An Efficient Deep Learning Approach for Approximating Parameter-to-Solution Maps of PDEs

In this paper, we consider approximating the parameter-to-solution maps of parametric partial differential equations (PPDEs) using deep neural networks (DNNs). We propose an efficient approach combining reduced collocation methods (RCMs) and DNNs. In the approximation analysis section, we rigorously derive sharp upper bounds on the complexity of the neural networks. These bounds only depend on the reduced basis dimension rather than the high-fidelity discretization dimension, thereby theoretically guaranteeing the computational efficiency of our approach. In numerical experiments, we implement the RCM using radial basis function finite differences (RBF-FD) and proper orthogonal decomposition (POD), and propose the POD-DNN algorithm. We consider various types of PPDEs and compare the accuracy and efficiency of different solvers. The POD-DNN has demonstrated significantly accelerated inference speeds compared with conventional numerical methods owing to the offline-online computation strategy. Furthermore, by employing the reduced basis methods (RBMs), it also outperforms standard DNNs in computational efficiency while maintaining comparable accuracy.