  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.
