Deep-OSG: Deep learning of operators in semigroup

Chen, Junfeng1; Wu, Kailiang1,2,3

2023-11-15

10.1016/j.jcp.2023.112498

JOURNAL OF COMPUTATIONAL PHYSICS   Impact Factor & Quartile

0021-9991

1090-2716

This paper proposes a novel deep learning approach for learning operators in semigroup, with applications to modeling unknown autonomous dynamical systems using time series data collected at varied time lags. It is a sequel to the previous flow map learning (FML) works [Qin et al. (2019) [29]], [Wu and Xiu (2020) [30]], and [Chen et al. (2022) [31]], which focused on learning single evolution operator with a fixed time step. This paper aims to learn a family of evolution operators with variable time steps, which constitute a semigroup for an autonomous system. The semigroup property is very crucial and links the system's evolutionary behaviors across varying time scales, but it was not considered in the previous works. We propose for the first time a framework of embedding the semigroup property into the data-driven learning process, through a novel neural network architecture and new loss functions. The framework is very feasible, can be combined with any suitable neural networks, and is applicable to learning general autonomous ODEs and PDEs. We present the rigorous error estimates and variance analysis to understand the prediction accuracy and robustness of our approach, showing the remarkable advantages of semigroup awareness in our model. Moreover, our approach allows one to arbitrarily choose the time steps for prediction and ensures that the predicted results are well selfmatched and consistent. Extensive numerical experiments demonstrate that embedding the semigroup property notably reduces the data dependency of deep learning models and greatly improves the accuracy, robustness, and stability for long-time prediction.(c) 2023 Elsevier Inc. All rights reserved.

Deep learning Neural network Learning ODE and PDE Operator learning Semigroup Flow map learning

SCI

English

First ; Corresponding

Shenzhen Science and Technology Program[RCJC20221008092757098] ; National Natural Science Foundation of China[12171227]

Computer Science ; Physics

Computer Science, Interdisciplinary Applications ; Physics, Mathematical

WOS:001086047800001

ACADEMIC PRESS INC ELSEVIER SCIENCE

PHYSICS

Web of Science

1.Southern Univ Sci & Technol, Dept Math, Shenzhen 518055, Peoples R China
2.Southern Univ Sci & Technol, SUSTech Int Ctr Math, Shenzhen 518055, Peoples R China
3.Natl Ctr Appl Math Shenzhen NCAMS, Shenzhen 518055, Peoples R China

Chen, Junfeng,Wu, Kailiang. Deep-OSG: Deep learning of operators in semigroup[J]. JOURNAL OF COMPUTATIONAL PHYSICS,2023,493.

Chen, Junfeng,&Wu, Kailiang.(2023).Deep-OSG: Deep learning of operators in semigroup.JOURNAL OF COMPUTATIONAL PHYSICS,493.

Chen, Junfeng,et al."Deep-OSG: Deep learning of operators in semigroup".JOURNAL OF COMPUTATIONAL PHYSICS 493(2023).