A Novel Rigorous Mathematical Programming Approach to Construct Phenomenological Models

摘要

The automated construction of physical laws from raw experimental measurements poses a great challenge in modern modelling and remains an open question. The work here presents a novel generalized Mixed-Integer Nonlinear Programming (MINLP) approach, which constitutes a rigorous theoretical formulation that best fits the given data. The proposal is based on the use of generic representation of analytical functions as binary evaluation trees which are Directed Acyclic Graphs (DAG) utilized to allow the construction of a superstructure out of which the optimal fitting model can be identified by solving the resulting (non-convex) MINLP problems. The trees are constructed in a way that their nodes are comprised of a linear combination of basic atomic functions, either arithmetic or unary, weighted by binary decision variables. Both single-input single-output (SISO) and multiple-input multiple-output systems are considered, as well as more complex models comprised of differential equations or even described by series summation of algebraic terms. The aim and contribution proposed methodology in this paper is to present the most general theoretical formulatioon of how models are constructed for systems quantification via analytical function forms, irrespective of the source of data. The constructed formulation is shown to contain all formulations thus far presented in the open literature, comprising a starting point either for direct fitting or for the derivation of simplified approaches.