Deep Learning-based Reduced Order Models (DL-ROMs) provide nowadays a well-established class of accurate surrogate models for complex physical systems described by parametrized PDEs, by nonlinearly compressing the solution manifold into a handful of latent coordinates. Until now, design and application of DL-ROMs mainly focused on physically parameterized problems. Within this work, we provide a novel extension of these architectures to problems featuring geometrical variability and parametrized domains, namely, we propose Continuous Geometry-Aware DL-ROMs (CGA-DL-ROMs). In particular, the space-continuous nature of the proposed architecture matches the need to deal with multi-resolution datasets, which are quite common in the case of geometrically parametrized problems. Moreover, CGA-DL-ROMs are endowed with a strong inductive bias that makes them aware of geometrical parametrizations, thus enhancing both the compression capability and the overall performance of the architecture. Within this work, we justify our findings through a thorough theoretical analysis, and we practically validate our claims by means of a series of numerical tests encompassing physically-and-geometrically parametrized PDEs, ranging from the unsteady Navier-Stokes equations for fluid dynamics to advection-diffusion-reaction equations for mathematical biology.

翻译：基于深度学习的降阶模型（DL-ROMs）通过将解流形非线性压缩为少量潜坐标，为参数化偏微分方程描述的复杂物理系统提供了一类成熟的精确代理模型。迄今为止，DL-ROMs的设计与应用主要集中于物理参数化问题。本文提出了一种新颖的架构扩展，使其能够处理具有几何可变性和参数化域的问题，即我们提出了连续几何感知DL-ROMs（CGA-DL-ROMs）。特别地，该架构的空间连续性特质使其能够处理多分辨率数据集——这在几何参数化问题中相当常见。此外，CGA-DL-ROMs具备强大的归纳偏置，使其能够感知几何参数化，从而同时增强架构的压缩能力和整体性能。本文通过系统的理论分析论证了这些发现，并借助涵盖物理-几何参数化偏微分方程的一系列数值实验进行实证验证，包括流体动力学中的非定常Navier-Stokes方程以及数学生物学中的对流-扩散-反应方程。