COMBINED GALERKIN AND REGRESSION BASED ALGORITHM FOR PARAMETER DEPENDANT PDE'S

Résumé fourni par la source

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. In this work, we extend and deepen the new framework for the reduction of nonlinear models known as the Nonlinear Compressive Reduced Basis Method (NLCRBM). Our approach combines nonlinear 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 supports 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.

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