基于指数拟合的数据驱动识别反应扩散动力学：从有限非局部噪声测量中获取主导特征值和初始条件模式
Data-driven identification of reaction-diffusion dynamics from finitely many non-local noisy measurements by exponential fitting
AI总结：
本文提出了一种基于指数拟合的数据驱动方法，用于从有限非局部噪声测量中准确估计反应扩散方程的主导特征值和初始条件模式。
摘要：
给定一个具有未知右侧项的反应扩散方程，我们考虑一个非线性反问题，即从有限数量的非局部噪声测量中估计相关的主导特征值和初始条件模式。我们定义了一个重建准则，并且对于足够小的噪声，我们证明了所求近似的存在性和唯一性，并推导了一阶条件数的闭合表达式，以及在测量样本数量固定而采样间隔长度趋于无穷大的情况下其渐进行为的界限。在计算所求估计量时，我们的模拟显示，指数拟合算法ESPRIT在第一阶上是最优的，因为其第一阶条件数具有与所考虑情形中解析条件数相同的渐进行为。
英文摘要：
Given a reaction-diffusion equation with unknown right-hand side, we consider a nonlinear inverse problem of estimating the associated leading eigenvalues and initial condition modes from a finite number of non-local noisy measurements. We define a reconstruction criterion and, for a small enough noise, we prove the existence and uniqueness of the desired approximation and derive closed-form expressions for the first-order condition numbers, as well as bounds for their asymptotic behavior in a regime when the number of measured samples is fixed and the inter-sampling interval length tends to infinity. When computing the sought estimates numerically, our simulations show that the exponential fitting algorithm ESPRIT is first-order optimal, as its first-order condition numbers have the same asymptotic behavior as the analytic condition numbers in the considered regime.
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