标题： CFL型条件及离散时间稀疏全阶模型推理的理论见解 显示英文标题

标题： A CFL-type Condition and Theoretical Insights for Discrete-Time Sparse Full-Order Model Inference

摘要： 在这项工作中，我们研究了通过稀疏全阶模型（sFOM）对离散时间动力系统进行数据驱动的推理。我们首先制定了涉及的最小二乘（LS）问题，并讨论了正则化的必要性，指出了通常使用的$l_2$正则化与推断出的离散时间 sFOM 稳定性之间的联系。然后，我们提供了理论见解，考虑了构成 sFOM 的数值方案的一致性和稳定性属性，并通过一维线性扩散和线性平流的说明性测试案例来举例说明。对于线性平流，我们解析地推导了一个“采样 CFL”条件，该条件规定了训练数据中空间和时间离散化步长比的一个界限，以确保推断出的 sFOM 的稳定性。最后，我们研究了两个非线性问题的 sFOM 推理，即二维 Burgers 测试案例和振荡顶盖驱动腔中的不可压缩流动，并探讨了理论发现与推断出的非线性 sFOM 特性之间的关系。 显示英文摘要

摘要： In this work, we investigate the data-driven inference of a discrete-time dynamical system via a sparse Full-Order Model (sFOM). We first formulate the involved Least Squares (LS) problem and discuss the need for regularization, indicating a connection between the typically employed $l_2$ regularization and the stability of the inferred discrete-time sFOM. We then provide theoretical insights considering the consistency and stability properties of the inferred numerical schemes that form the sFOM and exemplify them via illustrative, 1D test cases of linear diffusion and linear advection. For linear advection, we analytically derive a "sampling CFL" condition, which dictates a bound for the ratio of spatial and temporal discretization steps in the training data that ensures stability of the inferred sFOM. Finally, we investigate the sFOM inference for two nonlinear problems, namely a 2D Burgers' test case and the incompressible flow in an oscillating lid driven cavity, and draw connections between the theoretical findings and the properties of the inferred, nonlinear sFOMs.