Gappy AE: A Nonlinear Approach for Gappy Data Reconstruction using Auto-Encoder

TL;DR

This work introduces GappyAE, a nonlinear-manifold data reconstruction framework that overcomes the limitations of Gappy POD for sparse measurements by learning a low-dimensional latent manifold with a sparse, shallow auto-encoder. Reconstruction is performed by finding latent coordinates that minimize the measurement residual via Gauss-Newton, using a nonlinear decoder to map back to the full field, $\boldsymbol x \approx \boldsymbol x_{ref}+\boldsymbol g(\hat{\boldsymbol x})$, with $\hat{\boldsymbol x}_0=\boldsymbol h(\boldsymbol x_0-\boldsymbol x_{ref})$ and $\boldsymbol h\approx \boldsymbol g^{-1}$. The method integrates four sampling strategies (DEIM, S-OPT, uniform, LHS) applied to a residual-based framework to select measurement locations, and demonstrates across 2D diffusion, radial advection, and wave problems that GappyAE achieves $\mathcal{O}(10)$ to $\mathcal{O}(100)$ times lower reconstruction error than Gappy POD, while remaining effective with very sparse data. Overall, GappyAE advances digital-twin and real-time state estimation by enabling accurate field reconstruction from sparse observations, albeit with higher online computation than linear POD-based methods. The results highlight practical impact for sensor-limited, high-dimensional systems, and point to future work on retraining with noisy data and developing measurement-sensitivity strategies tailored to nonlinear decoders.

Abstract

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.

Figures (35)

• Figure 2.1: Graphical Summary of Gappy POD process from 1 to 3.
• Figure 2.2: Graphical Summary of Gappy AE process from 1 to 3.
• Figure 3.1: Diffusion simulation solutions from the initial to the final time for the two endpoints of $\mu$
• Figure 3.2: Accuracy of Gappy AE for diffusion problem with noiseless measurements on the boundary
• Figure 3.3: Accuracy of Gappy AE for diffusion problem with noisy measurements on the boundary