用于非线性场重建的间隙概率流形分解

中文摘要 AI 辅助

本文提出了间隙概率流形分解（Gappy PMD），这是一种从极稀疏测量值重建高维场的非线性方法。Gappy PMD 在通过概率流形分解（PMD）学习到的非线性流形上重建场。我们进一步提出了一种用于基于降阶模型（ROM）的场重建的可微点选择方法（DPS）。在基于 ROM 的重建框架内使用可微无网格插值，使全场重建误差相对于采样位置可微，并直接优化这些位置。此外，还对 Gappy PMD 进行了理论误差分析，将平方重建误差分解为两个正交部分：一部分垂直于重建流形，另一部分由稀疏采样和观测噪声引起。在采样算子的稳定性条件下，该误差随 PMD 近似误差和噪声一同消失。在三个数值测试案例上对 Gappy PMD 进行了评估：圆柱绕流、顶盖驱动空腔流和后向台阶流。对于相同的降阶维度和采样点，Gappy PMD 达到的平均相对 L² 误差比 Gappy POD 低一到两个数量级。使用 DPS 优化采样点进一步提高了重建精度和鲁棒性。

英文摘要

This paper proposes gappy probabilistic manifold decomposition (Gappy PMD), a nonlinear method for reconstructing high-dimensional fields from extremely sparse measurements. Gappy PMD reconstructs the field on the nonlinear manifold learned by probabilistic manifold decomposition (PMD). We further propose a differentiable point selection method for reduced-order model (ROM)-based field reconstruction (DPS). Using differentiable meshless interpolation within the ROM-based reconstruction framework, DPS makes the full-field reconstruction error differentiable with respect to the sampling locations and directly optimizes these locations. In addition, a theoretical error analysis for Gappy PMD is also given. It splits the squared reconstruction error into two orthogonal parts: one normal to the reconstruction manifold and the other induced by sparse sampling and observation noise. Under a stability condition on the sampling operator, this error vanishes with the PMD approximation error and the noise. The Gappy PMD is evaluated on three numerical test cases: flow past a cylinder, lid-driven cavity flow, and backward-facing step flow. For the same reduced dimension and sampling points, Gappy PMD attains mean relative $L^2$ errors one to two orders of magnitude below Gappy POD. Optimizing the sampling points with DPS further improves reconstruction accuracy and robustness.

发表机构

• Tongji University（同济大学）

机构由 AI 辅助整理，请以论文原文为准。