Generative Reduced Basis Method

TL;DR

This paper introduces a generative reduced basis method that enhances reduced order models for parametrized PDEs by generating larger snapshot sets through nonlinear transformations, leading to more accurate and certifiable solutions.

Contribution

The paper proposes a novel generative RB approach with a snapshot generation technique and error certification, improving model accuracy over traditional methods.

Findings

Generative RB achieves higher accuracy than proper orthogonal decomposition.

The method provides tight a posteriori error estimates.

Numerical experiments confirm improved approximation quality.

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.