论文标题
从PDE的单个解决方案中可以学到多少？
How much can one learn from a single solution of a PDE?
论文作者
论文摘要
线性进化PDE $ \ partial_t u（x，x，t）= - \ Mathcal {l} u $，其中$ \ Mathcal {l} $是独立于时间的强烈椭圆形操作员，作为一个示例，以显示是否可以超级置于单个（或一个有限的解决方案）的（或plose）解决方案的快照。我们的研究表明，这取决于特征值的增长率，$μ_n$，$ \ MATHCAL {l} $在$ n $中。当该语句为真时，一种简单的数据驱动方法，用于降低模型的模型和PDE任意解决方案的近似，而无需知道基础PDE的设计。提出了数值实验以证实我们的分析。
Linear evolution PDE $\partial_t u(x,t) = -\mathcal{L} u$, where $\mathcal{L}$ is a strongly elliptic operator independent of time, is studied as an example to show if one can superpose snapshots of a single (or a finite number of) solution(s) to construct an arbitrary solution. Our study shows that it depends on the growth rate of the eigenvalues, $μ_n$, of $\mathcal{L}$ in terms of $n$. When the statement is true, a simple data-driven approach for model reduction and approximation of an arbitrary solution of a PDE without knowing the underlying PDE is designed. Numerical experiments are presented to corroborate our analysis.
