PDENet
swMATH ID:  36963 
Software Authors:  Zichao Long, Yiping Lu, Xianzhong Ma, Bin Dong 
Description:  PDENet: Learning PDEs from Data. 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 feedforward deep network, called PDENet, 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 PDENet 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 PDENet 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 PDENet with some existing networks in computer vision such as NetworkInNetwork (NIN) and Residual Neural Network (ResNet). Numerical experiments show that the PDENet 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. 
Homepage:  https://arxiv.org/abs/1710.09668 
Source Code:  https://github.com/ZichaoLong/PDENet 
Dependencies:  Python 
Keywords:  Numerical Analysis; arXiv_math.NA; Machine Learning; arXiv_cs.LG; arXiv_cs.NE; arXiv_stat.ML; PDEs from Data 
Related Software:  DGM; Adam; TensorFlow; PyTorch; DeepONet; ImageNet; AlexNet; torchdiffeq; DeepXDE; UNet; MgNet; FPINNs; Keras; darch; VAMPnets; SINDy; DiffSharp; GitHub; NETT; PRMLT 
Cited in:  91 Publications 
Standard Articles
1 Publication describing the Software, including 1 Publication in zbMATH  Year 

PDENet 2.0: learning PDEs from data with a numericsymbolic hybrid deep network. Zbl 1454.65131 Long, Zichao; Lu, Yiping; Dong, Bin 
2019

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