Generative Reduced Basis Method

TL;DR

The paper addresses the limitations of classical linear RB approaches in representing high-dimensional and non-smooth solution manifolds for parametrized PDEs. It introduces a generative RB framework that enlarges the snapshot space via multivariate nonlinear transformations, forming enriched spaces $\\Phi_N^{M_1}$ and $\\Psi_N^{M_2}$ that converge more rapidly to the solution manifold $\\mathcal{M}$ than standard RB spaces. A Galerkin projection onto the generative spaces, coupled with inexpensive a posteriori error estimates and an offline-online decomposition, yields accurate reduced-order models with reliable error bounds. Numerical experiments on convection-diffusion and reaction-diffusion problems demonstrate substantial accuracy gains and tight error control, highlighting the method’s potential for high-dimensional parametric ROMs and real-time evaluation.

Abstract

We present a generative reduced basis (RB) approach to construct reduced order models for parametrized partial differential equations. Central to this approach is the construction of generative RB spaces that provide rapidly convergent approximations of the solution manifold. We introduce a generative snapshot method to generate significantly larger sets of snapshots from a small initial set of solution snapshots. This method leverages multivariate nonlinear transformations to enrich the RB spaces, allowing for a more accurate approximation of the solution manifold than commonly used techniques such as proper orthogonal decomposition and greedy sampling. The key components of our approach include (i) a Galerkin projection of the full order model onto the generative RB space to form the reduced order model; (ii) a posteriori error estimates to certify the accuracy of the reduced order model; and (iii) an offline-online decomposition to separate the computationally intensive model construction, performed once during the offline stage, from the real-time model evaluations performed many times during the online stage. The error estimates allow us to efficiently explore the parameter space and select parameter points that maximize the accuracy of the reduced order model. Through numerical experiments, we demonstrate that the generative RB method not only improves the accuracy of the reduced order model but also provides tight error estimates.

Figures (13)

• Figure 1: Plots of $u(x,\mu)$ defined in (\ref{['ex1u']}) as a function of $x$ for different values of $\mu$.
• Figure 2: The error metric for the standard RB space $W_N$ and the generative RB spaces $\Phi_N^{M_1}$ and $\Psi_N^{M_2}$ as a function of $N$ for $L= \min(4, N)$.
• Figure 3: The error metric for the standard RB space $W_N$ and the generative RB spaces $\Phi_N^{M_1}$ and $\Psi_N^{M_2}$ as a function of $N$ for $L= \min (8, N)$.
• Figure 4: Plots of $u(\bm x, \bm \mu)$ defined in (\ref{['ex2u']}) for four different values of $\bm \mu$.
• Figure 5: The error metric for the standard RB space $W_N$ and the generative RB spaces $\Phi_N^{M_1}$ and $\Psi_N^{M_2}$ as a function of $N$ for $L= \min(5, N)$.