We introduce a framework for generating, organizing, and reasoning with computational knowledge. It is motivated by the observation that most problems in Computational Sciences and Engineering (CSE) can be described as that of completing (from data) a computational graph representing dependencies between functions and variables. Functions and variables may be known, unknown, or random. Data comes in the form of observations of distinct values of a finite number of subsets of the variables of the graph. The underlying problem combines a regression problem (approximating unknown functions) with a matrix completion problem (recovering unobserved variables in the data). Replacing unknown functions by Gaussian Processes (GPs) and conditioning on observed data provides a simple but efficient approach to completing such graphs. Since the proposed framework is highly expressive, it has a vast potential application scope. Since the completion process can be automatized, as one solves $\sqrt{\sqrt{2}+\sqrt{3}}$ on a pocket calculator without thinking about it, one could, with the proposed framework, solve a complex CSE problem by drawing a diagram. Compared to traditional kriging, the proposed framework can be used to recover unknown functions with much scarcer data by exploiting interdependencies between multiple functions and variables. The Computational Graph Completion (CGC) problem addressed by the proposed framework could therefore also be interpreted as a generalization of that of solving linear systems of equations to that of approximating unknown variables and functions with noisy, incomplete, and nonlinear dependencies. Numerous examples illustrate the flexibility, scope, efficacy, and robustness of the CGC framework and show how it can be used as a pathway to identifying simple solutions to classical CSE problems (digital twin modeling, dimension reduction, mode decomposition, etc.).
翻译:我们引入了计算知识生成、组织和推理的框架。 它的动机是观察到计算科学与工程(CSE)中的大多数问题可以被描述为完成(来自数据)一个计算图的问题。 函数和变量可能是已知的, 未知的, 也可能是随机的。 数据的形式是观察图表变量子集数量有限的不同值。 潜在的问题包括一个回归问题( 匹配不完全的功能 ), 以及一个简单的矩阵完成问题( 覆盖数据中未观察到的变量 ) 。 由 Gausian 进程( GPs) 取代一个未知的函数, 代表函数的完成( 从数据中找到) 。 由 Gaussian 进程( GPs) 和对所观察到的数据进行调节为简单而有效的方法。 由于拟议框架的CSE 的复杂解释性( 使用不精确的模型), 并且用一个不精确的模型来显示一个不易变异的功能 。 比较过程可以自动化, 作为解析的系统 {2\ qrrtal {qration {{{{{{{{{{{{{{{}}}} 。 在口中, 它可以可以不思考一个简单的计算一个简单的计算器, 它可以在一个不考虑它, 。