This subject is in close connection with Arthur Marmin's PhD thesis (Rational models optimized exactly for chemical processes improvement), co-supervised by Marc Castella, Jean-Christophe Pesquet and Laurent Duval. The context of the subject is modeling of experimental data with a parametric family of functions.
Environment: CentraleSupelec/INRIA Saclay/TelecomSudParis/IFP Energies nouvelles (DataIA funding)
For a long time, polynomial and rational functions have been playing a key role in that respect (calibration, saturation, system modeling, interpolation, signal reconstruction..)Their use is classically confined to least-square or least-absolute value regression. However, it is highly desirable in real-world applications to incorporate:
Basically, from the modeling perspective, any cost function of practical interest can be approximated as accurately as desired by a polynomial. Nonetheless, optimizing multivariate rational function under polynomial constraints is a difficult problem when no convexity property holds. Recent mathematical breakthroughs have made it possible to solve problems of this kind in an exact manner by building a hierarchy of convex relaxations. The objective of this internship is to pursue the developments applied so far to signal restoration. We address here the quest for best model selection in experimental data fitting. We minimize a criterion composed of two terms. The first one is a fit measure between the model and recorded measurements. The second terms are sparsity-promoting (approximating the l0 count measure) and interval bound penalizations, weighted by positive parameters We choose terms as rational functions. The latter will be dealt with by subclasses (e.g. polynomials, homographic functions). The proposed methodology will be evaluated on the many models and experimental data available at IFP Energies nouvelles. A particular attention will be paid to the usability to data practitioners, by embedding the algorithms into an user-friendly interface, with help in choosing hyperparameters.
Online reference: Model learning with rational optimization and constraints.
(c) GdR 720 ISIS - CNRS - 2011-2020.