Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Simone Brivio, Stefania Fresca, Andrea Manzoni

TL;DR

This work proposes Continuous Geometry-Aware DL-ROMs, a novel extension of these architectures to problems featuring geometrical variability and parametrized domains that 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.

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.

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

TL;DR

This work proposes Continuous Geometry-Aware DL-ROMs, a novel extension of these architectures to problems featuring geometrical variability and parametrized domains that 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.

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.

Paper Structure

This paper contains 22 sections, 5 theorems, 58 equations, 15 figures, 4 tables.

Table of Contents

  1. Introduction
  2. ROMs: an historical perspective.
  3. Surrogate models for geometrically parameterized problems.
  4. Main contributions and work outline.
  5. Geometrically parametrized PDEs: formulation & properties
  6. Abstract setting
  7. Synthetic data generation
  8. An infinite-dimensional setting for dimensionality reduction
  9. CGA-DL-ROMs in an infinite-dimensional setting
  10. Characterization of CGA basis functions
  11. The compressive capabilities of CGA basis functions
  12. Modeling CGA basis functions with neural networks
  13. Numerical experiments
  14. Stationary flow in a stenotic channel
  15. Hyper-elasticity equations
  16. ...and 7 more sections

Key Result

Lemma 4.1

Letting $u \in L^2(\mathcal{T} \times \mathcal{P} \times \mathcal{G}; L^2(\Omega(\boldsymbol{\xi})))$, we define the POD cost functional as where $W_N = \{\{w_n\}_{n=1}^N \in L^2(\tilde{\Omega}) : (w_n,w_m)_{L^2(\tilde{\Omega})}=1, \quad \forall n,m \in \{1,\ldots,N\}\}$. Then, that is, over all possible linear finite-dimensional subspaces $W_N$, the projection error attains its minimum for the

Figures (15)

  • Figure 1: Visualization of a set of parametric domains obtained through a diffeomorphism $Z(\cdot)$; here the geometrical parameter $\xi$ regulates the radius of the circular hole included in each domain.
  • Figure 2: Example of parametric domains with different resolutions (the geometrical parameter $\xi$ is the radius of the hole).
  • Figure 3: Solution of a physically and geometrically parametrized advection diffusion reaction equation on parametric domains (first row) and action of the morphing operator $\mathcal{Z}_{\boldsymbol{\xi}}^{-1}$(second row).
  • Figure 4: Schematic representation of the CGA-DL-ROM architecture.
  • Figure 5: Stenosis test case: visualization of the effect of the geometrical parameters on the domain shape.
  • ...and 10 more figures

Theorems & Definitions (9)

  • Lemma 4.1
  • Lemma 4.2
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • Lemma B.1
  • proof
  • proof
  • proof