Résumé
A broad class of problems in science and engineering involves the repeated solution of partial differential equations (PDEs) for different parameter values. Linear reduced order models are a powerful tool to decrease the computational cost of these simulations by approximating the solutions in a low-dimensional space. They have proven highly effective in many settings; however, they often perform poorly for transport-dominated PDEs, where key solution features such as translations cannot be accurately represented in a linear subspace.
To overcome these limitations, several nonlinear reduced order models have recently been proposed, including approaches based on quadratic or polynomial mappings and neural networks. In this talk, I will present an alternative metric-based approach to nonlinear model order reduction. The central idea is to replace linear combinations in low-dimensional spaces with barycenters taken with respect to a suitably chosen metric, computed from a small number of representative solutions. In particular, I will provide constructions based on the Wasserstein distance from optimal transport, which is well adapted to capturing translations. I will also show how the choice of metric can be adapted to incorporate physical constraints, such as sparsity or prescribed marginals. The proposed methodology will be illustrated through numerical examples involving the approximation of electronic densities and pair densities arising in quantum chemistry.
