Abstract :
Solving parameterized partial differential equations (PDEs) efficiently remains a major challenge in computational science, especially when the parameter-to-solution map is highly nonlinear and the (linear) Kolmogorov $N$-width decays slowly. Although linear reduced basis methods (RBM) perform well in low-complexity regimes, their efficiency deteriorates in cases where the solution manifold cannot be approximated accurately by (low dimensional) linear subspaces. Nonlinear approaches, particularly those based on autoencoders and neural networks, offer enhanced representational power but often lack robustness, interpretability, and error control. Recently, different approaches have proposed to exploit a quadratic representation of the coefficients of high modes of the reduced basis in terms of the first ones. We propose an explanation for this anszatz an explain why this is not sufficient that leads to the Nonlinear Compressive Reduced Basis Method (NLCRBM) that combines non-linear manifold compression with Galerkin projection to retain physical structure while achieving efficient dimensionality reduction. The method applies to a broad range of parameterized PDEs and includes adaptive mechanisms for error monitoring and manifold refinement. We illustrate its advantages through representative numerical experiments and discuss its potential to overcome key limitations of both linear RBM and data-driven reduction techniques.