Solving Time-Dependent Parametric PDEs by Multiclass Classification-Based Reduced Order Model (CSIAM Transactions on Applied Mathematics, 2023)
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 solutionas time evolves. Based on this fact, we classify the dataset according to the magnitudeof the solutions and construct corresponding subnets dependent on different typesof data. Then we train a classifier to integrate different subnets together to obtain theMC-ROM. When subsets have the same architecture, we can use transfer learning techniques to accelerate offline training. Numerical experiments show that the MC-ROMimproves the generalization ability of the DL-ROM both for diffusion- and convection-dominant problems, and maintains the DL-ROM’s advantage of good approximationability. We also compare the approximation accuracy and computational efficiencyof 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 itscomputational performance is more efficient than the POD’s.