Handling geometrical variability in nonlinear reduced order modeling through Continuous Geometry-Aware DL-ROMs

TL;DR

This paper introduces Continuous Geometry-Aware DL-ROMs, an extension of deep learning-based reduced order models that effectively handle geometrical variability in parametrized PDEs, improving compression and performance.

Contribution

The paper presents a novel architecture, CGA-DL-ROMs, that incorporates geometrical parametrizations and handles multi-resolution datasets for complex physical problems.

Findings

Enhanced compression and performance in geometrically parametrized PDEs

Effective handling of multi-resolution datasets

Validated on fluid dynamics and biological PDEs

Abstract

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.