• Open Access

Robust discovery of partial differential equations in complex situations

Hao Xu and Dongxiao Zhang
Phys. Rev. Research 3, 033270 – Published 21 September 2021

Abstract

Data-driven discovery of partial differential equations (PDEs) has achieved considerable development in recent years. Several aspects of problems have been resolved by sparse regression-based and neural-network-based methods. However, the performance of existing methods lacks stability when dealing with complex situations, including sparse data with high noise, high-order derivatives, and shock waves, which bring obstacles to calculating derivatives accurately. Therefore, a robust PDE discovery framework, called the robust deep-learning genetic algorithm (R-DLGA), that incorporates the physics-informed neural network (PINN) is proposed in this paper. In the framework, preliminary results of potential terms provided by the DLGA are added into the loss function of the PINN as physical constraints to improve the accuracy of derivative calculation. It assists in optimizing the preliminary result and obtaining the ultimately discovered PDE by eliminating the error compensation terms. The stability and accuracy of the proposed R-DLGA in several complex situations are examined for proof and concept, and the results prove that the proposed framework can calculate derivatives accurately with the optimization of the PINN and possesses surprising robustness for complex situations, including sparse data with high noise, high-order derivatives, and shock waves.

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  • Received 30 May 2021
  • Accepted 30 August 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.033270

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Fluid DynamicsStatistical Physics & ThermodynamicsInterdisciplinary PhysicsNonlinear Dynamics

Authors & Affiliations

Hao Xu1,* and Dongxiao Zhang2,3,†

  • 1Beijing Innovation Center for Engineering Science and Advanced Technology (BIC-ESAT), Energy & Resources Engineering (ERE), and State Key Laboratory for Turbulence and Complex Systems (SKLTCS), College of Engineering, Peking University, Beijing 100871, People's Republic of China
  • 2School of Environmental Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, People's Republic of China
  • 3Intelligent Energy Lab, Peng Cheng Laboratory, Shenzhen 518000, People's Republic of China

  • *390260267@pku.edu.cn
  • zhangdx@sustech.edu.cn

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Vol. 3, Iss. 3 — September - November 2021

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