PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network
This work addresses the challenge of data-driven PDE discovery for researchers in scientific computing and machine learning, offering a flexible method but building incrementally on prior work.
The authors tackled the problem of discovering time-dependent partial differential equations (PDEs) from observed dynamic data with minimal prior knowledge, proposing PDE-Net 2.0, a numeric-symbolic hybrid deep network that learns differential operators and nonlinear response functions, and demonstrated its potential to uncover hidden PDEs and predict long-term dynamics even in noisy environments.
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.