, Vancouver, Canada.
Long, Zichao ; Lu, Yiping ; Ma, Xianzhong ; Dong, Bin. / PDE-Net : Learning PDEs from data. Paper presented at 6th International Conference on Learning Representations, ICLR 2018, Vancouver, Canada.
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