  We propose the POD-DNN, a novel algorithm leveraging deep neural networks
(DNNs) along with radial basis functions (RBFs) in the context of the proper
orthogonal decomposition (POD) reduced basis method (RBM), aimed at
approximating the parametric mapping of parametric partial differential
equations on irregular domains. The POD-DNN algorithm capitalizes on the
low-dimensional characteristics of the solution manifold for parametric
equations, alongside the inherent offline-online computational strategy of RBM
and DNNs. In numerical experiments, POD-DNN demonstrates significantly
accelerated computation speeds during the online phase. Compared to other
algorithms that utilize RBF without integrating DNNs, POD-DNN substantially
improves the computational speed in the online inference process. Furthermore,
under reasonable assumptions, we have rigorously derived upper bounds on the
complexity of approximating parametric mappings with POD-DNN, thereby providing
a theoretical analysis of the algorithm's empirical performance.
