论文标题
使用数据恢复部分微分方程中未知过程的方法
Methods to Recover Unknown Processes in Partial Differential Equations Using Data
论文作者
论文摘要
我们研究了使用观测数据识别嵌入时间依赖性偏微分方程（PDE）的未知过程的问题，并应用于对流扩散类型PDE。我们首先进行理论分析并得出条件，以确保问题的溶解度。然后，我们提出一组数值方法，包括盖尔金型算法和搭配类型算法。对算法的分析以及它们的实现细节进行了介绍。 Galerkin算法更适合于实际情况，尤其是具有嘈杂数据的情况，因为它可以避免使用衍生/梯度数据。然后提出各种数值示例，以证明数值方法的性能和特性。
We study the problem of identifying unknown processes embedded in time-dependent partial differential equation (PDE) using observational data, with an application to advection-diffusion type PDE. We first conduct theoretical analysis and derive conditions to ensure the solvability of the problem. We then present a set of numerical approaches, including Galerkin type algorithm and collocation type algorithm. Analysis of the algorithms are presented, along with their implementation detail. The Galerkin algorithm is more suitable for practical situations, particularly those with noisy data, as it avoids using derivative/gradient data. Various numerical examples are then presented to demonstrate the performance and properties of the numerical methods.
