PDE-Net: Learning PDEs from Data

10/26/2017
by   Zichao Long, et al.
0

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.

READ FULL TEXT

page 11

page 14

research
11/30/2018

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

Partial differential equations (PDEs) are commonly derived based on empi...
research
09/08/2020

Neural Time-Dependent Partial Differential Equation

Partial differential equations (PDEs) play a crucial role in studying a ...
research
03/10/2023

Neural Partial Differential Equations with Functional Convolution

We present a lightweighted neural PDE representation to discover the hid...
research
02/27/2022

DeepPropNet – A Recursive Deep Propagator Neural Network for Learning Evolution PDE Operators

In this paper, we propose a deep neural network approximation to the evo...
research
01/28/2023

MetaNO: How to Transfer Your Knowledge on Learning Hidden Physics

Gradient-based meta-learning methods have primarily been applied to clas...
research
05/15/2023

Finite Expression Methods for Discovering Physical Laws from Data

Nonlinear dynamics is a pervasive phenomenon observed in various scienti...
research
10/20/2018

BCR-Net: a neural network based on the nonstandard wavelet form

This paper proposes a novel neural network architecture inspired by the ...

Please sign up or login with your details

Forgot password? Click here to reset