Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data

Xu，Hao1; Zhang，Dongxiao2,3,4; Wang，Nanzhe1

2021-11-15

10.1016/j.jcp.2021.110592

JOURNAL OF COMPUTATIONAL PHYSICS   影响因子和分区

0021-9991

1090-2716

Data-driven discovery of partial differential equations (PDEs) has attracted increasing attention in recent years. Although significant progress has been made, certain unresolved issues remain. For example, for PDEs with high-order derivatives, the performance of existing methods is unsatisfactory, especially when the data are sparse and noisy. It is also difficult to discover heterogeneous parametric PDEs where heterogeneous parameters are embedded in the partial differential operators. In this work, a new framework combining deep-learning and integral form is proposed to handle the above-mentioned problems simultaneously, and improve the accuracy and stability of PDE discovery. In the framework, a deep neural network is firstly trained with observation data to generate meta-data and calculate derivatives. Then, a unified integral form is defined, and the genetic algorithm is employed to discover the best structure. Finally, the values of parameters are calculated, and whether the parameters are constants or variables is identified. Numerical experiments proved that our proposed algorithm is more robust to noise and more accurate compared with existing methods due to the utilization of integral form. Our proposed algorithm is also able to discover PDEs with high-order derivatives or heterogeneous parameters accurately with sparse and noisy data.

Deep-learning Integral form Noisy data PDE discovery

SCI ; EI

英语

通讯

National Natural Science Foundation of China[51520105005,"U1663208"] ; National Science and Technology Major Project of China["2017ZX05009-005","2017ZX05049-003"]

Computer Science ; Physics

Computer Science, Interdisciplinary Applications ; Physics, Mathematical

WOS:000722631000004

ACADEMIC PRESS INC ELSEVIER SCIENCE

20213510833261

Deep neural networks ; Genetic algorithms ; Mathematical operators ; Numerical methods ; Partial differential equations

Calculus:921.2 ; Numerical Methods:921.6

PHYSICS

2-s2.0-85113587915

Scopus

1.BIC-ESAT,ERE,SKLTCS,College of Engineering,Peking University,Beijing,100871,China
2.Guangdong Provincial Key Laboratory of Soil and Groundwater Pollution Control,School of Environmental Science and Engineering,Southern University of Science and Technology,Shenzhen,518055,China
3.State Environmental Protection Key Laboratory of Integrated Surface Water-Groundwater Pollution Control,School of Environmental Science and Engineering,Southern University of Science and Technology,Shenzhen,518055,China
4.Intelligent Energy Lab,Peng Cheng Laboratory,Shenzhen,518000,China

Xu，Hao,Zhang，Dongxiao,Wang，Nanzhe. Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data[J]. JOURNAL OF COMPUTATIONAL PHYSICS,2021,445.

Xu，Hao,Zhang，Dongxiao,&Wang，Nanzhe.(2021).Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data.JOURNAL OF COMPUTATIONAL PHYSICS,445.

Xu，Hao,et al."Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data".JOURNAL OF COMPUTATIONAL PHYSICS 445(2021).