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A brief review of Reduced Order Models using intrusive and non-intrusive techniques

Guglielmo Padula, Michele Girfoglio, Gianluigi Rozza

TL;DR

The paper surveys reduced order modeling approaches for parameterized PDEs, contrasting intrusive Galerkin ROMs with non-intrusive data-driven surrogates such as PINN, DDNN, RBF, DMD, and GPR, and discusses geometry parametrization via FFD and dimensionality reduction via Active Subspaces. It demonstrates that online speedups up to $10^5$ are achievable, with method performance varying by problem type (e.g., DMD+RBF accuracy in fluids, POD-Galerkin with larger RBs for cylinders, RBF on geometry). Through academic benchmarks (lid-driven cavity, flow past a cylinder, Stanford Bunny) and an industrial refrigerator case, the work illustrates the breadth of ROM techniques and their potential for real-time control and optimization in engineering contexts.

Abstract

Reduced Order Models (ROMs) have gained a great attention by the scientific community in the last years thanks to their capabilities of significantly reducing the computational cost of the numerical simulations, which is a crucial objective in applications like real time control and shape optimization. This contribution aims to provide a brief overview about such a topic. We discuss both an intrusive framework based on a Galerkin projection technique and non-intrusive approaches, including Physics Informed Neural Networks (PINN), purely Data-Driven Neural Networks (DDNN), Radial Basis Functions (RBF), Dynamic Mode Decomposition (DMD) and Gaussian Process Regression (GPR). We also briefly mention geometrical parametrization and dimensionality reduction methods like Active Subspaces (AS). Then we present some results related to academic test cases as well as a preliminary investigation related to an industrial application.

A brief review of Reduced Order Models using intrusive and non-intrusive techniques

TL;DR

The paper surveys reduced order modeling approaches for parameterized PDEs, contrasting intrusive Galerkin ROMs with non-intrusive data-driven surrogates such as PINN, DDNN, RBF, DMD, and GPR, and discusses geometry parametrization via FFD and dimensionality reduction via Active Subspaces. It demonstrates that online speedups up to are achievable, with method performance varying by problem type (e.g., DMD+RBF accuracy in fluids, POD-Galerkin with larger RBs for cylinders, RBF on geometry). Through academic benchmarks (lid-driven cavity, flow past a cylinder, Stanford Bunny) and an industrial refrigerator case, the work illustrates the breadth of ROM techniques and their potential for real-time control and optimization in engineering contexts.

Abstract

Reduced Order Models (ROMs) have gained a great attention by the scientific community in the last years thanks to their capabilities of significantly reducing the computational cost of the numerical simulations, which is a crucial objective in applications like real time control and shape optimization. This contribution aims to provide a brief overview about such a topic. We discuss both an intrusive framework based on a Galerkin projection technique and non-intrusive approaches, including Physics Informed Neural Networks (PINN), purely Data-Driven Neural Networks (DDNN), Radial Basis Functions (RBF), Dynamic Mode Decomposition (DMD) and Gaussian Process Regression (GPR). We also briefly mention geometrical parametrization and dimensionality reduction methods like Active Subspaces (AS). Then we present some results related to academic test cases as well as a preliminary investigation related to an industrial application.
Paper Structure (13 sections, 25 equations, 4 figures, 3 tables)

This paper contains 13 sections, 25 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Reduced Order Models
  3. An intrusive method: the Galerkin Projection
  4. Non-intrusive methods
  5. Physics Informed Neural Networks
  6. Purely Data-Driven Neural Networks
  7. Radial Basis Function
  8. Dynamic Mode Decomposition
  9. Gaussian Process Regression
  10. Geometrical Parametrization and Active Subspace Method
  11. Academic benchmarks
  12. Towards real world applications: an industrial case
  13. Conclusion

Figures (4)

  • Figure 1: Lid driven cavity: velocity magnitude at $t = 10$ s for $\mu=0.001$ computed by FOM (left) and by DMD+RBF (right).
  • Figure 2: Flow past a cylinder: velocity magnitude at $t = 1$ s for $\mu=0.5$ computed by FOM (top) and by the POD-Galerkin model (with 15 basis functions)
  • Figure 3: Parametrized Stanford Bunny: plot of the solution for a new value of $\mu$.
  • Figure 4: Industrial application: temperature in different sections of the fridge system computed by FOM (left) and by ROM (right).