Analysis of the Decoder Width for Parametric Partial Differential Equations

TL;DR

This paper provides a theoretical analysis of decoder width in Meta-Auto-Decoder (MAD) for parametric PDEs, demonstrating its potential to efficiently approximate complex solution sets across various PDE types.

Contribution

It offers the first theoretical bounds on decoder width for MAD applied to multiple PDEs, including challenging advection and shape-dependent elliptic equations.

Findings

Decoder widths decay rapidly, indicating efficient approximation.

MAD shows promise for complex PDEs with variable parameters.

Theoretical bounds support MAD's effectiveness in reduced order modeling.

Abstract

Recently, Meta-Auto-Decoder (MAD) was proposed as a novel reduced order model (ROM) for solving parametric partial differential equations (PDEs), and the best possible performance of this method can be quantified by the decoder width. This paper aims to provide a theoretical analysis related to the decoder width. The solution sets of several parametric PDEs are examined, and the upper bounds of the corresponding decoder widths are estimated. In addition to the elliptic and the parabolic equations on a fixed domain, we investigate the advection equations that present challenges for classical linear ROMs, as well as the elliptic equations with the computational domain shape as a variable PDE parameter. The resulting fast decay rates of the decoder widths indicate the promising potential of MAD in addressing these problems.