Physics-constrained robust learning of open-form partial differential equations from limited and noisy data

Du，Mengge1; Chen，Yuntian2,3; Nie，Longfeng4; Lou，Siyu3; Zhang，Dongxiao2,5

2024-05-01

10.1063/5.0204187

Physics of Fluids   影响因子和分区

1070-6631

1089-7666

Unveiling the underlying governing equations of nonlinear dynamic systems remains a significant challenge. Insufficient prior knowledge hinders the determination of an accurate candidate library, while noisy observations lead to imprecise evaluations, which in turn result in redundant function terms or erroneous equations. This study proposes a framework to robustly uncover open-form partial differential equations (PDEs) from limited and noisy data. The framework operates through two alternating update processes: discovering and embedding. The discovering phase employs symbolic representation and a novel reinforcement learning (RL)-guided hybrid PDE generator to efficiently produce diverse open-form PDEs with tree structures. A neural network-based predictive model fits the system response and serves as the reward evaluator for the generated PDEs. PDEs with higher rewards are utilized to iteratively optimize the generator via the RL strategy and the best-performing PDE is selected by a parameter-free stability metric. The embedding phase integrates the initially identified PDE from the discovering process as a physical constraint into the predictive model for robust training. The traversal of PDE trees automates the construction of the computational graph and the embedding process without human intervention. Numerical experiments demonstrate our framework's capability to uncover governing equations from nonlinear dynamic systems with limited and highly noisy data and outperform other physics-informed neural network-based discovery methods. This work opens new potential for exploring real-world systems with limited understanding.

SCI ; EI

英语

其他

PHYSICS

2-s2.0-85193280875

Scopus

1.College of Engineering,Peking University,Beijing,100871,China
2.Ningbo Institute of Digital Twin,Eastern Institute of Technology,Ningbo,Zhejiang,315200,China
3.Eastern Institute for Advanced Study,Eastern Institute of Technology,Ningbo,Zhejiang,315200,China
4.School of Environmental Science and Engineering,Southern University of Science and Technology,Shenzhen,518055,China
5.National Center for Applied Mathematics Shenzhen (NCAMS),Southern University of Science and Technology,Shenzhen,Guangdong,518000,China

Du，Mengge,Chen，Yuntian,Nie，Longfeng,et al. Physics-constrained robust learning of open-form partial differential equations from limited and noisy data[J]. Physics of Fluids,2024,36(5).

Du，Mengge,Chen，Yuntian,Nie，Longfeng,Lou，Siyu,&Zhang，Dongxiao.(2024).Physics-constrained robust learning of open-form partial differential equations from limited and noisy data.Physics of Fluids,36(5).

Du，Mengge,et al."Physics-constrained robust learning of open-form partial differential equations from limited and noisy data".Physics of Fluids 36.5(2024).