ADMOS 2023

Error estimation for surrogate models with noisy small-sized training sets

  • Wackers, Jeroen (Centrale Nantes / CNRS)
  • Pehlivan Solak, Hayriye (Centrale Nantes / CNRS)
  • Pellegrini, Riccardo (CNR-INM)
  • Serani, Andrea (CNR-INM)
  • Diez, Matteo (CNR-INM)

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Simulation-driven design optimization often uses surrogate models, where a limited number of simulations for different designs provides a dataset, through which an approximate model is fitted. The shape optimization is then performed over this surrogate model. For efficiency, modern approaches often construct the datasets adaptively, adding simulation points one by one where they are most likely to discover the optimum design. For reliable optimization, uncertainty estimation of the surrogate model is a necessity. Knowledge of the uncertainty allows to establish the regions where the optimum could be located; therefore, underestimating the uncertainty leads to adaptive sampling which concentrates on suboptimal regions and may miss the true optimum. Stochastic Radial Basis Functions (SRBF) provide an efficient uncertainty estimation, derived from the spread in a range of RBF fits with different kernels. We extend the SRBF approach by considering two complications: the number of data can be very small (less than 5 points, which corresponds to the start of an adaptive sampling process) and the data may be noisy. This requires three additions to the original SRBF uncertainty: - When it is estimated that locally, too few data are available for the RBF fits to accurately represent the true simulation response, the SRBF estimation is replaced by a default uncertainty estimation based only on the distances between the sampling points. _ The noise is filtered with least-squares fitting of the surrogate model using less RBF centers than sampling points. Like for SRBF, a range of RBF centers is tested and a noise-filtering uncertainty is derived from the spread of the data fits. - Even with perfect noise filtering, for a small dataset the local mean of the data may not correspond to the true simulation response. This introduces a mean-value uncertainty which is estimated with the Central Limit Theorem. In this paper, the new uncertainty estimation is presented and applied to analytical test problems and the shape optimisation of airfoils. The strengths and weaknesses of the approach are discussed, and potential areas for further development and improvement are identified.