Abstract:

A novel gappy technology, gappy autoencoder with proper orthogonal decomposition (Gappy POD-AE), is proposed for reconstructing physical fields from sparse data. High-dimensional data are reduced via proper orthogonal decomposition (POD), and low-dimensional data are used to train an autoencoder (AE). By integrating the POD operator with the decoder, a nonlinear solution form is established and incorporated into a new maximum-a-posteriori (MAP)-based objective for online reconstruction. The numerical results on the two-dimensional (2D) Bhatnagar-Gross-Krook-Boltzmann (BGK-Boltzmann) equation, wave equation, shallow-water equation, and satellite data show that Gappy POD-AE achieves higher accuracy than gappy proper orthogonal decomposition (Gappy POD), especially for the data with slowly decaying singular values, and is more efficient in training than gappy autoencoder (Gappy AE). The MAP-based formulation and new gappy procedure further enhance the reconstruction accuracy.

Key words: data reconstruction, gappy technology, proper orthogonal decomposition (POD), autoencoder (AE), maximum-a-posteriori (MAP)