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

TL;DR

Parametrized PDEs require repeated high-cost solves for many parameter values, motivating a nonlinear, non-intrusive reduced basis framework with online adaptation. The method offline-leverages a nonlinear dimension reduction and a hypernet-driven reconstruction to map parameters to a low-dimensional representation, while online refinement uses a lightweight PINN-like training to correct predictions efficiently. Across Burgers' equation and a high-dimensional lid-driven cavity test, the nonlinear RB consistently outperforms linear non-intrusive RB, with online adaptation substantially improving accuracy in challenging or data-scarce regimes and reducing online costs relative to full-order solves. The approach offers robust, fast predictions for complex parametric PDEs and opens avenues for time-dependent extensions and operator-learning integrations.

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.

Figures (14)

• Figure 1: The offline training procedure and the network architecture.
• Figure 2: The online training procedure and the network architecture.
• Figure 3: Exact solutions of Eq. \ref{['Ex1 eq']} for $\kappa = 1$ (left) and $\kappa = 9$ (right).
• Figure 4: Relative errors of the offline-computed solutions as a function of the number of basis functions. Left for the case $\kappa = 1$, and right for the case $\kappa = 9$.
• Figure 5: Histograms of the relative errors of the offline-computed solutions over 100 test samples. Left for the case $\kappa = 1$ using 8 basis functions; right for the case $\kappa= 9$ using 16 basis functions.