Nonlinear model reduction with Neural Galerkin schemes on quadratic manifolds

Leveraging nonlinear parametrizations for model reduction can overcome the Kolmogorov barrier that affects transport-dominated problems. In this work, we build on the reduced dynamics given by Neural Galerkin schemes and propose to parametrize the corresponding reduced solutions on quadratic manifolds. We show that the solutions of the proposed quadratic-manifold Neural Galerkin reduced models are locally unique and minimize the residual norm over time, which promotes stability and accuracy. For linear problems, quadratic-manifold Neural Galerkin reduced models achieve online efficiency in the sense that the costs of predictions scale independently of the state dimension of the underlying full model. For nonlinear problems, we show that Neural Galerkin schemes allow using separate collocation points for evaluating the residual function from the full-model grid points, which can be seen as a form of hyper-reduction. Numerical experiments with advecting waves and densities of charged particles in an electric field show that quadratic-manifold Neural Galerkin reduced models lead to orders of magnitude speedups compared to full models.