Partial differential equation-based numerical solution frameworks for initial and boundary value problems have attained a high degree of complexity. Applied to a wide range of physics with the ultimate goal of enabling engineering solutions, these approaches encompass a spectrum of spatiotemporal discretization techniques that leverage solver technology and high performance computing. While high-fidelity solutions can be achieved using these approaches, they come at a high computational expense and complexity. Systems with billions of solution unknowns are now routine. The expense and complexity do not lend themselves to typical engineering design and decision-making, which must instead rely on reduced-order models. Here we present an approach to reduced-order modelling that builds off of recent graph theoretic work for representation, exploration, and analysis on 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 first order dynamics, polynomial expansions, and Taylor series. Some aspects of the non-local calculus related to consistency of the models are explored. Details on the numerical implementations and the software library that has been developed for non-local calculus on graphs is described. Finally, we present examples of applications to various quantities of interest in mechano-chemical systems.
翻译:最初和边界价值问题的局部差异方程式数字解决方案框架已经达到高度复杂程度。 应用于一系列以扶持工程解决方案为最终目标的广泛物理学,这些方法包含一系列利用解决方案技术和高性能计算法的时空离散技术。 虽然可以使用这些方法实现高不忠解决方案,但它们的计算成本和复杂性很高。 数十亿个解决方案未知的系统现已是例行作业。 费用和复杂性并不适合于典型的工程设计和决策,而必须依靠减序模型。 我们在这里介绍了一个减序模型,根据最近关于物理系统的计算状态(Banerjee et al.,Comp. Meth. App. Mech. Eng., 351, 501-530, 2019)的图表,从最近图表理论理论理论、探索和分析中构建的缩序模型。 我们开发了非本地的计算模型的某些方面,而目前各种图表应用的卡路图应用中,我们所开发的卡路图应用的卡路里模型,最终探索了我们所开发的卡路里模型应用的卡路里模型。