Handling geometrical variability in nonlinear reduced order modeling through Continuous Geometry-Aware DL-ROMs
This work addresses the problem of efficiently modeling complex physical systems with varying geometries for researchers and engineers in computational physics, representing an incremental extension of existing DL-ROM methods.
The authors tackled the challenge of extending deep learning-based reduced order models (DL-ROMs) to handle geometrical variability and parametrized domains in physical systems, proposing Continuous Geometry-Aware DL-ROMs (CGA-DL-ROMs) that improve compression and performance, validated through numerical tests on PDEs like Navier-Stokes and advection-diffusion-reaction equations.
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.