PDE-Net swMATH ID: 36963 Software Authors: Zichao Long, Yiping Lu, Xianzhong Ma, Bin Dong Description: PDE-Net: 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 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. Homepage: https://arxiv.org/abs/1710.09668 Source Code: https://github.com/ZichaoLong/PDE-Net 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; PyTorch; TensorFlow; DeepONet; DeepXDE; AlexNet; ImageNet; torchdiffeq; U-Net; FPINNs; MgNet; NSFnets; Keras; L-BFGS; PPINN; DiffSharp; PhyGeoNet; darch; VAMPnets Cited in: 111 Documents Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year PDE-Net 2.0: learning PDEs from data with a numeric-symbolic hybrid deep network. Zbl 1454.65131Long, Zichao; Lu, Yiping; Dong, Bin 2019 all top 5 Cited by 277 Authors 9 Xiu, Dongbin 6 Jentzen, Arnulf 5 Chen, Zhen 5 Wu, Kailiang 4 Zhang, Dongxiao 3 Arridge, Simon R. 3 Darbon, Jerome 3 Hutzenthaler, Martin 3 Karniadakis, George Em 3 Koutsourelakis, Phaedon-Stelios 3 Lu, Jianfeng 3 Meng, Tingwei 3 Qin, Tong 3 Xu, Hao 3 Ying, Lexing 2 Adams, Nikolaus A. 2 Ayed, Ibrahim 2 Bazant, Martin Z. 2 Beck, Christian 2 Becker, Sebastian 2 Chang, Haibin 2 Churchill, Victor 2 de Bézenac, Emmanuel 2 Dong, Bin 2 Gallinari, Patrick 2 Glasner, Karl B. 2 Grohs, Philipp 2 Hesthaven, Jan S. 2 Hornung, Fabian 2 Kang, Sung Ha 2 Khoo, Yuehaw 2 Kruse, Thomas 2 Li, Yingzhou 2 Liao, Wenjing 2 Liu, Yingjie 2 Liu, Yongming 2 Lu, Lu 2 Perdikaris, Paris G. 2 Ray, Deep 2 Schaeffer, Hayden 2 Sheng, Hailong 2 Stuart, Andrew M. 2 Wang, Jianxun 2 Wang, Nanzhe 2 Xiao, Heng 2 Yang, Chao 2 Zhang, Linan 1 Alla, Alessandro 1 Aragon-Calvo, Miguel A. 1 Baharifard, Fatemeh 1 Bao, Feng 1 Barati Farimani, Amir 1 Bekkers, Erik J. 1 Bennett, Kane C. 1 Bezgin, Deniz A. 1 Bocquet, Marc 1 Bortz, David M. 1 Both, Gert-Jan 1 Botsas, Themistoklis 1 Braatz, Richard D. 1 Braunwarth, Ramon 1 Brunton, Steven L. 1 Cai, Shengze 1 Cai, Weihua 1 Calandra, Henri 1 Cao, Ba Trung 1 Chen, XiaoQian 1 Chen, Yifan 1 Chen, Yu 1 Chen, Yuntian 1 Cheng, Xiaoliang 1 Cheng, Xiuyuan 1 Cheridito, Patrick 1 Choi, Youngsoo 1 Chou, Ching-Shan 1 Choudhury, Subham 1 Cotter, Andrew 1 Cox, Ben T. 1 Cui, Tao 1 Cyr, Eric C. 1 Dona, Jérémie 1 Dong, Guozhi 1 Dower, Peter M. 1 Du, Qiang 1 Duits, Remco 1 Fan, Yuwei 1 Farchi, Alban 1 Felsberger, Lukas 1 Forgione, Marco 1 Funke, Simon W. 1 Gao, Han 1 Gao, Yi 1 Germann, Timothy Clark 1 Ghodsi, Ali 1 Gibou, Frédéric 1 Gratton, Serge 1 Gulian, Mamikon 1 Guo, Zhiwei 1 Gupta, Maya R. 1 Hähnel, Philipp ...and 177 more Authors all top 5 Cited in 44 Serials 40 Journal of Computational Physics 8 SIAM Journal on Scientific Computing 7 Computer Methods in Applied Mechanics and Engineering 7 Communications in Computational Physics 3 Journal of Machine Learning Research (JMLR) 2 Applied Mathematics and Computation 2 Physica D 2 Journal of Scientific Computing 2 Journal of Mathematical Imaging and Vision 2 Computational and Applied Mathematics 2 Advances in Computational Mathematics 2 Multiscale Modeling & Simulation 1 Computers & Mathematics with Applications 1 Bulletin of Mathematical Biology 1 Chaos, Solitons and Fractals 1 Journal of Computational and Applied Mathematics 1 Mathematics and Computers in Simulation 1 SIAM Journal on Numerical Analysis 1 MCSS. Mathematics of Control, Signals, and Systems 1 European Journal of Applied Mathematics 1 Machine Learning 1 MSCS. Mathematical Structures in Computer Science 1 SIAM Journal on Applied Mathematics 1 SIAM Review 1 ETNA. Electronic Transactions on Numerical Analysis 1 European Journal of Control 1 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 1 Optimization Methods & Software 1 Computational Geosciences 1 Foundations of Computational Mathematics 1 Journal of Numerical Mathematics 1 SIAM Journal on Applied Dynamical Systems 1 Acta Numerica 1 Journal of Statistical Mechanics: Theory and Experiment 1 Discrete and Continuous Dynamical Systems. Series S 1 Science China. Mathematics 1 GEM - International Journal on Geomathematics 1 Journal of Agricultural, Biological, and Environmental Statistics 1 Journal of the Operations Research Society of China 1 Research in the Mathematical Sciences 1 Journal of Nonlinear and Variational Analysis 1 Communications on Applied Mathematics and Computation 1 SN Partial Differential Equations and Applications 1 Foundations of Data Science all top 5 Cited in 24 Fields 72 Computer science (68-XX) 71 Numerical analysis (65-XX) 40 Partial differential equations (35-XX) 17 Fluid mechanics (76-XX) 10 Statistics (62-XX) 9 Dynamical systems and ergodic theory (37-XX) 9 Biology and other natural sciences (92-XX) 7 Operations research, mathematical programming (90-XX) 4 Calculus of variations and optimal control; optimization (49-XX) 4 Mechanics of deformable solids (74-XX) 4 Systems theory; control (93-XX) 4 Information and communication theory, circuits (94-XX) 3 Probability theory and stochastic processes (60-XX) 3 Geophysics (86-XX) 2 Statistical mechanics, structure of matter (82-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 Mathematical logic and foundations (03-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Category theory; homological algebra (18-XX) 1 Functions of a complex variable (30-XX) 1 Approximations and expansions (41-XX) 1 Operator theory (47-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Classical thermodynamics, heat transfer (80-XX) Citations by Year