Data-driven hybrid modelling of waves at mid-frequencies range: Application to forward and inverse Helmholtz problems

In this paper, we introduce a novel hybrid approach that leverages both data and numerical simulations to address the challenges of solving forward and inverse wave problems, particularly in the mid-frequency range. Our method is tailored for efficiency and accuracy, considering the computationally intensive nature of these problems, which arise from the need for refined mesh grids and a high number of degrees of freedom. Our approach unfolds in multiple stages, each targeting a specific frequency range. Initially, we decompose the wave field into a grid of finely resolved points, designed to capture the intricate details at various wavenumbers within the frequency range of interest. Importantly, the distribution of these grid points remains consistent across different wavenumbers. Subsequently, we generate a substantial dataset comprising 1,000 maps covering the entire frequency range. Creating such a dataset, especially at higher frequencies, can pose a significant computational challenge. To tackle this, we employ a highly efficient enrichment-based finite element method, ensuring the dataset’s creation is computationally manageable. The dataset which encompasses 1000 different values of the wavenumbers with their corresponding wave simulation will be the basis to train a fully connected neural network. In the forward problem the neural network is trained such that a wave pattern is predicted for each value of the wavenumber. To address the inverse problem while upholding stability, we introduce latent variables to reduce the number of physical parameters. Our trained deep network undergoes rigorous testing for both forward and inverse problems, enabling a direct comparison between predicted solutions and their original counterparts. Once the network is trained, it becomes a powerful tool for accurately solving wave problems in a fraction of the CPU time required by alternative methods. Notably, our approach is supervised, as it relies on a dataset generated through the enriched finite element method, and hyperparameter tuning is carried out for both the forward and inverse networks.