Solving Time-Dependent Parametric PDEs by Multiclass Classification-Based Reduced Order Model

Abstract

In this paper, we propose a network model, the multiclass classificationbased 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 diffusiondominant 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 convectiondominant 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 convectiondominant 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.

An Adaptive Method Based on Local Dynamic Mode Decomposition for Parametric Dynamical Systems

“…A large class of methods consider the dynamical low rank (DLR) approximation (see [17][18][19][20][21]) which allows both the deterministic and stochastic basis functions to evolve in time. Other strategies based on deep learning (DL) algorithms were proposed in [22][23][24] to construct the efficient surrogate model for time-dependent parametrized PDEs. In this contribution, we try to combine dynamic mode decomposition with the local Taylor approximation to construct an efficient and reliable approximation of input-output relationship (i.e.…”