论文标题
在有限加权图上非本地演算的数值分析，并应用于动态系统的减少阶
Numerical analysis of non-local calculus on finite weighted graphs, with application to reduced-order modelling of dynamical systems
论文作者
论文摘要
我们提出了一种减少订购建模的方法，该方法源于对物理系统计算状态的表示，探索和分析的最新图理论工作（Banerjee等，Comp。Meth。App。App。Mech。Eng。，351，301-530，2019）。我们通过利用多项式扩展和泰勒系列来扩展有限加权图上的非本地演算以构建此类模型。在图表上非本地演算的一般框架中，图边缘的权重与图形的嵌入式相关，因此与衍生物的定义相关。在先前的通信中（Duschenes和Garikipati，Arxiv：2105.01740），我们已经表明，与相应的差异性衍生性定义相比，径向对称，连续的边缘权重得出，例如高斯函数，导致不一致的衍生物。考虑到每个图顶点的局部邻域的嵌入，我们从有限差异方法中汲取灵感。鉴于此过程，我们确保在这种情况下非本地衍生物的一致性，这是数值应用的关键要求。我们表明，在基础数据中没有对称性假设的情况下，我们可以在所选数量的尺寸中实现任何所需的衍生物准确性顺序。最后，我们提出了使用此非本地演算提取降低模型的两个示例应用，以从抛物线偏微分方程的普通微分方程的形式进行逐渐更大的复杂性。
We present an approach to reduced-order modelling that builds off recent graph-theoretic work for representation, exploration, and analysis of computed states of physical systems (Banerjee et al., Comp. Meth. App. Mech. Eng., 351, 501-530, 2019). We extend a non-local calculus on finite weighted graphs to build such models by exploiting polynomial expansions and Taylor series. In the general framework for non-local calculus on graphs, the graph edge weights are intricately linked to the embedding of the graph, and consequently to the definition of the derivatives. In a previous communication (Duschenes and Garikipati, arXiv:2105.01740), we have shown that radially symmetric, continuous edge weights derived from, for example Gaussian functions, yield inconsistent results in the resulting non-local derivatives when compared against the corresponding local, differential derivative definitions. Taking inspiration from finite difference methods, we algorithmically compute edge weights, considering the embedding of the local neighborhood of each graph vertex. Given this procedure, we ensure the consistency of the non-local derivatives in this setting, a crucial requirement for numerical applications. We show that we can achieve any desired orders of accuracy of derivatives, in a chosen number of dimensions without symmetry assumptions in the underlying data. Finally, we present two example applications of extracting reduced-order models using this non-local calculus, in the form of ordinary differential equations from parabolic partial differential equations of progressively greater complexity.
