Data-driven Calibration of Penalties for Least-Squares Regression
The Journal of Machine Learning Research
Dynamical systems identification using Gaussian process models with incorporated local models
Engineering Applications of Artificial Intelligence
Slope heuristics: overview and implementation
Statistics and Computing
IEEE Transactions on Information Theory
Runge-Kutta neural network for identification of dynamical systems in high accuracy
IEEE Transactions on Neural Networks
Identification and control of dynamical systems using neural networks
IEEE Transactions on Neural Networks
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Thermal engineering deals with the estimation of the temperature at different spatial points and different instants for a given set of boundary and initial conditions. For this purpose, the reference model is a numerical simulation model but it is time-consuming. Consequently we build a surrogate model in order to replace it. This surrogate model is a recursive multilayer perceptron, independent of the boundary conditions and parametrized by the statistical learning of multidimensional temporal trajectories computed with the reference model. It emulates the outputs of the reference model over time from the only knowledge of initial conditions and exogenous variables. Moreover this model is able to predict these outputs in steady state, even if its formulation is time-dependent. A new methodology is proposed so as to overcome the learning problem associated to the very weak number of trajectories available for the surrogate model construction. The first step attempts to build a more robust surrogate model by considering it as the average of local models resulting from the V-folds cross-validation technique. This new kind of multilayer perceptron is much more robust and accurate, in particular when the learning dataset is very small. The second step consists in the creation of a new learning dataset which is made up of each time observation coming from each trajectory. In this way, we artificially obtain a sizeable sample allowing all the classic neural networks constructions. Furthermore, many approaches exist in order to select the best hidden neurons number but most of them are costly or require a lot of observations. We consider here a non-asymptotic approach based on the minimization of a penalized criterion providing accurate results in an economical computational way. In order to calibrate precisely the penalty term, we use the slope heuristic or the dimension jump, recently introduced in a regression framework. The validation of the method is performed on a toy function. The prediction ability of the surrogate model built with the new methodology is successfully compared to usual constructions on a simplified problem and then applied to thermal engineering.