Generative reduced basis method

We present a generative reduced basis (RB) approach for the rapid and reliable solution of parametrized linear partial differential equations. Central to this approach is the construction of generative RB spaces that provide rapidly convergent approximations of the solution manifold. We propose 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, thereby enabling a more accurate approximation of the solution manifold than commonly used dimensionality reduction techniques such as proper orthogonal decomposition and greedy sampling. We employ the generative RB spaces to construct reduced order models and compute a posteriori error estimates. The error estimates allow us to efficiently explore the parameter space and select parameter points that improve the efficiency and 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.