Flow Map Learning for Unknown Dynamical Systems: Overview, Implementation, and Benchmarks

Flow map learning (FML), in conjunction with deep neural networks (DNNs), has shown promises for data driven modeling of unknown dynamical systems. A remarkable feature of FML is that it is capable of producing accurate predictive models for partially observed systems, even when their exact mathematical models do not exist. In this paper, we present an overview of the FML framework, along with the important computational details for its successful implementation. We also present a set of well defined benchmark problems for learning unknown dynamical systems. All the numerical details of these problems are presented, along with their FML results, to ensure that the problems are accessible for cross-examination and the results are reproducible

Numerical integration formulas of degree two,

Abstract Numerical integration formulas in n-dimensional nonsymmetric Euclidean space of degree two, consisting of n + 1 equally weighted points, are discussed, for a class of integrals often encountered in statistics. This is an extension of Stroud's theory [A.H. Stroud, Remarks on the disposition of points in numerical integration formulas, Math. Comput. 11 (60) (1957) 257-261; A.H. Stroud, Numerical integration formulas of degree two, Math. Comput. 14 (69) (1960) 21-26]. Explicit formulas are given for integrals with nonsymmetric weights. These appear to be new results and include the Stroud's degree two formula as a special case