A neural-network based nonlinear non-intrusive reduced basis method with online adaptation for parametrized partial differential equations

Abstract: We propose a nonlinear, non-intrusive reduced basis method with online adaptation for efficient approximation of parametrized partial differential equations. The method combines neural networks with reduced-order modeling and physics-informed training to enhance both accuracy and efficiency. In the offline stage, reduced basis functions are obtained via nonlinear dimension reduction, and a neural surrogate is trained to map parameters to approximate solutions. The surrogate employs a nonlinear reconstruction of the solution from the basis functions, enabling more accurate representation of complex solution structures than linear mappings. The model is further refined during the online stage using lightweight physics-informed neural network training. This offline-online framework enables accurate prediction especially in complex scenarios or with limited snapshot data. We demonstrate the performance and effectiveness of the proposed method through numerical experiments.