Solving time-dependent parametric PDEs by multiclass classification-based reduced order model

TL;DR

This paper introduces MC-ROM, a classification-based reduced order model that improves the approximation and efficiency of solving time-dependent parametric PDEs, especially for diffusion- and convection-dominant problems, by classifying solution behaviors.

Contribution

The paper proposes a novel MC-ROM that classifies solution data and constructs specialized subnets, enhancing generalization and efficiency over existing deep learning ROMs and POD methods.

Findings

MC-ROM outperforms DL-ROM in generalization for various PDEs.

MC-ROM achieves higher accuracy than POD in low-dimensional spaces.

MC-ROM offers better computational efficiency than POD for diffusion problems.

Abstract

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) 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.