Approximating field variables and data vectors from sparse samples is a key challenge in computational science. Widely used methods such as gappy proper orthogonal decomposition and empirical interpolation rely on linear approximation spaces, limiting their effectiveness for data representing transport-dominated and wave-like dynamics. To address this limitation, we introduce quadratic manifold sparse regression, which trains quadratic manifolds with a sparse greedy method and computes approximations on the manifold through novel nonlinear projections of sparse samples. The nonlinear approximations obtained with quadratic manifold sparse regression achieve orders of magnitude higher accuracies than linear methods on data describing transport-dominated dynamics in numerical experiments.

翻译：从稀疏样本中近似场变量与数据向量是计算科学中的关键挑战。广泛使用的方法（如间隙本征正交分解和经验插值）依赖于线性近似空间，这限制了其在处理表征输运主导和类波动动力学数据时的有效性。为解决这一局限，我们提出了二次流形稀疏回归方法。该方法通过稀疏贪婪算法训练二次流形，并利用稀疏样本的新型非线性投影计算流形上的近似解。数值实验表明，在处理描述输运主导动力学的数据时，二次流形稀疏回归获得的非线性近似精度比线性方法高出数个数量级。