Quadratic-manifold Neural Galerkin reduced models give locally unique, residual-minimizing trajectories and, for linear full models, online cost independent of the full dimension.
Empirical sparse regression on quadratic manifolds
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Approximating field variables and data vectors from sparse samples is a key challenge in computational science. Widely used methods such as gappy proper orthogonal decomposition and empirical interpolation rely on linear approximation spaces, limiting their effectiveness for data representing transport-dominated and wave-like dynamics. To address this limitation, we introduce quadratic manifold sparse regression, which trains quadratic manifolds with a sparse greedy method and computes approximations on the manifold through novel nonlinear projections of sparse samples. The nonlinear approximations obtained with quadratic manifold sparse regression achieve orders of magnitude higher accuracies than linear methods on data describing transport-dominated dynamics in numerical experiments.
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Nonlinear model reduction with Neural Galerkin schemes on quadratic manifolds
Quadratic-manifold Neural Galerkin reduced models give locally unique, residual-minimizing trajectories and, for linear full models, online cost independent of the full dimension.