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.

翻译：近期，元自动解码器（MAD）被提出作为一种求解参量偏微分方程（PDE）的新型降阶模型（ROM），该方法的最优性能可通过解码器宽度进行量化。本文旨在提供与解码器宽度相关的理论分析。我们考察了若干参量PDE的解集，并估算了相应解码器宽度的上界。除了固定域上的椭圆型和抛物型方程外，我们还研究了对经典线性ROM构成挑战的平流方程，以及以计算域形状为可变PDE参数的椭圆型方程。所得解码器宽度的快速衰减速率表明MAD在解决这些问题上具有广阔前景。