  We introduce a novel data reconstruction algorithm known as Gappy
auto-encoder (Gappy AE) to address the limitations associated with Gappy proper
orthogonal decomposition (Gappy POD), a widely used method for data
reconstruction when dealing with sparse measurements or missing data. Gappy POD
has inherent constraints in accurately representing solutions characterized by
slowly decaying Kolmogorov N-widths, primarily due to its reliance on linear
subspaces for data prediction. In contrast, Gappy AE leverages the power of
nonlinear manifold representations to address data reconstruction challenges of
conventional Gappy POD. It excels at real-time state prediction in scenarios
where only sparsely measured data is available, filling in the gaps
effectively. This capability makes Gappy AE particularly valuable, such as for
digital twin and image correction applications. To demonstrate the superior
data reconstruction performance of Gappy AE with sparse measurements, we
provide several numerical examples, including scenarios like 2D diffusion, 2D
radial advection, and 2D wave equation problems. Additionally, we assess the
impact of four distinct sampling algorithms - discrete empirical interpolation
method, the S-OPT algorithm, Latin hypercube sampling, and uniformly
distributed sampling - on data reconstruction accuracy. Our findings
conclusively show that Gappy AE outperforms Gappy POD in data reconstruction
when sparse measurements are given.
