Dl-pde: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data

Xu，Hao1; Chang，Haibin1; Zhang，Dongxiao2,3,4

2021-03-01

10.4208/CICP.OA-2020-0142

Communications in Computational Physics   Impact Factor & Quartile

1815-2406

1991-7120

In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is to discover unknown physics and corresponding equations. However, prior to achieving this goal, major challenges remain to be resolved, including learning PDE under noisy data and limited discrete data. To overcome these challenges, in this work, a deeplearning based data-driven method, called DL-PDE, is developed to discover the governing PDEs of underlying physical processes. The DL-PDE method combines deep learning via neural networks and data-driven discovery of PDE via sparse regressions. In the DL-PDE, a neural network is first trained, then a large amount of meta-data is generated, and the required derivatives are calculated by automatic differentiation. Finally, the form of PDE is discovered by sparse regression. The proposed method is tested with physical processes, governed by the diffusion equation, the convectiondiffusion equation, the Burgers equation, and the Korteweg-de Vries (KdV) equation, for proof-of-concept and applications in real-world engineering settings. The proposed method achieves satisfactory results when data are noisy and limited.

Data-driven discovery Deep neural network Machine learning Noisy data Sparse regression

SCI

English

Corresponding

WOS:000614555300002

2-s2.0-85100289162

Scopus

1.BIC-ESAT,ERE,and 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 SurfaceWater-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,Chang，Haibin,Zhang，Dongxiao. Dl-pde: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data[J]. Communications in Computational Physics,2021,29(3):698-728.

Xu，Hao,Chang，Haibin,&Zhang，Dongxiao.(2021).Dl-pde: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data.Communications in Computational Physics,29(3),698-728.

Xu，Hao,et al."Dl-pde: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data".Communications in Computational Physics 29.3(2021):698-728.