TY - JOUR

T1 - Physics-informed machine learning for reduced-order modeling of nonlinear problems

AU - Chen, Wenqian

AU - Wang, Qian

AU - Hesthaven, Jan S.

AU - Zhang, Chuhua

N1 - Funding Information:
The first author is financially supported by Xi'an Jiaotong University Graduate Short-term Academic Visiting Program, National Key Research and Development Project of China [Grant number 2016YFB0200901 ], National Science and Technology Major Project of China [Grant number 2017-II-0006-0020 ].
Publisher Copyright:
© 2021 The Author(s)

PY - 2021/12/1

Y1 - 2021/12/1

N2 - A reduced basis method based on a physics-informed machine learning framework is developed for efficient reduced-order modeling of parametrized partial differential equations (PDEs). A feedforward neural network is used to approximate the mapping from the time-parameter to the reduced coefficients. During the offline stage, the network is trained by minimizing the weighted sum of the residual loss of the reduced-order equations, and the data loss of the labeled reduced coefficients that are obtained via the projection of high-fidelity snapshots onto the reduced space. Such a network is referred to as physics-reinforced neural network (PRNN). As the number of residual points in time-parameter space can be very large, an accurate network – referred to as physics-informed neural network (PINN) – can be trained by minimizing only the residual loss. However, for complex nonlinear problems, the solution of the reduced-order equation is less accurate than the projection of high-fidelity solution onto the reduced space. Therefore, the PRNN trained with the snapshot data is expected to have higher accuracy than the PINN. Numerical results demonstrate that the PRNN is more accurate than the PINN and a purely data-driven neural network for complex problems. During the reduced basis refinement, the PRNN may obtain higher accuracy than the direct reduced-order model based on a Galerkin projection. The online evaluation of PINN/PRNN is orders of magnitude faster than that of the Galerkin reduced-order model.

AB - A reduced basis method based on a physics-informed machine learning framework is developed for efficient reduced-order modeling of parametrized partial differential equations (PDEs). A feedforward neural network is used to approximate the mapping from the time-parameter to the reduced coefficients. During the offline stage, the network is trained by minimizing the weighted sum of the residual loss of the reduced-order equations, and the data loss of the labeled reduced coefficients that are obtained via the projection of high-fidelity snapshots onto the reduced space. Such a network is referred to as physics-reinforced neural network (PRNN). As the number of residual points in time-parameter space can be very large, an accurate network – referred to as physics-informed neural network (PINN) – can be trained by minimizing only the residual loss. However, for complex nonlinear problems, the solution of the reduced-order equation is less accurate than the projection of high-fidelity solution onto the reduced space. Therefore, the PRNN trained with the snapshot data is expected to have higher accuracy than the PINN. Numerical results demonstrate that the PRNN is more accurate than the PINN and a purely data-driven neural network for complex problems. During the reduced basis refinement, the PRNN may obtain higher accuracy than the direct reduced-order model based on a Galerkin projection. The online evaluation of PINN/PRNN is orders of magnitude faster than that of the Galerkin reduced-order model.

KW - Feedforward neural network

KW - Nonlinear PDE

KW - Physics-informed machine learning

KW - Reduced-order modeling

UR - http://www.scopus.com/inward/record.url?scp=85114241678&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2021.110666

DO - 10.1016/j.jcp.2021.110666

M3 - Article

AN - SCOPUS:85114241678

VL - 446

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

M1 - 110666

ER -