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.

中文翻译：

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Gappy AE：使用自动编码器进行 Gappy 数据重建的非线性方法

我们引入了一种称为 Gappy 自动编码器 (Gappy AE) 的新颖数据重建算法，以解决与 Gappy 固有正交分解 (Gappy POD) 相关的限制，Gappy POD 是处理稀疏测量或丢失数据时广泛使用的数据重建方法。 Gappy POD 在准确表示以缓慢衰减的 Kolmogorov N 宽度为特征的解决方案方面具有固有的限制，这主要是由于它依赖于线性子空间进行数据预测。相比之下，Gappy AE 利用非线性流形表示的力量来解决传统 Gappy POD 的数据重建挑战。它擅长在只有稀疏测量数据的情况下进行实时状态预测，有效填补空白。此功能使 Gappy AE 特别有价值，例如对于数字孪生和图像校正应用。为了证明 Gappy AE 具有稀疏测量的卓越数据重建性能，我们提供了几个数值示例，包括 2D 扩散、2D 径向平流和 2D 波动方程问题等场景。此外，我们还评估了四种不同的采样算法（离散经验插值法、S-OPT 算法、拉丁超立方采样和均匀分布采样）对数据重建精度的影响。我们的研究结果最终表明，当给出稀疏测量时，Gappy AE 在数据重建方面优于 Gappy POD。