TY - JOUR
T1 - Solving Time-Dependent Parametric PDEs by Multiclass Classification-Based Reduced Order Model
AU - Cui , Chen
AU - Jiang , Kai
AU - Shu , Shi
JO - CSIAM Transactions on Applied Mathematics
VL - 1
SP - 13
EP - 40
PY - 2023
DA - 2023/01
SN - 4
DO - http://doi.org/10.4208/csiam-am.SO-2021-0042
UR - https://global-sci.org/intro/article_detail/csiam-am/21334.html
KW - Parametric partial differential equation, reduced order model, deep learning, generalization ability, classification, computational complexity.
AB - <p style="text-align: justify;">In this paper, we propose a network model, the multiclass classification-based reduced order model (MC-ROM), for solving time-dependent parametric partial differential equations (PPDEs). This work is inspired by the observation of applying the deep learning-based reduced order model (DL-ROM) [14] to solve diffusion-dominant PPDEs. We find that the DL-ROM has a good approximation for some special model parameters, but it cannot approximate the drastic changes of the solution
as time evolves. Based on this fact, we classify the dataset according to the magnitude
of the solutions and construct corresponding subnets dependent on different types
of data. Then we train a classifier to integrate different subnets together to obtain the
MC-ROM. When subsets have the same architecture, we can use transfer learning techniques to accelerate offline training. Numerical experiments show that the MC-ROM
improves the generalization ability of the DL-ROM both for diffusion- and convection-dominant problems, and maintains the DL-ROM’s advantage of good approximation
ability. We also compare the approximation accuracy and computational efficiency
of the proper orthogonal decomposition (POD) which is not suitable for convection-dominant problems. For diffusion-dominant problems, the MC-ROM has better approximation accuracy than the POD in a small dimensionality reduction space, and its
computational performance is more efficient than the POD’s.</p>