Recent advances in Meta-model of Optimal Prognosis

TL;DR

This work addresses the need for efficient surrogate models in expensive nonlinear simulations by introducing the Meta-model of Optimal Prognosis (MOP), which automatically selects the best meta-model (polynomial regression or Moving Least Squares) and reduces the input space via significance and importance filters. The authors define the coefficient of prognosis ($CoP$) as a cross-validated measure to assess surrogate quality and demonstrate that cross-validation provides reliable estimates for small samples. The framework is implemented in optiSLang, enabling automatic evaluation of multiple configurations and reliable identification of key variables, often achieving $CoP$ values near $99\%$ while maintaining simple, interpretable models. The results indicate that MOP can enhance design exploration by efficiently combining variable reduction with robust prognosis, supporting faster and more reliable optimization in nonlinear, high-dimensional problems, with ongoing work on dependent inputs, coupling detection, and bootstrapping of $CoP$.

Abstract

In real case applications within the virtual prototyping process, it is not always possible to reduce the complexity of the physical models and to obtain numerical models which can be solved quickly. Usually, every single numerical simulation takes hours or even days. Although the progresses in numerical methods and high performance computing, in such cases, it is not possible to explore various model configurations, hence efficient surrogate models are required. Generally the available meta-model techniques show several advantages and disadvantages depending on the investigated problem. In this paper we present an automatic approach for the selection of the optimal suitable meta-model for the actual problem. Together with an automatic reduction of the variable space using advanced filter techniques an efficient approximation is enabled also for high dimensional problems.

Figures (11)

• Figure 1: Original model response function $z(x,y)$
• Figure 2: Polynomial least square approximation with quadratic basis
• Figure 3: Classical MLS approximation depending on the influence radius $D$
• Figure 4: Interpolating MLS approximation with regularized weighting
• Figure 5: Convergence of coefficient of determination and the coefficient of prognosis depending on the number of support points of a linear polynomial and MLS approximation of a coupled nonlinear two-dimensional model